Amit Chouhan

INVESTIGATION INTO FRACTIONAL DIFFERINTEGRAL OPERATORS, AND THEIR APPLICATION INTO VARIOUS DISCIPLINES

A THESIS Submitted for the Award of Ph.D. degree of University of Kota, Kota (Mathematics-Faculty of Science)

AMIT CHOUHAN

Under the supervision of Dr. Satish Saraswat (M.Sc., Ph.D. ) Lecturer Department of Mathematics Government College Kota, Kota – 324001(India)

UNIVERSITY OF KOTA, KOTA (2013) Dr. Satish Saraswat Lecturer, (M.Sc., Ph.D.) Department of Mathematics

Govt. College Kota, Kota -324001.

CERTIFICATE

I feel great pleasure in certifying that the thesis entitled

“INVESTIGATION INTO FRACTIONAL DIFFERINTEGRAL

OPERATORS, AND THEIR APPLICATION INTO VARIOUS

DISCIPLINES”, embodies a record of the results of investigations carried out by

Mr. Amit Chouhan under my guidance. I am satisfied with the analysis of data, interpretation of results and conclusions drawn.

He has completed the residential requirement as per rules.

I recommend the submission of thesis.

Date : (Dr. Satish Saraswat) Research Supervisor

DECLARATION

I hereby declare that the (i) The thesis entitled “ INVESTIGATION INTO FRACTIONAL DIFFERINTEGRAL OPERATORS, AND THEIR APPLICATION INTO VARIOUS DISCIPLINES ” submitted by me is an original piece of research work, carried out under the supervision of Dr. Satish Saraswat. (ii) The above thesis has not been submitted to this university or any other university for any degree.

Date: Signature of Candidate

(Amit Chouhan) ACKNOWLEDGEMENTS

I express my heartful gratitude to the “ ALMIGHTY GOD ” for his blessing to complete this piece of work. I wish to express my unfeigned indebtedness to my research supervisor Dr. Satish Saraswat , Department of Mathematics, Government College, Kota for his constant inspiration, supervision and able guidance in making this endeavor a success. His meritorious discussions and remarks on the subject have made this arduous task comprehensive. I wish to express my gratitude to Prof. Sunil Bhargava , Principal, Dr. H.C. Jain and Mrs. Hemlata Loya , Vice Principals, Government College, Kota for their constant encouragement and providing all the necessary facilities for conducting the present research work. I am grateful to Shri S.N. Mathur , Head, Department of Mathematics, Government College, Kota and all the learned faculty members of the Department of mathematics for their constant encouragement and invaluable suggestions provided to me during the entire period of the present study. My sincere thanks and gratitude are to Dr. S.D. Purohit , Assistant Professor of Mathematics, Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur for his learned comments, fruitful discussions and help on the subject. I would like to take this opportunity to express my deepest gratitude to my parents Shri S.R. Chouhan and Mrs. Manju Chouhan , parents-in-law Shri R.P. Parihar and Mrs. Pushpa Parihar for their motivation and co-operation given to me at every step in this work from the beginning. Words will ever remain inadequate to express the sense of reverence, veneration and gratitude for my wife Mrs. Richa Chouhan and daughter Ku. Navya Chouhan who have willingly undergone all hardships of suffering to sustain my spirit and to support to attain my ambition.

Date: (AMIT CHOUHAN) CONTENTS

Chapter Name Page No.

1. A Brief Survey of the Work Done And Recent 1-22 Developments in the Theory of Fractional Calculus Operators and Special Functions

1.1 Prologue 1 1.2 Fractional Integral and Differential Operators 3 1.3 Erdélyi-Kober Fractional Operators 5 1.4 Saigo Operators 7 1.5 Gauss’s Hypergeometric Function 8 1.6 Generalized Hypergeometric Function 9 1.7 Fox’s H – Function 10 1.8 Wright Generalized Hypergeometric Function 11 1.9 Mittag-Leffler Function 12 1.10 M-Series 14 1.11 Laplace Transform of Fractional Operators and Mittag- 16 Leffler Functions 1.12 Fractional Kinetic Equations 18 1.13 Further Results on Fractional Calculus and its 20 Applications

2. Fractional Differintegral Operators and 23-38 Generalized Mittag-Leffler Function

2.1. Prologue 23 ,, 26 2.2. Fractional Calculus of , ,, 30 2.3. , In Term of Other Functions ,, 32 2.4. Integral Transforms of , 2.5. Saigo Operator and Generalized Mittag-Leffler Function 34 2.6. Concluding Remarks 37

3. Fractional Calculus of a Function of Generalized 39-54 Mittag-Leffler Function

3.1. Prologue 39 3.2. Fractional Operators and Generalized Mittag-Leffler 40 Function 3.3. Main Results 43 3.4. Concluding Remarks 54

4. Fractional Calculus of Generalized M-Series 55-68

4.1. Prologue 55 4.2. Main Results 56 4.3. Concluding Remarks 68

5. On Solution of Generalized Fractional Kinetic 69-96 Equations

5.1 Prologue 69 SECTION - A 5.2 Introduction 71 5.3 Main Result 74 SECTION - B 5.4 Introduction 78 5.5 Main Result 79 5.6 Alternative Method 88 5.7 Generalized Fractional Kinetic Equation Which Involve 91 an Integral Operator Containing Generalized Mittag- Leffler Function in its Kernel

6. Alternative Method for Solving Generalized 97-107 Fractional Differential Equation 6.1 Prologue 97 6.2 Main Result 98 6.3 Applications of the Main Results 103 6.4 Concluding Remarks 106

Bibliography 108-124

CHAPTER - 1

A BRIEF SURVEY OF THE WORK DONE AND RECENT DEVELOPMENTS IN THE THEORY OF FRACTIONAL CALCULUS OPERATORS AND SPECIAL FUNCTIONS

SUMMARY

This chapter contains a brief review on the recent developments and investigations carried out by various authors in the field of fractional calculus and their applications. Definitions of various fractional differential and integral operators, special functions, M-series, and fractional kinetic equations have also been included in this chapter. Most of these functions and operators will be needed in presenting the results of the subsequent chapters.

1.1 PROLOGUE:

Fractional calculus is the field of mathematical analysis which deals with the investigations and applications of integrals and derivatives of arbitrary order, which we shall term as differintegral operators. It is also known by several other names such as Generalized Integral and Differential Calculus and Calculus of Arbitrary Order. Fractional calculus can be categorized as applicable Mathematics. During the last three decades scientists have found many applications of fractional calculus in the field of physics, chemistry, quantitative biology, engineering, image

1 and signal processing, rheology, diffusion and transport theory etc. (Podlubny, 1999, Kilbas et al. , 2006).

The prominent mathematicians like Abel (1826), Liouville (1832), Grünwald (1867), Letnikov (1868, 1872), Weyl (1917), Littlewood (1925), Kober (1940), Zygmund (1945), Riesz (1949), Erdélyi (1940), Erdélyi and Kober (1965) have made fundamental discoveries in fractional calculus up to the middle of 20th century. In last three decades numerous research papers have been published related to the fractional calculus and applied the theory of fractional operators in obtaining certain remarkable results. That is fractional integrals, fractional derivatives and fractional differential equations. The research papers contributed by Raina and Koul (1979), Raina and Kiryakova (1983), Srivastava and Panda (1984), Banerji and Chaudhary (1996), Saigo and Saxena (1998), Kilbas and Saigo (1998), Chaurasia and Gupta (1999), Gupta et al. (1999), Chaurasia and Godika (2001a, 2001b), Gutpa et al. (2002), Chaurasia and Srivastava (2006, 2007b), Benchohra and Slimani (2009), Kilbas and Zhukovskaya (2009), Jaimini and Saxena (2010), Haubold et al. (2011), Purohit and Kalla (2011), Saxena et al. (2013), Kumar (2013) and Jaimini and Gupta (2013) are worth mentioning. A detailed account of theory and exposition of the fundamentals of fractional calculus can be found in books by Oldham and Spanier (1974), Nishimoto (1984, 1987, 1989, 1991), Saigo (1984), Samko et al. (1993) and Miller and Ross (1993). Caputo (1969), in his book systematically used his original definition of fractional differentiation for formulating and solving problems of viscoelasticity .

1.2 FRACTIONAL INTEGRAL AND DIFFERENTIAL OPERATORS:

Fractional integration is an immediate generalization of repeated integration. If be a finite interval on the = , (−∞<<<∞) real axis , then the Riemann-Liouville fractional integrals and ℝ of order are defined by ∈ ℂ ( () > 0)

1 ()() = ( )() = ( − ) () , > (1.2.1) Γ() and

1 ()() = ( )() = ( − ) () , < (1.2.2) Γ()

respectively. These integrals are called the left-sided and the right-sided Riemann – Liouville fractional integrals and are convergent for a wide class of functions . The limit may be real and complex. When = 0 equation (1.2.1) is equivalent to Riemann’s definition, and when = −∞ we have Liouville’s definition.

The Riemann – Liouville fractional derivatives and of order are defined by ∈ ℂ ( () ≥ 0) ()() = ( )() 1 = ( − ) () , ( = () + 1; > ) Γ( − )

(1.2.3) and

()() = − ( )() 1 = − ( − ) () , ( = () + 1; < ) Γ( − )

(1.2.4)

3 respectively, where [ ] denotes the integral part of . () ()

It can be observed that Riemann – Liouville fractional integration and fractional differentiation operators (1.2.1), (1.2.2) and (1.2.3), (1.2.4) of the power functions and , generate ( − ) ( − ) () > 0 power functions of same form as given below

(( − ) )()

Γ() = ( − ) ( () > 0) (1.2.5) Γ( + ) (( − ) )()

Γ() = ( − ) ( () > 0) (1.2.6) Γ( + ) and

(( − ) )()

Γ() = ( − ) ( () ≥ 0) (1.2.7) Γ( − ) (( − ) )()

Γ() = ( − ) ( () ≥ 0). (1.2.8) Γ( − )

The Weyl integral of of order , denoted by () ∈ ℂ ( () > 0) . , is defined by

. . ( )() = ()() = ( )() 1 = ( − ) (). (−∞ < < ∞) (1.2.9) Γ() The Weyl derivative of of order , denoted by () ∈ ℂ ( () > 0) . , is defined by . . ()() = ( )() = − ( )()

1 = − ( − ) () (1.2.10) Γ( − ) where, . = () +1;−∞<<∞<

Hilfer (2000) generalized the Riemann – Liouville fractional derivative operators in (1.2.3) and (1.2.4) by introducing the right sided fractional

derivative , and the left sided fractional derivative operator , of order and type with respect to x as follows: (0 < < 1) , (0 ≤ ≤ 1)

, () ()() ± () = ±± (± ) () (1.2.11)

Fractional integrals of functions of several complex variables have been developed by Riesz (1949), which has been used in the solution of partial differential equations (Baker and Copson, 1950). The relation between fractional integrals and Laplace transform has been studied by Doetsch (1943) and Widder (1941). Kober (1940, 1941) has considered the connections of fractional integrals with Mellin transform, whereas Erdélyi and Kober (1940) have considered fractional integrals with the Hankel-transform. In the theory of Fourier series fractional integrals are used by Zygmund (1959). Further Bora, Kalla and Saxena (1970) and Bora and Saxena (1971) have investigated the relationship between Riemann - Liouville fractional integral and integral transforms.

1.3 ERDÉLYI-KOBER FRACTIONAL OPERATORS:

Operators of fractional integration of a general kind with a higher degree of mathematical sophistication have been introduced by Kober (1940) which are given in a slight variant form by Erdélyi (1950-1951). These operators are the generalization of Riemann - Liouville and Weyl operatos.

The Erdélyi-Kober operators of first kind and of second kind () are defined by means of the following relations: ()

, , () = = ()

Γ = ( − ) () ,, ∈ ℂ; () > 0 (1.3.1) () and

, () = Γ = ( − ) () ,, ∈ ℂ; () > 0 (1.3.2) () provided that

() ∈ (0, ∞), () > − , () > − , + = 1, ≥ 1.

When , (1.3.1) reduces to Riemann – Liouville operator, i.e. , = 0 , = .

For , (1.3.2) reduces to the Weyl operator for the function = 0 i.e. , , () , = ().

Saxena and Kiryakova (1992) derived certain relations between H- function transforms in term of Erdélyi – Kober operators, which extend the work of Nishimoto and Saxena (1990) and Saxena et al. (1994). Relations connecting Erdélyi – Kober operators and generalized Laplace transforms were derived by Saxena and Gupta (1993) generalizing the results given earlier by Saxena (1967).

1.4 SAIGO OPERATORS:

Saigo operators are further extension of both the Riemann–Liouville and Erdélyi–Kober fractional integration in terms of Gauss’s hypergeometric function. Saigo (1978) defined the left sided generalized

fractional integral operator ,, and the right sided generalized fractional integral operator ,, as follows:

,, () = ( − ) +,−;;1− () Γ()

() > 0 (1.4.1) , , = (), () ≤ 0; = (−) + 1 (1.4.2)

and

,, ∞ ()

Γ 1 = ( − ) +,−;;1− () , ()

() > 0 (1.4.3) ,, = (−1) (), () ≤ 0,

= (−) + 1 (1.4.4) where, is Gauss’s hypergeometric function and its details are given (. ) in section 1.5.

When , (1.4.1) and (1.4.3) reduce to the Riemann – Liouville = − and Weyl fractional operators for as given below: () > 0

, , () = ()() 1 = ( − ) () , () > 0 (1.4.5) Γ()

and

, , . ( )() = ( )() 1 = ( − ) (), () > 0. (1.4.6) Γ()

For , the operators defined by (1.4.1) and (1.4.2) reduce to = 0 Erdélyi - Kober operators, defined by (1.3.1) and (1.3.2) respectively.

In a series of papers, Saigo (1978, 1979, 1980, 1981), Saigo et al. (1992a), Saigo and Raina (1991), Srivastava and Saigo (1987), Saigo and Saxena (1998), and others obtained several interesting properties of these operators and then applied in many problems.

1.5 GAUSS’S HYPERGEOMETRIC FUNCTION:

Gauss introduced the hypergeometric function by means of the following infinite series representation, Rainville (1960) ∞

()() (, ; ; ) = (1.5.1) () !

where, is neither zero nor a negative integer, and z < 1 () denotes the Pochhamer’s symbol or shifted factorial defined as:

1 ; = 0 () = (1.5.2) ( + 1)( + 2) … ( + − 1) ; ∈ .

Salvatore Pincherle studied Mellin - Barnes type integrals (Mainardi and Pagnini, 2003). These integrals have been employed in defining the Gauss hypergeometric function by Barnes in 1908. Barnes (1908) introduced the definition of hypergeometric function in the from:

Γ ∞ Γ Γ Γ 1 Γ (Γ) () (Γ − ) ( − ) (, ; ; −) = ∞ 2 () () ( − )

(1.5.3) where the poles of , at the points are Γ() = − ( = 0, 1 , 2 ,. . .) separated from those of at the points Γ( − ) = + ( = 0, 1 , and at the points and 2 , . . .) Γ(b − s) =+ ( =0, 1 , 2 ,. . .) . The importance of this definition lies in the fact that on arg (−z) < account of Mellin inversion formula, the coefficients of in (1.5.3) z gives the Mellin transform of . Thus, we have (⋅)

Γ()Γ()Γ( − )Γ( − ) (, ; ; −) = (1.5.4) Γ()Γ()Γ( − )

where and . In () > 0,(−) > 0,(−) > 0 arg (−z) < 1936, Dixon and Ferrar derived the asymptotic expansion of Mellin- Barnes integrals. Swaroop (1964) studied this function by defining the hypergeometric function transform, whose kernel is the Gauss’s hypergeometric function. Saxena (1966) and Kalla and Saxena (1969) employed the hypergeometric function in defining operators of fractional integration. Mathai and Saxena (1966) have used several properties hypergeometric function associated with this function.

1.6 GENERALIZED HYPERGEOMETRIC FUNCTION:

By increasing the number of parameters in the numerator as well as in the denominator of (1.5.1), the generalized hypergeometric function can be defined in the form:

() ( , … , ; , … , ; ) = () ! Γβ 1 Γ(α − s) … Γα − sΓ(s) = (−z )ds, Γα 2 π i Γ(β − s) … Γβ − s

(1.6.1)

where for convergence, or ( and ). The path of p ≤ q p = q + 1 (z) < 1 integration is indented, if necessary, in such a manner that the poles of at the points are separated from those of Γ(s) s = − υ (υ = 0, 1 , 2 ,…) at the points . An Γ(α − s) α = s + v v = 0, 1 , 2 ,…,j=1,… p empty product is always interpreted as unity. Poles of the integrand in (1.6.1) are assumed to be simple.

1.7 FOX’S H – FUNCTION:

Fox (1961) defined the H-function by means of a Mellin-Barnes type integral in the following manner:

, (, ) , (, ), … , (, ) , = , (, ) (, ), … , (, )

1 = χ(s)z ds (1.7.1) 2πi where

Γ( + ) Γ(1 − − ) () = (1.7.2) Γ(1 − − ) Γ( + )

and an empty product is always interpreted as unity, with , , , ∈ A’s and B’s are all positive numbers, the contour 1≤≤;0≤≤, L extends from to , such that the poles of for − ∞ + ∞ Γ( + ) are to its right and those of for . = 1, … , Γ(1 − − ) = 1, … , to the left of it . The poles of the integrand are assumed to be simple.

It has been shown by Braksma (1964) that the integral (1.7.1)

converges if ∗ where () < ,

∗ = − + − > 0, and

= − ≥ 0.

A detailed account of the H-function is available from the monographs of Mathai and Saxena (1978), Srivastava et al. (1982) and Prudnikov et al. (1990).

1.8 WRIGHT GENERALIZED HYPERGEOMETRIC FUNCTION:

Wright (1935, 1940) defined generalized hypergeometric function by means of the series representation in the form

(, ), … , , ; Γ( + ) () = = , (, ), … , , ; Γ + !

(1.8.1) where , , ∈ ℂ, , ∈ ℝ, ≠ 0, ≠ 0; =1,… ,; = 1,… ,,

− > −1. The relation connecting Wright’s function and the H-function () given in the monograph of Mathai and Saxena (1978) as

(, ), … , , ; (, ), … , , ;

, (1 − , ), … , 1 − , = , − (1.8.2) (0,1), (1 − , ), … , 1 − ,

where , represents the Fox’s H- function. ,(. )

1.9 MITTAG-LEFFLER FUNCTION:

The Swedish mathematician Gosta Mittag-Leffler (1903) introduced the function , defined by ()

∞

Γ 1 () = , (, ∈ ℂ, () > 0) (1.9.1) ( + 1)

The Mittag-Leffler function (1.9.1) is considered as a generalization of the exponential function since it reduces immediately to the exponential α function , when = 1. Its importance has been realized = () during the last two decades due to its involvement in the problems of applied sciences such as physics, engineering, chemistry, biology, fluid flow, probability, statistical distribution theory, reaction-diffusion in complex systems and anomalous diffusion etc. Mittag-Leffler function occurs naturally in the solution of fractional order differential or integral equations. A generalization of was studied by Wiman (1905) who defined () the function as follows: ,() ∞

Γ 1 ,() = ( + )

(,, ∈ ℂ, () > 0, () > 0) (1.9.2) The function (1.9.2) is studied by Wiman (1905), Agarwal (1953), Humbert (1953) and Humbert and Agrawal (1953). The main results in the classical theory of the functions and can be found in the () ,() hand book of Erdélyi et al. (1955). Mittag-Leffler function with three parameters is introduced by Prabhakar (1971) in the form ∞

Γ () ,() = ( + )!

,,, ∈ ℂ, (), () > 0 (1.9.3)

12 where is the Pochhammer symbol (1.5.2). () Prabhakar (1971) also introduces an integral operator involving the generalised Mittag-Leffler function (1.9.3) in its kernel as follows:

,,; () = ( − ) ,(( − ) )() (1.9.4) with , and applied the results ,,, ∈ ℂ, () > 0, () > 0 obtained to prove the existence and uniqueness of the solution for the corresponding integral equation of the first kind

( − ) ,(( − ) )() = (), ( < < ) (1.9.5) on a finite interval [ a,b ] of the real axis . The fractional ℝ = (−∞, ∞) integral operator (1.9.4) was further investigated by Kilbas et al. (2004).

Shukla and Prajapati (2007) introduced the function , which is ,() defined for and as: z, α, β, γ ∈ ℂ, min Re (α), Re (β) > 0 q ∈ (0,1) ∪ N

, () ,() = . (1.9.6) Γ( + )! It follows from (1.9.6) that ∞

, Γ 1 ,() = () = (1.9.7) ( + 1) and ∞ ∞ , Γ Γ , () = = = . (1.9.8) ( + 1) ( + 1)

A detailed account of various properties, generalizations, and application of the Mittag-Leffler function is available from the research monographs due to Dzherbashyan (1966), Caputo and Mainardi (1971), Blair (1974), Bagley and Torvik (1984), Gorenflo and Vessella (1991), Gorenflo and Rutman (1994), Kilbas and Saigo (1995), Luchko and Srivastava (1995), Gorenfloet et al. (1998), Luchko (1999), Haubold and

Mathai (2000), Srivastava and Saxena (2001), Saxena et al. (2003), Kilbas et al. (2002, 2004), Saxena and Saigo (2005), Kiryakova (2008), , Mathai et al. (2006), Haubold et al. (2007), Shukla and Prajapati (2007, 2009), Saxena and Kalla (2008), Chaurasia and Pandey (2010), Camargo (2012), Prajapati et al. (2013) and others.

1.10 M-SERIES:

M-series which is an extension of both Mittag-Leffler function and generalized hypergeometric function , and these functions have recently found essential applications in solving problems in physics, biology, engineering and applied sciences. Sharma and Jain (2009) introduced the generalized M-series as the function defined by means of the power series:

, , … , ; , , … ; = ; ;

( ) … = () = ;,, ∈ ℂ (1.10.1) () … Γ( + )

where are the known Pochammer (), () > 0, , symbols (1.5.2). The series (1.10.1) is defined when none of the parameters is a negative integer or zero; if any ′ , = 1,2,… ,, numerator parameter is a negative integer or zero, then the series terminates to a polynomial in z. The series in (1.10.1) is convergent for all

z if , it is convergent for if and ≤ z < δ = α = + 1 divergent, if . When and , the series can > + 1 = + 1 z = δ converge on conditions depending on the parameters.

Some special cases of the -function are the following: ()

(i) The Mittag-Leffler function (1.9.2) is obtained when there is no upper and lower parameters in (1.10.1) , we have ( = = 0) ,() = (− ; −; ) ∞

Γ 1 . = ; () = (− ; −; ) (1.10.2) ( + )

where is the one parameter Mittag-Leffler function (), = 1 (1.9.1).

(ii) The generalized Mittag- Leffler function (1.9.3) is obtained from (1.10.1) for : ==1; =∈ℂ; = 1 Γ () ,() = ( + ) ! () Γ = = ( ; 1; ) (1.10.3) (1) ( + )

(iii) The generalized M-series can be represented as a special case of the Wright generalized hypergeometric function (1.8.1) and the Fox H- function(1.7.1),

(, 1), … , , 1, (1,1); ; ; = (, 1), … , , 1, (, ); , 1 − , 1; 1 , (0,1) = , − ; (0,1), 1 − , 1 , (1 − , ) Γ Γ( ) = (1.10.4) ()

(iv) For , the generalized M-series is the M-series from Sharma (2008):

; ; = ()

() … = . (1.10.5) () … Γ( + 1)

M-series has gained recently an important role in the theory of differentiation of arbitrary order and in the solutions of fractional order differential equations. In view of its importance Sharma and Jain (2009) stuided the following representations of the generalized M-series in terms of the Wright generalized hypergeometric function and Fox's H- function, with formulas for fractional calculus operators :

() = ,1; , + 1; (1.10.6) Γ( + 1) and

() = ,1; ,1 − ; (1.10.7) Γ(1 − )

That is, a R-L fractional integral or derivative of a generalized M- series is again a generalized M-series with indices p, q increased to (p + 1), (q + 1).

1.11 LAPLACE TRANSFORM OF FRACTIONAL OPERATORS AND MITTAG-LEFFLER FUNCTIONS:

The theory of Laplace transform provides a powerful tool in deriving the solution of fractional differential equations governing certain physical problems. The Laplace transform of the function , denoted by , is () () defined by the equation

ℒ()() = ℒ(); = () = () (1.11.1) where , which may be symbolically written as (s) > 0

or () = ℒ(); () = ℒ (); provided that the function is continuous for . () ≥ 0

The Parseval theorem or the convolution theorem for Laplace transform is defined as

ℒ ( − )() () = ℒ()()ℒ()(). (1.11.2)

The Laplace transform of Riemann – Liouville fractional operators

and are given by Oldham and Spanier (1974) ()() ()() ℒ()()() = s ℒ()() − s (0 +)

( − 1 < < ) (1.11.3) and

ℒ()()() = ℒ ()ℒ()() Γ() = ℒ()() (1.11.4) respectively, where . , ∈ ℂ, () > 0, (s) > 0

Prabhakar (1971) introduced the Laplace transform formula for the generalized Mitta-Leffler function (1.10.3) as

ℒ ,( ) () = (1.11.5) ( − ) where ,, ∈ ℂ, () > 0, () > 0, < 1. Kilbas et al. (2004) obtained the following relations by applying Parseval theorem on combinations of various Mittag-Leffler functions:

(i) If , then ,,,,, ∈ ℂ, () > 0, () > 0, () > 0 ( − ) ,( − ) ,( )

= , (1.11.6)

(ii) If , then ,,,, ∈ ℂ, () > 0, () > 0, () > 0 ( − ) ,( − ) ,( )

= , (1.11.7)

(iii) If , then ,,,, ∈ ℂ, () > 0, () > 0, () > 0 ( − ) ,( − )

= Γ() , . (1.11.8)

The Laplace transform is the simplest well known transform and due to its popularity and wide applications, a number of generalizations of this transform have been studied from time to time in mathematical analysis. Some of the generalizations are given by Meijer (1940), Varma (1951), Bhise (1967). Among other persons who have enriched the theory of Laplace, the name of Bose (1950), Rathie (1953), Srinivasan (1963), Pathan (1967), Srivastava (1968) are worth mentioning.

More detailed information about Intgral transforms may be found in the books by Titchmarsh (1937) Ditkin and Prudnikov (1965), Doetsch (1974), Sneddon (1979), Beerends (2003) and Debnath and Bhatta (2010).

1.12 FRACTIONAL KINETIC EQUATIONS:

In last two decades fractional kinetic equations have been extensively used in describing and solving various problems of applied sciences, Podlubny (1999), Baleanu et al. (2012). The great use of mathematical physics in distinguished astrophysical problems has attracted astronomers and physicists to pay more attention to available mathematical tools that can be widely used in solving several problems of astrophysics/physics, Kilbas et al. (2006). A spherically symmetric non-rotating, self-gravitating

18 model of star like the sun is assumed to be in thermal equilibrium and hydrostatic equilibrium. The star is characterized by its mass, luminosity effective surface temperature, radius central density and central temperature. The stellar structures and their mathematical models are investigated on the basis of above characters and some additional information related to the equation of nuclear energy generation rate and the opacity.

Haubold and Mathai (2000) studied the solution of a simple kinetic equation of the type used for the computation of the change of the chemical composition in stars like the Sun. By integrating the standard kinetic equation

() = −(), ( > 0) (1.12.1) it is derived that (Haubold and Mathai, 2000)

() − = −( )() (1.12.2)

where is the standard Riemann integral operator. In the original paper of Haubold and Mathai (2000), the number density of species , is a function of time and is the number , = () ( = 0) = density of species i at time . By dropping the index i in (1.12.1), the = 0 solution of its generalized form

() − = − ( )() (1.12.3) is obtained by Haubold and Mathai (2000) as

∞ (−1Γ ) ( ) () = . (1.12.4) n=0 ( + 1)

Furthermore the solution of the generalized fractional kinetic equation

() − ,(− ) = − ( )() (1.12.5) is obtained by Saxena et al. (2004a) as

() = , (− ). (1.12.6)

In view of the usefulness and importance of the fractional kinetic equation in certain physical problems governing reaction-diffusion in complex systems and anomalous diffusion fractional kinetic equations are studied by Gloeckle and Nonnenmacher (1991), Saichev and Zaslavsky (1997), Chaurasia and Pandey (2008, 2010a), Chaurasia and Kumar (2010), Saxena et al. (2002, 2004a, 2004b). Recently, in a series of papers Saxena et al. (2006a, 2006b, 2006c, 2010), Haubold et al. (2011) have investigated the solution of certain fractional differintegral equations related to reaction diffusion equations. The solutions are developed in a compact form in terms of the H-function by the application of direct and inverse Laplace and Fourier transforms. Tripathi and Jain (2012) investigate the solution of fractional diffusion equation involving Mittag- Leffler functions. Khan (2013) studied multiple fractional kinetic differintegral equation and multiple fractional diffusion equation with i- function of r-variables.

1.13 FURTHER RESULTS ON FRACTIONAL CALCULUS AND ITS APPLICATIONS

Fractional calculus can be categorized as ‘Applicable Mathematics’. The first application of fractional calculus was made by Abel (1826) in the solution of an integral equation that arises in the formulation of the tautochronous problem. This problem deals with the determination of the shape of a frictionless plane curve through the origin in a vertical plane along which a particle of mass m can fall in a time that is independent of the starting position. Heaviside, in 1892, introduced the idea of fractional

20 derivatives in his study of electric transmission lines, Debnath and Bhatta (2010).

During the second half of the twentieth century, considerable amount of research in fractional calculus was published in engineering literature. Indeed, recent advances of fractional calculus are dominated by modern examples of applications in differential and integral equations, physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology and electrochemistry, Hilfer (2000). There is no doubt that fractional calculus has become an exciting new mathematical method of solution of diverse problems in mathematics, science, and engineering. In an article by Debnath (2003), he presented numerous new and recent applications of fractional calculus in mathematics, science, and engineering.

A comprehensive account of fractional calculus and its applications can be found in the monographs written by Kiryakova (1994), Nishimoto (1991), Oldham and Spainer (1974), Miller and Ross (1993) and Samko et al . (1993), Srivastava and Sxena (2001), Machado et al. (2011). In particular, the five volumes work of Nishimoto (1991) certain an interesting account of theory applications of fractional calculus.

Applications of fractional calculus in the solution integral equations are given by Fox (1971, 1972), and Saigo and Saxena (1998). Relations connecting Dirichlet averages with fractional calculus are derived by Deora et al. (1993, 1994, 1994a). Solution of certain fractional differential equations are dicussed by Jones (1993), Mainardi (1996), Luchko and Srivastva (1995) and Podlubny (1997, 1999).

Use of fractional calculus in the theory of convolution integral equations can be seen in the work of Srivastva and Bushman (1977, 1992). Srivastva and Manocha (1984) employed fractional calculus in the theory of generating functions. Some applications of fractional calculus operator

21 to certain classes of analytic and multivalent functions is explained by Srivasta and Owa (1989). Srivastva and Saxena (2001) presents detailed account of various operators of fractional integrations studied during last 30 years. Some theorems for the N – fractional calculus of Nishimoto (1984) is given by Purohit et al. (2002). Camargo et al. (2012) explained the generalized Mittag-Leffler function and its application in a fractional telegraph equation. Yang (2012) introduced the recent results of local fractional calculus and the applications of advanced local fractional calculus on the mathematical science and engineering problems.

One of the major advantages of fractional calculus is that it can be considered as a super set of integer-order calculus. Thus, fractional calculus has the potential to accomplish what integer-order calculus cannot. We believe that many of the great future developments will come from the applications of fractional calculus to different fields.

CHAPTER - 2

FRACTIONAL DIFFERINTEGRAL OPERATORS AND GENERALIZED MITTAG-LEFFLER FUNCTION

SUMMARY

The chapter envisages the derivation of certain properties of a generalized function of Mittag-Leffler type, written in the form ,, , where , , , , , ∈ , , including various , { , , , } > 0 ∈ 0,1 ∪ fractional integral operators like Riemann – Liouville operator. Laplace transform, Mellin transform and Mellin –Barnes integral representation for this function are established. Image of this function under the Saigo operator has also been obtained.

2.1 PROLOGUE:

In the integer order calculus, the exponential function plays a vital role. Similarly in the fractional order calculus, the Mittag-Leffler functions plays the important role.

The Swedish mathematician Gosta Mittag-Leffler (1903) introduced the function , by mean of the equation (1.9.1), which is considered as a generalization of the exponential function. Mittag - Leffler (1903, 1905) investigated certain properties of this function. Wiman (1905) who defined the two parameter Mittag-Leffler function , as a ,

23 generalization of , which is z. The function is studied, , among others, by Wiman (1905), Humbert (1953) and Humbert and Agarwal (1953) and others. The main properties of these functions are given in the book by Erdélyi et al. (1955) and a more comprehensive and a detailed account of Mittag-Leffler functions are presented in Dzherbashyan (1966).

The Mittag-Leffler function is generalized by Agarwal (1953). This function is particularly remarkable to the fractional order system theory due to its Laplace transform given by Agarwal. The function is defined as follows:

∞ , = Γ 2.1.1 +

By means of the series representation, a further generalization of the Mittag –Leffler function , was introduced by Prabhakar (1971) as , , defined by the equation (1.9.3). Indeed, it is interesting to note , that the generalized Mittag –Leffler function , itself is actually a , very specialized case of a rather extensively investigated Wright generalized hypergeometric function , equation (1.8.1), as indicated below (Srivastava and Saxena (2005))

1 , 1; , = Γ 2.1.2 , ;

Mittag-Lefﬂer functions are important in mathematical as well as in theoretical and applied physics (Hilfer, 2000). A primary reason for the recent surge of interest in these functions is their appearance in the solution of fractional order differintegral equations and especially in the study of the fractional generalization of the kinetic equation (Saxena et al. 2002, 2006a, 2006b, 2006c, 2010). Mainardi and Gorenflo (2000) studied a variety of fractional evolution processes i.e. a phenomenon

24 governed by an integro-dierential equation containing integrals and derivatives of fractional order in time, whose solutions turn out to be related to Mittag-Leffer type functions. Recently, Camargo et al. (2012) introduced some integral transforms associated with the generalized Mittag-Leffler function (1.9.3).

The classical Mittag-Leffler function has numerous applications in various scientific fields. Literatures suggested that many other generalizations of the classical Mittag-Leffler function are available to analyze the behavior of physical phenomenon. Particularly, Shukla and

Prajapati (2007) introduced the function , which is defined by the , equation (1.9.6). In that paper its various properties including usual differentiation and integration, Laplace transforms, Euler (Beta) transforms, Mellin transforms, Mellin –Barnes integral representation etc. are obtained.

Recently, Prajapati et al. (2013), studied generalized Mittag-Le er ﬄ function operator (1.9.6) and established Laplace and Mellin transforms of this operator.

In this chapter, we consider a further generalization of the Mittag-

Leffler function ,, , which is defined as , ,, , = 2.1.3 Γ + where, , ,,,, ∈ ,{ > 0, > 0, , } > 0 Γ and denotes the generalized Pochhammer Γ ∈ 0,1 ∪ = symbol (Rainville (1960)) which in particular reduces to

+ − 1 ∈ .

The generalized function ,, contains the Mittag-Leffler , functions given in the section (1.9). Note that

,, , , , = , ,, , , = , ,, , = , and

,, . , =

2.2 FRACTIONAL CALCULUS OF ,, : ,

In this section, we shall evaluate certain derivatives of integer order, and Riemann – Liouville fractional integral and derivatives of the function

,, . ,

THEOREM 1. Let ,,,, ∈ ,{ > 0, > 0, , , then for k N } > 0 ∈ ∞ ∈ ,, + + 1 , = Γ 2.2.1 + + + and

,, ,, [ , ] = , 2.2.2

PROOF OF (2.2.1): Differentiating equation (2.1.3) times with respect to z, we obtained ∞ ,, , = Γ + ∞ = Γ + ∞

− 1 … − + 1 = Γ +

26 now replacing from , we get ∞ + ,, [ + + − 1 … + 1] , = Γ + + using the definition (1.5.2) of Pochhamer’s symbol, we have ∞

+ 1 = Γ + + ∞ Γ Γ + + + 1 = Γ Γ Γ + + + + ∞ Γ Γ Γ Γ + + + + = Γ Γ Γ Γ + + + +

+ 1 × Γ ∞ + +

+ + 1 = Γ . + + + This completes the proof of equation (2.2.1).

PROOF OF (2.2.2): Again using (2.1.3) and term-by-term differentiation under the sign of summation, we get ∞ ,, [ , ] = Γ + ∞ = Γ + ∞ = Γ + ∞ × [ + − 1 + − 2 … + − ] = Γ × [ + − 1 + − 2 … + − ] + Γ + − × Γ + −

∞ Γ + = Γ Γ + + − ∞

= Γ + − ∞

= Γ + − finally by the definition (2.1.3) of generalized Mittag-Leffler function, we have

,, ,, [ , ] = , . This completes the proof of equation (2.2.2).

THEOREM 2. Let ∞ and > ∈ ℝ ∶= [0, , ,,,, ∈ , , then { , , , } > 0 ∈ ,, − , −

,, = − , − 2.2.3 and

,, − , −

,, − , − 2.2.4

PROOF OF (2.2.3): Multiplying the generalized Mittag-Leffler function (2.1.3) by

and applying Riemann – Liouville fractional integral operator − (1.2.1), we obtained

,, − , −

= − ∞

× Γ − +

∞ = Γ − + now by virtue of the formula (1.2.5), we obtained for > ,, − , − ∞ Γ + = Γ Γ × − + + + ∞

= − Γ [ − ] + + finally by the definition (2.1.3) of generalized Mittag-Leffler function, we have

,, − , −

,, = − , − . This completes the proof of equation (2.2.3).

REMARK 1. For in equation (2.2.3), we obtained = 0 ,, ,, , = , . 2.2.5

PROOF OF (2.2.4): Multiplying the generalized Mittag-Leffler function (2.1.3) by

and applying Riemann – Liouville fractional differential − operator (1.2.3), we obtained

,, − , − ,, = − , − by virtue of equation (2.2.3) above equation can be written as

,, − , − ,, = − , − finally applying equation (2.2.2), we get

,, − , −

,, = − , − This completes the proof of equation (2.2.4).

REMARK 2. For in equation (2.2.4), we obtained = 0 ,, ,, , = , . 2.2.6

2.3 ,, IN TERM OF OTHER FUNCTIONS: ,

In this section the generalized Mittag-Leffler function ,, is , represented in the form of Wright generalized hypergeometric function and Mellin-Barnes type integral. Finally its Fox’s H-function equivalent term is established.

The generalized Mittag-Leffler function ,, given in equation , (2.1.3), can be written in the form ∞ Γ Γ Γ ,, + + 1 , = Γ Γ Γ 2.3.1 + + ! now making use of the equation (1.8.1), ,, can be written in term of , the Wright generalized hypergeometric function as

,, Γ . , , 1,1; , = 2.3.2 Γ , , , ; where , ,,,, ∈ , { > 0, > 0, , } > 0 . ∈ 0,1 ∪

Now in order to write ,, in terms of Fox’s H-function, we first , express ,, as a Mellin-Barnes type integral in the following , theorem.

THEOREM 3. Let γ ,,,, ∈ ≠ 0, min{ > 0, > , then ,, represents Mellin-Barnes integral as 0, } > 0 ∈ , , . ,, 1 Γ ΓΓ1 − Γ − , = − 2.3.3 2 Γ Γ − Γ − where, , the contour of integration begins at ∞ and ends at arg < − ∞, and intended to separate all the poles to the left + = − ∈ and all the poles to the right. = ∈

PROOF: Evaluating the integral on the RHS of (2.3.3) as the sum of the residues at the poles , we get = 0,−1,−2,… . 1 ΓΓ1 − Γ − − 2 Γ − Γ − ΓΓ1 − Γ − = → − Γ − Γ − using the formula of gamma function, we have ΓΓ1 − = Γ − = → + − Γ − Γ − + Γ − = → → − Γ − Γ − Γ + = −1 − Γ + Γ + Γ + Γ Γ = Γ + Γ + Γ Γ Γ = Γ Γ + finally using the equation (2.1.3), we get

. 1 ΓΓ1 − Γ − Γ ,, − = , 2 Γ − Γ − Γ

i.e.

. ,, 1 Γ ΓΓ1 − Γ − , = − 2 Γ Γ − Γ − This completes the proof of equation (2.3.3).

Furthermore, on making use of (1.8.2), the Mittag-Leffler function

,, can be written in the form of Fox’s H-function as follows ,

,, Γ , 0,1, 1 − , , = , − . 2.3.4 Γ 0,1, 1 − , , 1 − ,

2.4 INTEGRAL TRANSFORMS OF ,, : ,

In the present section we established Laplace transform and Mellin

transform of the generalized Mittag-Leffler function ,, given in , equation (2.1.3). The Mellin transform (Sneddon (1979)) of the function is defined fz as

∗ M[; ] = = , > 0 2.4.1 and the inverse Mellin transform is written as

∗ 1 ∗ = M [ ; ] = 2.4.2 2 where L is a contour of integration that begins at and ends at . −∞ +∞

THEOREM 4. (Laplace transform)

,, ℒ , ;

Γ . , , , , 1,1; = 2.4.3 Γ , , , ; where , ∈ ,{ > 0, > 0, , } > 0 ∈ and Re(s) > 0 . 0,1 ∪

PROOF: By the definition of Laplace transform (1.11.1), we have

∞ −1 ,, − −1 ,, ℒ , ; = , 0 using the definition (2.1.3), we get

= Γ + −1 ,, ℒ , ; ∞ ∞ − +−1 = 2.4.4 =0 Γ + 0 by the definition of gamma function we know that

Γ = 2.4.5 using equation (2.4.5) in equation (2.4.4), we get

∞ Γ −1 ,, + ℒ , ; = + =0 Γ + Γ Γ + Γ + = Γ Γ + Γ + now making use of the definition of Wright generalized hypergeometric function (1.8.1), we finally obtained

−1 ,, Γ − . , , , , 1,1; ℒ , ; = . Γ 3 2 , , , ; This completes the proof of equation (2.4.3).

REMARK 3. Particularly when in equation (2.4.3), we = and = have

,, Γ . , , 1,1; ℒ , ; = . 2.4.6 Γ , ;

THEOREM 5. (Mellin transform)

γ δ Γ Γ Γ Γ γ α,β, ω s 1 − s − qs ω E , − z; = Γ Γ β α Γ δ 2.4.7 − s − qs where ω , ∈ , min{ > 0, > 0, , } > 0 ∈ and Re(s) > 0. 0,1 ∪ PROOF: From equation (2.3.3), we have

. ,, 1 Γ ΓΓ1 − Γ − , − = 2 Γ Γ − Γ − Γ . ,, 1 ∗ , − = Γ π z ds 2.4.8 2 where, Γ Γ Γ γ ∗ s 1 − s − qs ω = Γ β α Γ δ − s − qs equation (2.4.8) is in the form of inverse Mellin transform (2.4.2). So it can be written as

,, Γ ∗ , − = [ ; ] Γ now applying the Mellin transform (2.4.1), we obtained

,, Γ ΓΓ1 − Γ − , −; = Γ Γ − Γ − This completes the proof of equation (2.4.7).

2.5 SAIGO OPERATOR AND GENERALIZED MITTAG- LEFFLER FUNCTION:

THEOREM 6. Let ,, be the Saigo’s left-sided fractional integral , operator (1.10), then there holds the formula

,, ,, , ∞ Γ − ++ = Γ Γ 2.5.1 − + +++

34 the conditions for the validity of (5.1) are (i) and ω are any complex numbers, , , > 0 (ii) ρ and υ are arbitrary such that ρ υ + − + > 0.

PROOF: Applying Saigo operator (1.4.1) to the equation (3.2.4), we have

,, ,, , . = Γ − +,−;;1− ,, × , using the definition of Gauss’s hypergeometric function (1.5.1), and generalized Mittag-Leffler function (2.1.3), we obtained

,, ,, , ∞ + − = Γ − 1 − ! ∞

× Γ + interchanging the order of integration and summation, we get ∞ ∞ + − = Γ Γ ! + × − 1 − ∞ ∞ 1 + − = Γ Γ ! + × − now substituting , which yields =

∞ ∞ 1 + − = Γ Γ ! + × 1 − now using the definition of beta function i.e.

Γ Γ β , β = 1 − = Γ 2.5.2 + β in the above equation, we obtained ∞ ∞ 1 + − = Γ Γ ! + Γ Γ + + × Γ ∞ ∞ + + +

+ − = Γ ! + + + ∞ ∞ Γ + − + + = Γ Γ ! + + + + + ∞ ∞

+ − 1 = Γ ! + + + + now using the definition of Gauss’s hypergeometric function (1.5.1), we get ∞

. = +,−;++;1

1 × Γ 2.5.3 + + again by virtue of well known Gauss summation formula (Srivastava and Manocha, 1984), i.e. Γ Γ . − − , ; ; 1 = Γ Γ 2.5.4 − − in the equation (2.5.3), we obtained

∞ Γ Γ + + + − + = Γ Γ + − + + +

1 × Γ ∞ + + Γ − + + = Γ Γ . − + +++ This completes the proof of equation (2.5.1).

COROLLARY 1. Let in (2.5.1) and using definition (1.4.5), we = − have

,, ,, ,, , = , ∞ Γ ++ + = Γ Γ + + + ++ ∞

= Γ + + using the definition (2.1.3), we get

,, ,, , = , i.e. we obtained equation (2.2.5).

REMARK 4. For and we arrive at the following result = 1 = 1 given by Saxena and Saigo (2005).

, = , .

2.6 CONCLUDING REMARKS:

We conclude this chapter with the remark that as the exponential function plays the basis role in integer order calculus, so does Mittag- Leffler function has its role in the fractional calculus. The Mittag-Leffler function and its properties established in this chapter are of power-series

37 expansions and fit a variety of power law following processes. The results of this chapter are likely to find their importance when problems related to the solutions of the fractional differintegral equations are dealt with. The variants of Mittag-Leffler functions are developed in last four decades; several others may be developed in future to explain the physical processes of nature.

CHAPTER - 3

FRACTIONAL CALCULUS OF A FUNCTION OF GENERALIZED MITTAG-LEFFLER FUNCTION

SUMMARY

The principal aim of this chapter is to establish the function υ γ δ , and its properties by using fractional calculus. It has been shownE c, that, , , qthe fractional integral and differential operators transform such function with power multiplier into functions of the same form.

3.1 PROLOGUE:

In the chapter 2, we have illustrated the importance of the Mittag- Leffler function. The function υ γ δ , involving the generalized Ec, , , , q Mittag-Leffler function of the form , given by equation (2.1.3), ,, , is derived in the present chapter and its certain properties are established by using Riemann – Liouville operator, Hilfer operator, Erdélyi - Kober operator and Saigo operator. We applied the fractional integral and differential operators to transform such function with power multiplier into functions of the same form.

3.2 FRACTIONAL OPERATORS AND GENERALIZED MITTAG-LEFFLER FUNCTION:

In this section, the new functions υ γ δ and γ δ are derived by mean of Riemann-LiouvilleE c, , fractional, , q E integrationc, −μ, , and, q differentiation respectively of certain power series.

Consider the function

= 3.2.1 ! where , and c is the arbitrary, constant. ∈ , > 0, > 0, ∈ 0,1∪

Now applying the Riemann-Liouville fractional integral operator (1.2.1) of order υ to the function (3.2.1), we have

∞ Γ1 = − ! ∞ Γ1 = − ! this on the change of variable , yields

∞ = Γ1 = 1 − ! ∞ Γ Γ Γ1 Γυ n + 1 = ! υ + n + 1 the simplification of above equation gives, ∞ Γ = 3.2.2 + + 1 again using the definition (2.1.3) of generalised Mittag-Leffler function, the above equation may be written as,

,, = , . 3.2.3

We denote the function (3.2.3) as E t(c, υ, γ, δ, q), i.e. υ γ δ ,, Ec, , , , q = , . 3.2.4 Now applying the Riemann-Liouville fractional differential operator (1.2.3) of order µ to the function (3.2.1), we have

= ∞ Γ 1 = − − ! ∞ Γ 1 = − − ! this on the change of variable , yields = ∞ Γ 1 = − !

× 1 − ∞ Γ Γ Γ 1 Γ − n + 1 = − ! − + n + 1 on simplification, we obtained ∞ Γγ ct = t δ − + n + 1 the above equation reduces to ∞ Γ = − + 1 again using the definition (2.1.3) of generalised Mittag-Leffler function, the above equation may be written as,

,, = , . 3.2.5

We denote the function (3.2.5) as E t(c,-µ, γ, δ, q), i.e. µ γ δ ,, Ec, − , , , q = , 3.2.6

LEMMA 1. If , then the >0,0< <1,0≤ ≤1, > 0 following result holds true for the fractional derivative operator , defined by (1.2.11) Γ

, Γ = 3.2.7 − PROOF: We observe from the equation (1.2.5) that Γ

Γ = differentiating both side with respect[1 −to x, we1 − get + ]

Γ = [1 − 1 − + ] Γ

[1Γ − 1 − + − 1] = [1 − Γ1 − + ]

Γ = which in light of the definition[1 − (1.2.11) 1 − of Hilfer+ − operator, 1] yield

, = Γ

Γ = again using[ the1 − equation 1 − (1.2.5), + −we 1] obtained Γ

, Γ = Γ [1 − 1 − + − 1]

Γ [1 − 1 − + − 1] [] × finally on[ simplification,1 − 1 − we+−1+1− obtained ]

Γλ λ υ , Γ λ υ = x . This completes the proof of equation (3.2.7).−

3.3 MAIN RESULTS:

In this section we have investigated certain properties of the functions υ γ δ and µ γ δ , defined already via equations (3.2.4) and E c, (3.2.6), , , q respectively,E c, − using, , , q Riemann – Liouville operator, Hilfer operator, Erdélyi - Kober operator and Saigo operator.

THEOREM 1. If , and c is an arbitrary constant, then ∈ , for fractional > 0, integr >al 0,operator ∈ 0,1∪ of order λ, the following equations holds

E, , , , = E,+,,, 3.3.1 E, , , , = E,−,,, 3.3.2

, [ ] E , , , , =Γ E ,−,,, 3.3.3

1 Γ . , , 1,1; ℒ[E, , , , ; ] = 3.3.4 , ; PROOF OF (3.3.1): Applying Riemann-Liouville fractional integral operator (1.2.1) to the equation (3.2.4), we get

Γ1 E, , , , = − E, , , , Γ1 ,, = − , using the equation (2.1.3) above equation become

∞ Γ1 Γ = − + + 1

∞ Γ1 Γ = − + + 1 now substituting x = tz , which yields

E, , , , ∞ λ υ Γ λ Γ = 1 − z z dz + + 1 ∞ Γ λ Γ Γ λ Γ Γλ + + 1 = ∞ + + 1 + + + 1 Γ = + + + 1 again by virtue of equation (2.1.3), above equation gives

,, E, , , , = , finally using equation (3.2.4), we obtain

λ υ γ δ E, , , , = E, , , , = Ec, + , , , q. This completes the proof of equation (3.3.1).

PROOF OF (3.3.2): Applying Riemann-Liouville fractional differential operator (1.2.3) of order to the function (3.2.4), we have E, , , , = E, , , , Γ 1 = t − x E, , , , dx − Γ 1 ,, = t − x , dx − using the equation (2.1.3), above equation become

∞ Γ 1 Γ = t − x dx − + + 1

∞ Γ 1 Γ = t − x dx − + + 1 now substituting x = tz , which yields

E, , , , ∞ Γ 1 Γ = − + + 1

λ υ × 1 − z z dz ∞ Γ λ Γ Γ 1 Γ Γ − λ + + 1 = × − ∞ + + 1 − + + + 1 Γ λ = ∞ − + + + 1 Γ λ = −∞ + + + 1 Γ λ = − + + 1 again by virtue of equation (2.1.3), above equation gives

,, E, , , , = , finally using equation (3.2.4), we obtain

υ λ γ δ E, , , , = E, , , , = Ec, − , , , q. This completes the proof of equation (3.3.2).

PROOF OF (3.3.3):

Applying the fractional derivative operator defined by (1.2.11) to , the equation (3.2.4), we obtained

, , ,, [E, , , , ] = , using the equation (2.1.3), above equation become

∞ , , Γ [E, , , , ] = ∞ + + 1 Γ , = + + 1 now by virtue of the equation (3.2.7), we get ∞ Γ λ , Γ Γ + +λ 1 [E, , , , ] = ∞ + + 1 + − + 1 υ λ Γ λ = t + − + 1 again by virtue of equation (2.1.3), above equation gives

, ,, [E, , , , ] = , finally using equation (3.2.4), we obtain υ λ γ δ , This completes the[E proof, , ,of ,equation ] = E (3.3.3).c, − , , , q .

PROOF OF (3.3.4): Taking Laplace transform of equation (3.2.4), we have

,, ℒE, , , , ; = ℒ 1,+1 ; now using the definition (2.1.3) of generalized Mittag-Leffler function, we obtained ∞ Γ ℒE, , , , ; = ℒ ; ∞ =0 + + 1 Γ + = ℒ ; ∞=0 + + 1 Γ Γ + + 1 = ++1 =0 +∞ + 1 s 1 c = +1 ∞s =0 Γ Γ Γ Γ υ1 Γ + 1 + c = + !

46 finally using Wright generalized hypergeometric function (1.8.1), we get Γ γ

υ1 Γ , qδ, 1,1; c ℒE, , , , ; = +1 1 . 2 This completes the proof of equation (3.3.4). , q ; s

THEOREM 2. If , and c is an,, arbitrary ∈ , constant > 0, then for > fractional 0, > integ 0,ral ∈ 0,1operator ∪ of order µ, the following equations holds

E, −, , , = E,−,,, 3.3.5 E, −, , , = E,−−,,, 3.3.6

, [E , −, , , ] = EΓ,−−,,, 3.3.7 . 1 Γ , , 1,1; ℒEt, −, , , ; = 1− 1 3.3.8 2 , ;

PROOF OF (3.3.5): Applying Riemann-Liouville fractional integral operator (1.2.1) to the equation (3.2.6), we get

Γ1 E, −, , , = − E, −, , , Γ1 ,, = − , using the equation (2.1.3) above equation become

∞ Γ1 Γ = − 1 − + ∞ Γ1 Γ = − 1 − + now substituting x = tz , which yields

E, −, , , ∞ λ Γ λ Γ = 1 − z z dz 1 − +

∞ Γ λ Γ Γ λ Γ Γλ 1 − + = ∞ 1 − + + 1 − + Γ = + 1 − + again by virtue of equation (2.1.3), above equation gives

,, E, −, , , = , finally using equation (3.2.6), we obtain

λ γ δ E, , , , = E, −, , , = Ec, − , , , q. This completes the proof of equation (3.3.5).

PROOF OF (3.3.6): Applying Riemann-Liouville fractional differential operator (1.2.3) of order to the function (3.2.6), we have E, −, , , = E, −, , , Γ 1 = t − x E, −, , , dx − Γ 1 ,, = t − x , dx − using the equation (2.1.3), above equation become

∞ Γ 1 Γ = t − x dx − 1 − + ∞ Γ 1 Γ = t − x dx − 1 − + now substituting x = tz , which yields

E, −, , , ∞ Γ 1 Γ = − 1 − +

λ × 1 − z z dz ∞ Γ λ Γ Γ 1 Γ Γ − λ 1 − + = × − ∞ 1 − + − + 1 − + Γ λ = ∞ − + 1 − + Γ λ = −∞ + 1 − + Γ λ = 1 − − + again by virtue of equation (2.1.3), above equation gives

,, E, −, , , = , finally using equation (3.2.6), we obtain

E, −, , , λ γ δ This completes the= proof E of, equation −, , , (3.3.6). = E c, − − , , , q.

PROOF OF (3.3.7): Applying the fractional derivative operator defined by (1.2.11) to , the equation (3.2.6), we obtained

, , ,, [E, −, , , ] = , using the equation (2.1.3), above equation become ∞ , , Γ [E, −, , , ] = 1 − +

∞ Γ , = + + 1 now by virtue of the equation (3.2.7), we get

, [E, −, , ,∞ ] Γ λ Γ Γ 1 − + λ = ∞ 1 − + 1 − + − λ Γ λ = t 1 − + − again by virtue of equation (2.1.3), above equation gives

, ,, [E, −, , , ] = , finally using equation (3.2.6), we obtain λ γ δ , [ ] This completes Ethe ,proof −, ,of ,equation = E (3.3.7).c, − − , , , q .

PROOF OF (3.3.8): Taking Laplace transform of equation (3.2.6), we have

− ,, ℒE, −, , , ; = ℒ 1,1− ; now using the definition (2.1.3) of generalized Mittag-Leffler function, we obtained ∞ − Γ ℒE, −, , , ; = ℒ ; ∞ =0 1 − + Γ −+ = ℒ ; ∞=0 1 − + Γ Γ 1 − + = 1−+ =0 1 −∞ + s 1 c = 1− ∞s =0 Γ Γ Γ Γ 1 Γ + 1 + c = + !

50 finally using Wright generalized hypergeometric function (1.8.1), we get Γ γ

1 Γ , qδ, 1,1; c ℒE, −, , , ; = 1− 1 . 2 This completes the proof of equation (3.3.8). , q ; s

THEOREM 3. If , and c is an arbitrary constant, then ∈ , for fractional > 0, integr >al 0,operator ∈ 0,1∪ of order λ, the following equations holds λ η υ γ δ , Γ E c, , , , q υ Γγ . + + 1, 1, , , 1,1; = t 3.3.9 and + + + 1, 1 , + 1, , , ; λ β η υ γ δ , , ∞ Ec, , , , q υ λ β υ λ γ δ + − = t Ec, + + k, , , q 3.3.10 ! PROOF OF (3.3.9): Applying Erdélyi-Kober operators of first kind (1.3.1) to the equation (3.2.4), we have

, Γ E, , , , = − E, , , , Γ ,, = − , using the definition (2.1.3) of generalized Mittag-Leffler function, above equation reduces to

∞ Γ Γ = − + + 1 ∞ Γ Γ = − + + 1 now substituting , which yields =

, E , , , , ∞ Γ Γ = 1 − + + 1 on simplification, we get ∞ Γ Γ Γ Γ Γλ + + + 1 = ∞ + + 1 + + + + 1 Γ Γ Γ λ + + + 1 = ∞ + + 1 + + + + 1 Γ Γ Γ Γ

Γ Γ λ + + +Γ 1 + Γ 1 + = + + + + 1 + + 1 + ! finally using Wright generalized hypergeometric function (1.8.1), we get

, E , , , , Γ

Γ . + + 1, 1, , , 1,1; = This completes the proof of equation + + (3.3.9). + 1, 1 , + 1, , , ;

REMARK 1. Setting in (3.3.9), it give rise to the equation (3.3.1) as = 0 . , , E, , , , = E, , , , = E,+ ,,,

PROOF OF (3.3.10): Applying Saigo operator (1.4.1) to the equation (3.2.4), we have

,, E, , , , Γ . = − +,−;;1− × E, , , , Γ . ,, = − +,−;;1− , using the definition of Gauss’s hypergeometric function (1.5.1), and generalized Mittag-Leffler function (2.1.3), we obtained

,, E , , , , ∞ Γ + − = − 1 − ! ∞ Γ × ∞ ∞ + + 1 Γ + − Γ = ! + + 1 × − 1 − ∞ ∞ Γ1 + − Γ = ! + + 1 × − now substituting , which yields =∞ ∞ Γ + − Γ = ! + + 1 × 1 − ∞ ∞ Γ + − Γ = ! + + 1 Γ Γ

Γ + + + 1 × ∞ ∞ ++++1 + − Γ = × ∞ ! ∞ ++++1

+ − Γ = ∞ ! ∞ ++++1

+ − Γ = ! ++++1

again using the definition of generalized Mittag-Leffler function (2.1.3), we obtained ∞

+ − ,, = , ! finally by virtue of equation (3.2.4), we obtained ∞

+ − = E,++,,, ! This is the proof of equation (3.3.10).

REMARK 2. Setting and in (3.3.10), it give rise to the equation (3.3.1) as = 0 = − . ,, E , , , , = E , , , , = E ,+ ,,, 3.4 CONCLUDING REMARKS:

Using the function υ γ δ and µ γ δ obtained by Ec, , , , q Ec, − , , , q operating on the generalized Mittag-Leffler function , various ,, , theorems and relevant properties have been illustrated fairly adequately. The technique will be used in deriving and illustrating other properties in future.

CHAPTER - 4

FRACTIONAL CALCULUS OF GENERALIZED M-SERIES

SUMMARY

This chapter deals with fractional calculus of the generalized M-series. Certain relations that exist between M-series and the Riemann-Liouville fractional integrals and derivatives are investigated. It has been shown that the fractional integration and differentiation operators transform such functions with power multipliers into the functions of the same form.

4.1 PROLOGUE:

Sharma and Jain (2005) introduced the generalized M-series by means of the equation (1.10.1). M-series is a further extension of both Mittag-

Leffler function and generalized hypergeometric function . , and these functions have recently found essential applications in solving problems in

engineering and applied sciences. Some special cases of the - () function i.e. M-series are given in the equations (1.10.2), (1.10.3) and (1.10.4). The object of the present chapter is to evaluate the Riemann-Liouville fractional integrals and derivatives involving generalized M-series. The Saigo operator (1.4.1) involving Gauss’s hypergeometric function (1.6.1) is also applied on generalized M-series to obtain certain new results. The

applications of the main results in deriving some well established results are also discussed in the form of remarks.

4.2 MAIN RESULTS:

Here we discuss the fractional calculus of generalized M-series using

the fractional calculus operators , , , and ,, defined in the chapter 1.

THEOREM 1. Let and be the left sided α>0,β>0,γ>0,a∈R operator of Riemann-Liouville fractional integral (1.2.1). Then there holds the formula

; ; ()

= ; ; (4.2.1) PROOF: Let we represent L.H.S. of equation (4.2.1) as K i.e.

≡ ; ; () now using Riemann-Liouville fractional integral formula given in (1.4), we obtained

≡ ; ; () 1 = Γ ( − ) ; ; by the definition of generalised M-series (1.10.1), we get ∞ 1 ( ) … = Γ ( − ) Γ ( ) ( + ) … ∞ 1 ( ) … = Γ ( − ) Γ ( ) ( + ) …

∞ 1 (a ) … a a = Γα Γ (x − t) t dt ( ) (βn + γ) b … b to sove the integral we let

= ⇒ = hence, ∞ 1 (a ) … a a K = Γα Γ ( ) (b) … b βn + γ × (1 − ) by the definition of beta function we know that

Γ Γ β (1 − ) = Γ ( + β) using above formula in K, we obtained

1 () … Γ Γ( + ) = ( ) ( ) Γ () … Γ + Γ + + ( ) … = ( ) () … Γ + + again using definition of M-series i.e . the equation (1.10.1), finally we obtained

= ; ; . This completes the proof of (4.2.1).

REMARK 1. If we put in (4.2.1), we ==1,=∈,=1 obtained

, () = ,

which is well derived result given by Saxena and Saigo (2005, p.145, Eq.14).

THEOREM 2. Let and be the right sided α>0,β>0,γ>0,a∈R operator of Riemann-Liouville fractional integral (1.2.2). Then there holds the formula

; ; ()

= ; ; (4.2.2)

PROOF: Let we represent L.H.S. of equation (4.2.2) as K, i.e.

≡ ; ; () now using Riemann-Liouville fractional integral formula given in (1.2.2), we obtained

≡ ; ; () ∞

1 = ( − ) ; ; Γ by the definition of generalized M-series (1.10.1), we get ∞ ∞ 1 ( ) … = ( − ) Γ ( ) Γ( + ) … ∞ ∞ 1 ( ) … = ( − ) Γ ( ) Γ( + ) … ∞ ∞ 1 ( ) … = ( − ) Γ ( ) Γ( + ) … to solve the integral we let

−= ⇒=(1 + )

and = hence, ∞ 1 () … = ( ) Γ () … Γ + ∞

× ( ) [(1 + )]

∞ ∞ 1 ( ) … = × Γ ( ) Γ( + ) (1 + ) … by the definition of beta function we know that ∞ Γ Γ β (, β) = = Γ (1 + ) ( + β) using above formula in K, we obtained ∞ 1 ( ) … = (, + ) ( ) Γ () … Γ + ∞ Γ Γ 1 ( ) … (βn + γ) = Γ ( ) ( ) Γ () … Γ + + βn + γ ∞ ( ) … = ( ) () … Γ + + again using definition of M-series i.e . the equation (1.10.1), finally we obtained

= ; ; . This completes the proof of (4.2.2).

REMARK 2. If we put in (4.2.2), we ==1,=∈,=1 obtained

, () = , which is well derived result by Saxena and Saigo (2005, p.147, Eq.23).

THEOREM 3. Let and be the left sided α>0,β>0,γ>0,a∈R operator of Riemann-Liouville fractional derivative (1.2.3). Then there holds the formula

; ; ()

= ; ; (4.2.3)

PROOF: Let we represent L.H.S. of equation (4.2.3) as K, i.e.

≡ ; ; () now using Riemann-Liouville fractional derivative formula given in (1.2.3), we obtained

≡ ; ; () = ; ; () 1 = ( − ) Γ( − )

× ; ; where . Now by the definition of generalised M-series = [] + 1 (1.10.1), we get

1 = ( − ) Γ( − ) ∞ ( ) … × ( ) () … Γ +

∞ 1 ( ) … = ( − ) Γ( − ) ( ) Γ( + ) … ∞ 1 () … = ( ) Γ( − ) () … + × ( − ) to sove the integral we let

= ⇒ = hence,

∞ 1 () … = ( ) Γ( − ) () … Γ + × (1 − ) by the definition of beta function we know that

Γ Γ β (1 − ) = Γ ( + β) using above formula in K, we obtained ∞ 1 () … Γ( − ) Γ( + ) = ( ) ( ) Γ( − ) () … Γ + Γ − + + × ( ) … = ( ) () … Γ + + −

() … = ( ) () … Γ + + −

× ( + − + − 1)( + − + − 2) … ( + − ) () … = ( ) () … Γ + −

( ) … = ( ) () … Γ + − again using definition of M-series i.e . the equation (1.10.1), finally we obtained

= ; ; . This completes the proof of (4.2.3).

REMARK 3. If we put in (4.2.3), we ==1,=∈,=1 obtained the result given by Saxena and Saigo (2005, p.149, Eq.29).

THEOREM 4. Let with and α>0,β>0,γ>0, γ − α + α > 1 ∈ and let be the right sided operator of Riemann-Liouville fractional derivative (1.2.4). Then there holds the formula

; ; ()

= ; ; (4.2.4)

PROOF: Let we represent L.H.S. of equation (4.2.4) as K i.e.

≡ ; ; () now using Riemann-Liouville fractional integral formula given in (1.2.4), we obtained

≡ ; ; () = − ; ; () ∞ 1 = – ( − ) Γ( − )

× ; ; where . Now by the definition of generalised M-series = [] + 1 (1.10.1), we get ∞ 1 = – ( − ) Γ( − ) ∞ ( ) … × ( ) () … Γ + ∞ 1 = – ( − ) Γ( − ) ∞ () … × ( ) () … +

∞ 1 () … = ( ) ( ) Γ − () … Γ + ∞ × – ( − ) to solve the integral we let

= ⇒ = − hence, ∞ 1 () … = ( ) ( ) Γ − () … Γ + × – − − ∞ 1 () … = ( ) ( ) Γ − () … Γ + () × – (1 − )

() × − ∞ 1 () … = ( ) ( ) Γ − () … Γ + × – (1 − ) using the definition of beta function in K, we obtained ∞ 1 () … = ( ) ( ) Γ − () … Γ + Γ( − ) Γ( + − ) × – Γ( − + )

∞ () … Γ( + − ) = ( ) ( ) () … Γ + Γ − + × (−1) ( − − )( − − − 1) … (−βn − γ + 1) ]

∞ () … Γ( + − ) = ( ) ( ) () … Γ + Γ − +

× [( + − )( + − + 1) … (+−+(−1))]

∞ ( ) … Γ( + ) = ( ) ( ) () … Γ + Γ − +

() … = ( ) () … Γ + −

( ) … = ( ) () … Γ + − again using definition of M-series i.e . the equation (1.10.1), finally we obtained

= ; ; . This completes the proof of (4.2.4).

REMARK 4. If we put in (4.2.4), we ==1,=∈,=1 obtained the result given by Saxena and Saigo (2005, p.150, Eq.35).

THEOREM 5. Let >0,>0, >0,>0,>0, ( + ) ≥ 0 and ,, be the left sided generalized fractional integral operator (1.4.1). Then there holds the formula

,, ; ; () Γ( + + 1) = , − Γ(1 − ) Γ( − + 1) + 1, 1 ; ,++1,1−; (4.2.5) PROOF: Let we represent L.H.S. of equation (4.2.5) as K i.e.

,, ≡ ; ; () now using Riemann-Liouville fractional integral formula given in (1.4.1), we obtained

= ( − ) +,−γ;;1− Γ

× ; ; by the definition of generalized M-series (1.10.1), we get

= ( − ) +,−γ;;1− Γ ∞ () … × ( ) () … Γ + again by the use of Gaussian hypergeometric series (1.5.1), we get ∞ ( + )(−γ) = ( − ) 1 − Γ ()(!) ∞ () … × ( ) () … Γ + interchanging the order of integration and summations, we obtained ∞ ∞ ( + )(−γ) () … 1 = ( ) ( ) Γ (!) () … Γ + × ( − ) 1 −

∞ ∞ ( + )(−γ) () … = ( ) ( ) Γ (!) () … Γ + × ( − ) to sove the integral we let

= ⇒ = hence, ∞ ∞ ( + )(−γ) () … = ( ) ( ) Γ (!) () … Γ + × ( − ) ( ) ∞ ∞ ( + )(−γ) () … = ( ) ( ) Γ (!) () … Γ + × (1 − ) now using the definition of beta function in K, we obtained ∞ ∞ ( + )(−γ) () … = ( ) ( ) Γ (!) () … Γ + Γ( + 1)Γ( + ) × Γ( + + + 1) ∞ ∞ ( + )(−γ) ( ) … = ( ) (!) () … Γ + Γ( + 1) × Γ( + + + 1) Γ( + ) as, () = Γ ∞ ∞ ( + )(−γ) Γ( + 1) ( ) … = ( ) ( ) ( ) + + 1 (!) Γ + + 1 () … Γ + again by the use of Gaussian hypergeometric series (1.5.1), we get

∞

Γ( + 1) = (+,−γ;++1;1) ( ) Γ + + 1

() … × (4.2.6) ( ) () … Γ + now by the virtue of Gauss summation theorem (Srivastava and Manocha, 1984), we know that

Γ Γ( − − ) (, ; ; 1) = ; () > ( + ) (4.2.7) Γ( − )Γ( − ) using equation (4.2.7) in the equation (4.2.6), we get ∞

Γ( + + 1)Γ( − + γ + 1) Γ( + 1) = ( ) ( ) ( ) Γ − + 1 Γ + + γ + 1 Γ + + 1

() … × ( ) () … Γ +

Γ(γ − + 1) = Γ(1 − )Γ( + γ + 1) ∞ () … (γ − + 1)(1) × ( )( ) ( ) () … + γ + 1 1 − Γ + finally using definition of M-series, i.e . the equation (1.10.1), we obtained

Γ ( + + 1) = Γ Γ (1 − ) ( − + 1) × ,−+1,1 ; , + + 1,

1 − ; . This completes the proof of (4.2.5).

REMARK 5. If we put in our result (4.2.5), we arrive at the = − result given by Sharma and Jain (2009, p.451, Eq.10).

4.3 CONCLUDING REMARKS:

We conclude this chapter with the remarks that the results proved in this chapter are new and most likely to find some applications to the solutions of certain fractional differential and integral equations.

CHAPTER - 5

ON SOLUTION OF GENERALIZED FRACTIONAL KINETIC EQUATIONS

SUMMARY

This chapter is devoted to investigate certain generalized fractional kinetic differintegral equations using Laplace transform technique. Section-A presents the solution of a generalized fractional kinetic equation which involves an integral operator containing a generalized Mittag-Leffler function in its kernel. In section- B, Fractional kinetic differintegral equations involving M-series are also studied and results are obtained in the form suitable for numerical computation. Several special cases containing generalized Mittag-Leffler function are discussed. An alternative method is suggested for solving certain fractional differential equations.

5.1 PROLOGUE:

The kinetic equation of fractional order have been successfully used to determine certain physical phenomena governing diffusion in porous media, reaction and relaxation processes in complex system etc.; therefore, a large body of research in the solution of these equations has been published in the literature (Haubold and Mathai, 2000, Chechkina and Gonchar, 2002, Chaurasia et al. , 2010, Saxena et al., 2002, 2004a, 2004b, 2006a, 2006b, 2006c, 2010). Fractional kinetic equation for Hamiltonian chaos is discussed by Zaslavsky (1994). Solutions and applications of certain kinetic equations are studied by Saichev and Zaslavsky (1997).

Consider an arbitrary reaction characterized by a time dependent quantity . It is possible to equate the rate of change to a = () balance between the destruction rate and the production rate of , that is . In general, through feedback or other interaction = − + mechanisms, destruction and production depend on the quantity N itself: or . This dependence is complicated since the = () = () destruction or production at time t depends not only on but also on () the past history of the variable N. This may be formally (), < , represented by (Haubold and Mathai, 2000)

= −() + () (5.1.1) where denotes the function defined by ∗ ∗ ∗ , ( ) = ( − ), > 0 and are functional and equation (5.1.1) represents a functional- differential equation.

Haubold and Mathai (2000) studied the following special case of equation (5.1.1), when spatial fluctuation or inhomogeneities in quantity are neglected () = −() (5.1.2) with the initial condition that is the number density of ( = 0) = species at time and constant . Equation (5.1.2) is known = 0 > 0 as standard kinetic equation. The solution of the above standard kinetic equation (2) is given by

() = (5.1.3) Haubold and Mathai (2000) studied the generalized fractional kinetic equation (1.12.3) and obtained its solution given by mean of equation (1.12.4).

The solution of the following generalization of the fractional kinetic equation in terms of Mittag-Leffler function

() − ,(− ) = − ( )() (5.1.4)

is obtained by Saxena et al. (2002) as

() = ,(− ) − ,(− ) (5.1.5) −

which extended the work of Haubold and Mathai (2000). In another paper the solution (1.12.6) of the generalized fractional kinetic equation (1.12.5) is obtained by Saxena et al. (2004a). Kumar (2013) obtained the solutions of fractional kinetic equations are through a binomial type integral transform. Saxena et al. (2013) used Sumudu transform (Belgacem and Karaballi, 2005) to extend the work done by Saxena et al. (2004), and Chaurasia and Pandey (2008).

The main motive for this chapter is to evaluate the solutions of certain generalized fractional differintegral equations. Mainly the Laplace trasform technique is used to obtained the results. In section-B we used the method employed by Babenko (1986) for solving various types of fractional integral and differential equations. The results obtained are likely to find applications to the solutions of certain more differintegral equations. To present the results in an effective manner, we have divided this chapter into two sections.

SECTION – A

5.2 INTRODUCTION:

Prabhakar (1971) defined and investigated in detail the integral

operator given by (1.9.4) with ,,; () ,,, ∈ , () >

containing the function (1.9.3) in its kernel. The fractional 0, () > 0 integral operator (1.9.4) was further investigated by Kilbas et al. (2004). The object of this section is to present solution of a generalized fractional kinetic equation which involves an integral operator

. ,,; ()

LEMMA 1. If then ,,, ∈ , () > 0, () > 0

ℒ ,,; () = () (5.2.1) ( − )

PROOF:

Taking Laplace transform of equation (1.9.4) both side, we have

ℒ ,,; () = ℒ ( − ) ,(( − ) )() using convolution theorem given by (1.11.2), we get

= ℒ ,( ) ℒ() using equation (1.11.5), we obtained

= () ( − ) where . φ(s) = ℒφ(t) This completes the proof of (5.2.1).

LEMMA 2. If , ,,,,, ∈ ℂ, () > 0, () > 0, () > 0 then

( − ) ,( − ) ,( ) = , (5.2.2)

PROOF:

Applying convolution theorem for the Laplace transform given by (1.11.2), we get

ℒ ( − ) ,( − ) ,( ) ()

= ℒ ,( ) ℒ ,( )() = ( − ) ( − ) ()() = () ( − ) = ℒ ,( ) () now taking inverse Laplace transform both side, we obtained

( − ) ,( − ) ,( ) = , . This completes the proof of (5.2.2).

REMARK 1. Putting in equation (1.9.4), we have φ(t) = ,( ) ,,; ,( ) = ( − ) ,( − ) × ,( ) using equation (5.2.2), we get

,,; ,( ) = , (5.2.3)

REMARK 2. If we set in equation (5.2.2), we get = 1

( − ) ,( − ) ,( )

= , . (5.2.4) in particular putting in equation (1.9.4) and using φ(t) = ,( ) equation (5.2.4), we obtained

,,; ,( ) = , . (5.2.5)

5.3 MAIN RESULT:

THEOREM 1. If , and ∞ is min (), () > 0, > 0 ∈ (0, ) Lebesgue integrable function, then for the solution of the fractional kinetic equation

() − () = − ,,;() (5.3.1) there holds the formula

() = − ,,;(). (5.3.2) Number density of a given species at time t, N0 = N(0) is the number density of that species Where, N(t) denotes the at time t = 0 .

PROOF: Applying Laplace transform on both sides of (5.3.1), we get

ℒ() − ℒ() = − ℒ ,,;() and using (5.2.1), we get

() − ( ) = − () ( − ) which can be written as

1 + () = ( ) ( − ) thus

( ) () = 1 + ( − )

= 1 − (− ) ( ) ( − )

( ) = − ( ) (5.3.3) ( − ) again by virtue of (1.11.5), it is not difficult to see that

− = ℒ − , ( ) () ( − ) now applying inverse Laplace transform and using convolution theorem (1.11.2) and above result in (5.3.3), we obtain

() = − ( − ) , (( − ) )() finally by virtue of (1.9.4) we arrives at

() = − ,,;(). This completes the proof of theorem 1.

On setting in (5.3.1) we can deduce a particular case of () = solution (5.3.2) given by Corollary 1.

COROLLARY 1. If , and (), (), () > 0, > 0 ∞ is Lebesgue integrable function, then for the solution of the ∈ (0, ) fractional kinetic equation

() − = − ,,;() (5.3.4) there holds the formula

() = − Γ() , . (5.3.5)

PROOF:

Putting in equation (5.3.2), we get () = () = − ,,; (5.3.6) now using the following integral formula given by Kilbas et al. (2004, eq. 2.26)

( − ) ,( − ) = , it is obvious that

,,;( ) = Γ() , (5.3.7) using (5.3.7) in equation (5.3.6), we obtained

() = − Γ() , . This completes the proof of corollary 1.

REMARK 3. If, however , , then the following kinetic equation: = 1

() − = − ,,;() (5.3.8) has its solution in the space given by L(0, ∞)

() = − , . (5.3.9)

Again on setting in (5.3.1) we can deduce a () = ,( ) particular case of solution (5.3.2) given by Corollary 2.

COROLLARY 2. If , and (), (), () > 0, > 0 is Lebesgue integrable function, then for the solution of the ∈ (0, ∞) fractional kinetic equation

() − ,( ) = − ,,;() (5.3.10) there holds the formula

() = − , . (5.3.11) PROOF:

Putting in equation (5.3.2), we get () = ,( )

() = − ,,; ,( ) (5.3.12) now using equation (5.2.5), we have

,,; ,( ) = , (5.3.13) using (5.3.13) in equation (5.3.12), we obtained

() = − , . This completes the proof of corollary 2.

On setting , in (5.3.1) we can deduce a ( ) , particular case of solution= (5.3.2) ( given) by ∈ Coroll ary 3.

COROLLARY 3. If , and is Lebesgue integrable ( function,), () , then ( for) > the 0, solu >tion 0 of the fractional ∈ (0, ∞) kinetic equation

() − ,( ) = − ,,;() (5.3.14) there holds the formula

() = − , . (5.3.15)

PROOF:

Putting in equation (5.3.2), we get () = ,( )

() = − ,,;() = ,( ) (5.3.16) now using equation (5.2.3), we have

,,; ,( ) = , (5.3.17) using (5.3.17) in equation (5.3.16), we obtained

() = − , . This completes the proof of corollary 3.

Several more special cases of theorem 1 can also be obtained by substituting suitable special function for f(t).

SECTION-B

5.4 INTRODUCTION:

In the present section we are investigating certain generalized fractional kinetic differintegral equations. Several special cases involving Mittag-Leffler function and M-series are also presented. We have also consider a more generalized fractional kinetic equation. Its solution involving M-series and generalized fractional integral operator containing a generalized Mittag-Leffler function in its kernel is obtained.

5.5 MAIN RESULT:

The main objective of this part is to investigated the solution of certain fractional kinetic differintegral equations using Laplace transform technique.

Let denotes the number density of a given species at time t, () is the number density of that species at time t = 0 . = (0)

THEOREM 2. If , and , then min (), () > 0, > 0 ∈ (0, ∞) for the solution of the fractional kinetic differintegral equation

()() − () = − ( )() (5.5.1) with the initial condition

= , ( = 0,1,2,…,−1) (5.5.2) there holds the formula

() = ,,;()

+ ,− . (5.5.3) Where, = + 1. PROOF:

Applying Laplace transform on both side of (5.5.1), we get

− ℒ 0+() − ℒ0() = − ℒ0+() now using (1.11.3) and (1.11.4), we get

() − − ( ) = − ()

79 which can be written as

( + ) () = ( ) + thus we have

() = ( ) + (5.5.4) + + for first part in RHS of equation (5.5.4), using (1.11.5), we get

= ℒ ,(− ) + again by convolution property (1.11.2), we have

( ) +

= ℒ ( − ) ,(− ( − ) )() (5.5.5) for the second part in RHS of equation (5.3.4), using (1.11.5), we get

= ℒ ,(− ) + using (5.5.5) and above equation in (5.5.4), we get

() = ℒ ( − ) ,(− ( − ) )()

+ ℒ ,(− ) now taking Laplace inverse both side, we obtained

() = ( − ) ,(− ( − ) )()

+ ,(− ) using (5.5.2) and by virtue of (1.9.4) (for ), we arrive at the = 1 following solution

() = ,,;() + ,− . This completes the proof of theorem 2.

REMARK 4. If we set and in (5.5.3), we get the result = 0, = 0 obtained by Saxena et al. (2010).

If we set in (5.5.1), then we can deduce a particular case () = of solution (5.5.3) given by Corollary 4.

COROLLARY 4. If , then for the min (), (), () > 0, > 0 solution of the equation

()() − = − ( )() (5.5.6) holds the relation

() = Γ() ,(− )

+ ,(− ). (5.5.7) PROOF:

Putting in equation (5.5.3), we get () =

() = ,,;( )

+ ,− (5.5.8) now using the following integral formula given by Kilbas et al. (2004, eq. 2.26)

( − ) ,( − ) = , it is obvious that

,, ;( ) = Γ() ,(− ) (5.5.9) using (5.5.9) in equation (5.5.8), we obtained

() = Γ() ,(− )

+ ,(− ). This completes the proof of corollary 1.

If we set in (5.5.1), then we can deduce () = ,(− ) a particular case of solution (5.5.3) given by Corollary 5.

COROLLARY 5. If , then for the min (), (), () > 0, > 0 solution of the equation

()() − ,(− ) = − ( )() (5.5.10) holds the relation

() = ,(− )

+ ,(− ). (5.5.11) PROOF:

Putting in equation (5.5.3), we get () = ,(− )

() = ,,; ,(− )

+ ,− (5.5.12) now using equation (5.2.5) for , we have = 1

,, ; ,(− )

= ,(− ) (5.5.13) using (5.5.13) in equation (5.5.12), we obtained

() = ,(− ) + ,(− ). This completes the proof of corollary 5.

LEMMA 3. If , ,,,,, ∈ ℂ, () > 0, () > 0, () > 0 then

( − ) ,( − ) ,( ) = , (5.5.14) PROOF:

Applying convolution theorem for the Laplace transform given by (1.11.2), we get

ℒ ( − ) ,( − ) ,( ) () = ℒ ,( )ℒ ,( )() = ( − ) ( − ) ()() = () ( − ) = ℒ ,( )() now taking inverse Laplace transform both side, we obtained

( − ) ,( − ) ,( ) = , . This completes the proof of (5.5.14).

REMARK 5. Putting in equation (1.9.4) for , φ(t) = ,( ) = 1 we have

,,; ,( ) = ( − ) ,( − ) × ,( ) using equation (5.5.14), we get

,,; ,( ) = , (5.5.15)

Now on setting in (5.5.1) we can () = ,(− ) deduce a particular case of solution (5.5.3) given by corollary 6.

COROLLARY 6. If , then for the min (), (), () > 0, > 0 solution of the equation

()() − ,(− ) = − ( )() (5.5.16) holds the relation

() = ,(− )

+ ,(− ). (5.5.17) PROOF:

Putting in equation (5.5.3), we get () = ,(− )

() = ,, ; ,(− )

+ ,− (5.5.18) now using equation (5.5.15), we have

,, ; ,(− )

= ,(− ) (5.5.19)

84 using (5.3.19) in equation (5.3.18), we obtained

() = ,(− )

+ ,(− ). This completes the proof of corollary 6.

REMARK 6. If we put and , the above corollary give rise to = 0 = 0 the solution of generalized fractional kinetic equation as obtained by Saxena et al. (2004a, theorem 1)

LEMMA 4. Let then , , , ∈ , ( (), (), () > 0),

,,; ( )

= ( ) ( ). (5.5.20) PROOF:

Using the operator (1.9.4) for ,we get = 1

,,; ( ) = ( − ) ,( − ) ( ) using formula (1.10.1) for M-series, we get

,,; ( )

() … = () … Γ( + ) × ( − ) ,( − ) using the formula (1.11.8) for , we get = 1

,,; ( )

() … = () … Γ( + ) × Γ( + ) ,( )

( ) … = , () … now using the definition (1.9.2) of Mittag-Leffler function, we obtain

,,; ( )

( ) … ( ) = () … Γ( + + + )

( ) … = ( ) () … Γ( + + + ) finally using formula (1.10.1) of M-series, we get

,,; ( ) = ( ) ( ). This completes the proof of (5.5.20).

THEOREM 3 . If , then for the solution of min (), () > 0, > 0 the fractional kinetic differintegral equation

()() − (− ) = − ( )() (5.5.21) with the initial condition (5.5.2), there holds the formula

() () = (− ) (− ) + ,− . (5.5.22)

Where, = + 1. PROOF:

Putting in equation (5.5.3), we get () = (− )

() = ,, ; (− )

+ ,− (5.5.23) now using equation (5.5.20), we have

,,; ( )

() = ( ) ( ) (5.5.24) using (5.3.24) in equation (5.3.23), we obtained

() () = (− ) (− ) + ,− . This completes the proof of theorem 3.

REMARK 7. For in theorem 3 we obtain corollary 5 in the following manner, p putting = q = 0 and using (1.10.1) in equation (5.5.22), we get p = q = 0

(− ) () = (− ) Γ( + ) + ( + ) + + + ,(− )

(− ) = Γ( + ) + + + ,(− )

(2) (− ) = Γ( + ) + + ! + ,(− )

= ,(− ) + ,(− ) this is equation (5.5.11).

REMARK 8. For and in theorem 3 we obtain = = 1, = = 1 corollary 6.

5.6 ALTERNATIVE METHOD:

Theorem 2 is solved here by using an alternative method. This method is employed by Babenko (1986) for solving various types of fractional integral and differential equations and further described by Podlubny (1999).

Applying fractional integral operator to the both side of (5.5.1) and using the following formula from Samko et al. (1993),

() = () − (5.6.1) Γ( − ) we have,

() 1 + () = () + Γ( − ) thus

() () = 1 + () () + 1 + Γ( − ) using (5.5.2) and binomial expansion, we get

() () = (− ) () () + (− ) Γ( − ) on making use of equation (2.35) from Samko et al. (1993), i.e.,

() () = Γ( − ) Γ( + ) + ( − ) we have,

() () = (− ) ()

− + (5.6.2) Γ( + ) + ( − ) for first part in RHS of equation (5.6.2), using (1.2.1), we get

() (− ) ()

(− ) () = ( − ) () Γ(( + ) + )

(− ( − ) ) = ( − ) () Γ(( + ) + )

89 now using the definition of two parameter Mittag-Leffler function i.e. (1.9.2), we get

() (− ) () = ( − ) ,− ( − ) () by virtue of (1.9.4) for , we obtained = 1

( ) (− ) () = ,, ;() (5.6.3) now using equation (5.6.3) in the equation (5.6.2), we get

() = ,, ;() − + Γ( + ) + ( − ) finally using the definition of two parameter Mittag-Leffler function i.e. (1.9.2), we get

() = ,,;() + ,− . This completes the proof of theorem 2.

Moreover, for the proof of theorem 3 via this alternative method, on putting in the equation (5.6.2),we get () = (− )

() () = (− ) (− ) − + Γ( + ) + ( − ) now using equation (4.2.1), we get

() () = (− ) (− ) + ,− . This completes the proof of theorem 3.

5.7 GENERALIZED FRACTIONAL KINETIC EQUATION WHICH INVOLVE AN INTEGRAL OPERATOR CONTAINING GENERALIZED MITTAG-LEFFLER FUNCTION IN ITS KERNEL:

THEOREM 4. If , then for the min (), (), () > 0, > 0 solution of the fractional kinetic equation

, ( )() − ( ) = − ,,;() (5.7.1) with initial condition,

()() (0 +) = (5.7.2) there holds the formula

() = − , ( ) +

( ) × − ( ) ( ). (5.7.3) ! PROOF:

Applying Laplace transform on both side of (5.7.1), we get

, ℒ( )() − ℒ ( )

= − ℒ ,,;() (5.7.4)

The Laplace transform of the Hilfer operator , is given by ( )() Hilfer (2000) as:

, ℒ ()() = ℒ()()

() ()() − (0 +), (0 < < 1) (5.7.5) using equation (5.7.5) in equation (5.7.4), we get

() ()() () − (0 +) − ℒ ( )

= − ℒ ,,;() now using equations (5.7.2) and (5.2.1), we get

() () − − ℒ ( ) = − () ( − ) thus

() + () = + ℒ ( ) ( − ) using formula (1.10.1) for M-series, we obtained

+ () ( − ) ∞ () ( ) … ( ) = + ℒ () … Γ( + ) ∞ () ( ) … () = + ℒ () … Γ( + ) ∞ () ( ) … Γ( + ) = + () … Γ( + ) thus we have

() () = + ( − )

() … + (5.7.6) () … + ( − ) again by virtue of (1.11.5), it is not difficult to see that

() + ( − ) = ℒ − , ( ) () and,

+ ( − ) = ℒ − ,( ) () taking Laplace inverse of equation (5.7.6) and using above two relations, we get

() = − , ( ) ( ) … + () …

× − ,( ) (5.7.7) again considering the second term of equation (5.7.7) with the definition (1.9.3), we have

( ) … − ,( ) () …

( ) … = − () … () ( ) × Γ(+2+++) !

() = − !

() … ( ) × (5.7.8) () … Γ(+2+++) now using definition (1.10.1) of M-series, we get

() = − ( ) ! finally substituting this result in (5.7.7), we obtained

() = − , ( ) +

( ) × − ( ) ( ). ! This completes the proof of theorem 4.

Since for p = q = 0 from equation (1.10.2) we have ( ) = . Hence if we set p = q = 0 in theorem 4 then we get the ,()( ) following particular case of the solution (5.7.3)

COROLLARY 7. If , then for the min (), (), () > 0, > 0 solution of the fractional kinetic equation

, ( )() − ,()( ) = − ,,;() (5.7.9) with initial condition (5.7.2), there holds the formula

() = − , ( )

+ − ,( ). (5.7.10) PROOF:

Putting p = q = 0 in equation (5.7.3), we get

() = − , ( ) +

( ) × − ( ) ( ) (5.7.11) ! using equation (1.10.2), we get

() = − , ( ) +

( ) × − ( ) ,( ) ! now using the definition (1.9.2) of two parameter Mittag-Leffler function, we obtained

() = − , ( ) +

( ) ( ) × − ( ) ! Γ(+2+++)

= − , ( ) +

() ( ) × − ! Γ(+)+2++

= − , ( ) +

() ( ) × − ! Γ( + 2 + + )

= − , ( ) +

( ) (1) ( ) × ! − ! ( − )! ! Γ( + 2 + + )

= − , ( ) +

( + 1) ( ) × − Γ + 2 + + ! finally using definition (1.9.3), we obtained

() = − , ( ) + − ,( ). This completes the proof of corollary 7.

A number of several special cases of theorem 4 can also be obtained by taking suitable values for parameters in M-series.

CHAPTER - 6

ALTERNATIVE METHOD FOR SOLVING GENERALIZED FRACTIONAL DIFFERENTIAL EQUATION

SUMMARY

This chapter presents an alternative simple method for deriving the solution of the generalized forms of the fractional differential equation and Volterra type differintegral equation. The solutions are obtained in a straight- forward manner by the application of Riemann-Liouville fractional integral operator and its interesting properties. As applications of the main results, solutions of certain generalized fractional kinetic equations involving generalized Mittag-Leffler function are also studied. Moreover, results for some particular values of the parameters are also pointed out.

6.1 PROLOGUE:

Fraction differential and integral equations are one of the corner stones of modeling in most scientific and engineering application, employed to model innumerable phenomenon for instance in solid state physics, nonlinear optics, fluid dynamics, mathematical biology and chemical kinetics. One may refer to the books by Samko et al. (1993), Podlubny (1999) and Hilfer (2000), and the recent papers Saxena et al. (2010) and Purohit and Kalla (2011) on the subject.

Due to the extensive applications of fractional differintegral equations in engineering and science, various techniques for solving these equations has been developed significantly (Podlubny, 1999). For the solution of fractional differential equation an iteration method is given by Samko et al. (1993). In particular, the solution and application of certain kinetic equations of fractional order are studied by Zaslavsky (1994), Saichev and Zaslavsky (1997), Saxena et al. (2006a, 2006b, 2006c), Haubold et al. (2011) using integral transform technique. Jaimini and Gupta (2013) established solutions of some fractional differential equations using the Laplace transform method. The object of this chapter is to investigate solution of certain class of generalized fractional differential equations by applying the technique similar to that used by Al-Saqabi and Tuan (1996) for solving general differintegral equation of Volterra’s type. The method extend the use of Riemann-Liouville fractional calculus operators.

6.2 MAIN RESULTS:

In this section, we shall evaluate the solution of certain generalized fractional differential equations.

Corresponding to the bounded sequence , let the function () is defined as () = (6.2.1)

THEOREM 1. If >0, >0,>0 , then there exists the unique solution of the fractional differential equation

() + () = ( − ) − ( − ) (6.2.2) given by

Γ( + ) () = ( − ) − ( − ) . (6.2.3) Γ( + ) PROOF:

Multiplying both sides of (6.2.2) by , we get (− )

() (− ) () − (− ) () = (− ) ( − ) − ( − ) now summing up from = 0 to ∞, it yields () (− ) () − (− ) () = (− ) ( − ) − ( − ) thus we have

() = (− ) ( − ) − ( − ) using (6.2.1), we obtain

() = (− ) ( − ) (− ) ( − ) = (− ) ( − ) on using (1.2.5) above equation becomes

Γ( + ) () () = (− ) ( − ) Γ( + + ) now using the following relation given by Srivastava and Manocha (1984)

(, ) = (, − ) (6.2.4) we obtained

() = ( − ) (− ) Γ( + ) × ( − )() Γ( − + ) + )

Γ( + ) = ( − ) − ( − ) . Γ( + ) Which completes the proof of theorem 1.

THEOREM 2. If υ>0,α>0, μ > 0, > 0 then there exists the unique solution of the Volterra’s type fractional differintegral equation

() + () = ( − ) − ( − ) (6.2.5) with the initial conditions

= ; ( =0,1,2,…,−1) (6.2.6) given by

() = ( − ) Γ( + ) + × − ( − ) Γ( + ) + + + ( − ) × ,− ( − ) . (6.2.7)

PROOF:

Applying the fractional integral operator to the both sides of (6.2.5), we get

() + () = ( − ) − ( − ) now using the following formula given by Samko et al. (1993)

() ( − ) = () − (6.2.8) Γ( − ) where be an integer such that = + 1, we obtain () () + () = ( − ) − ( − ) ( − ) + (6.2.9) Γ( − )

100 now applying () to the both sides of (6.2.9), we get (− )

() ()() (− ) () − (− ) () () = (− ) ( − ) − ( − ) () ( − ) + ( )(− ) Γ( − ) now summing up from = 0 to ∞, we get () ()() (− ) () − (− ) () () = (− ) ( − ) − ( − ) () ( − ) + ( )(− ) Γ( − ) using equations (1.2.5) and (6.2.6), we obtained

() () = (− ) ( − ) − ( − ) () (− ) ( − ) + Γ( + ) + ( − ) () = (− ) ( − ) − ( − ) () (− ) ( − ) + ( − ) Γ( + ) + ( − ) now applying definition (1.9.2), we obtained

() () = (− ) ( − ) − ( − ) + ( − ) ,− ( − ) using definition (6.2.1) in the first part of right hand side of above equation, we obtained

101

() () = (− ) ( − ) × A−c (t − a) + ( − ) × ,− ( − ) () () () = A(− ) ( − ) + ( − ) ,− ( − ) again applying definition (1.9.2), we obtained

() = (− ) Γ( + ) + × ( − )()() Γ( + )( + ) + + + ( − ) ,− ( − ) finally using (6.2.4), we obtain

() = ( − ) Γ( + ) + × − ( − ) Γ( + ) + + + ( − ) ,− ( − ) . This completes the proof of theorem 2.

REMARK 1. It is interesting to observe that, when and = 0 = 0 theorem 2 reduces to theorem 1.

102

6.3 APPLICATIONS OF THE MAIN RESULTS:

In this section, we consider some consequences and applications of the main results. By assigning suitable special values to the arbitrary sequence , our main results (theorems 1 and 2) can be applied to derive solutions of certain generalized fractional kinetic equation.

If is the number density of a given species at time and is the () number density of that species at time = , then following results holds.

THEOREM 3. If υ>0,μ>0,>0 , then there exists the unique solution of the fractional kinetic equation

() − ( − ) ,− ( − ) = − () (6.3.1) given by

() = ( − ) , − ( − ) (6.3.2) where is the Mittag-Leffler function defined by (1.9.3). , PROOF:

Let () , then by virtue of theorem 1, we have = ()!

() = ( − ) () Γ( + ) × −( − ) Γ( + )! Γ( + )

() − ( − ) = ( − ) Γ( + ) ! now by equation (1.5.2) we have Γ , thus we can (1) = ( − + 1) write

() = ( − ) () (1) ! − ( − ) × Γ( + ) Γ( − + 1) ! ! using the following formula, used by Carlitz (1977)

103

( + ) = () () (6.3.3) we get

( + 1) () = ( − ) − ( − ) Γ( + )! finally on using equaion (1.9.3), we obtained

() = ( − ) , − ( − ) . This completes the proof of theorem 3.

When = 0 , theorem 3 reduces to the following result given by Saxena et al. (2004a):

COROLLARY 1. If υ>0,>0,μ>0, then the unique solution of the integral equation

() − ,− = − () (6.3.4) is given by

() = , − . (6.3.5)

THEOREM 4. If υ>0,α>0,μ>0,>0, then there exists the unique solution of the fractional kinetic differintegral equation

() − ( − ) ,− ( − )

= − () (6.3.6) with the initial conditions

= ; ( =0,1,2,…,−1) (6.3.7) given by

() = ( − ) ,− ( − )

+ ( − ) ,− ( − ) . (6.3.8)

104

PROOF:

Let () then by virtue of theorem 2, we have = ()!

() () Γ( + ) + = ( − ) Γ( + ) + ! Γ( + ) + + × − ( − ) + ( − ) ,− ( − )

() − ( − ) = ( − ) Γ( + ) + + ! + ( − ) ,− ( − ) now by equation (1.5.2) we have Γ , thus we can (1) = ( − + 1) write

() = ( − ) () (1) ! − ( − ) × Γ( + ) + + Γ( − + 1) ! ! + ( − ) ,− ( − ) using the formula (6.3.3), we obtained

() = ( − ) ( + 1) × −( − ) Γ( + ) + + ! + ( − ) ,− ( − ) . finally, on making use of equation (1.9.3), we can easily arrive at

() = ( − ) ,− ( − )

+ ( − ) ,− ( − ) .

105

This completes the proof of theorem 4.

REMARK 2. Again, if we set and theorem 4 reduces to α = 0 b = 0 theorem 3.

COROLLARY 2. If 0 < α ≤ 1, υ>0,α>0,μ>0,>0, then there exists the unique solution of the fractional kinetic differintegral equation

(6.3.6), with the initial condition , given by = () = ( − ) ,− ( − ) + ( − ) ,− ( − ) . (6.3.9) The solution (6.3.9) can easily be obtained by putting = () + in equation (6.3.8) and then using the initial condition 1 = 1 = .

COROLLARY 3. If 1 < α ≤ 2, υ>0,α>0,μ>0,>0, then there exists the unique solution of the fractional kinetic differintegral equation

(6.3.6), with the initial condition , given by = ; = 0, 1 () = ( − ) ,− ( − ) + ( − ) ,− ( − ) + ( − ) ,− ( − ) . (6.3.10) The solution (6.3.9) can easily be obtained by putting = () + in equation (6.3.8) and then using the initial condition 1 = 2 = . ; = 0, 1

6.4 CONCLUDING REMARKS:

We conclude with the remark that, in this paper, we have used an alternative, simple and effective method for solving fractional differential and Volterra’s type fractional differintegral equations. It has been shown that by selecting bounded sequence, we found solutions of generalized Kinetic equations, as special cases of the main results. Further, as the

106 applications of these results, one can easily derive a number of graphical representations for solutions by setting numerical values to the parameters used in the main results.

107

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