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) by 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 R,S,, 26 2.2. Fractional Calculus of ̿P,Q ʚͮʛ R,S,, 30 2.3. ̿P,Q ʚͮʛ In Term of Other Functions R,S,, 32 2.4. Integral Transforms of ̿P,Q ʚͮʛ 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 . 2 1.2 FRACTIONAL INTEGRAL AND DIFFERENTIAL OPERATORS: Fractional integration is an immediate generalization of repeated integration. If be a finite interval on the ̈́ = ʞ͕, ͖ʟ (−∞<͕<͖<∞) real axis , then the Riemann-Liouville fractional integrals P and P ℝ ͚̓ͮ ͚̓ͯ of order are defined by ∈ ℂ (͙͌ ( ) > 0) 3 P ͯP 1 Pͯͥ (͚̓ͮ)(ͬ) = (̾ͮ ͚)(ͬ) = ǹ (ͬ − ͨ) ͚(ͨ)ͨ͘ ,ͬ > ͕ (1.2.1) Γ( ) and P ͯP 1 Pͯͥ (͚̓ͯ)(ͬ) = (̾ͯ ͚)(ͬ) = ǹ (ͨ − ͬ) ͚(ͨ)ͨ͘ , ͬ < ͖ (1.2.2) Γ( ) 3 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 P and P of ͚̾ͮ ͚̾ͯ order are defined by ∈ ℂ (͙͌ ( ) ≥ 0) ) P ͘ )ͯP (͚̾ͮ)(ͬ) = ƴ Ƹ (̓ͮ ͚)(ͬ) ͬ͘ ) 3 1 ͘ )ͯPͯͥ = ƴ Ƹ ǹ (ͬ − ͨ) ͚(ͨ)ͨ͘ , (͢ = ʞ͙͌ ( )ʟ + 1; ͬ > ͕) Γ(͢ − ) ͬ͘ (1.2.3) and ) P ͘ )ͯP (͚̾ͯ)(ͬ) = ƴ− Ƹ (̓ͯ ͚)(ͬ) ͬ͘ ) 1 ͘ )ͯPͯͥ = ƴ− Ƹ ǹ (ͨ − ͬ) ͚(ͨ)ͨ͘ , (͢ = ʞ͙͌ ( )ʟ + 1; ͬ < ͖) Γ(͢ − ) ͬ͘ 3 (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 P `ͯͥ (̓ͮ(ͨ − ͕) )(ͬ) Γ() `ͮPͯͥ = (ͬ − ͕) (͙͌ ( ) > 0) (1.2.5) Γ( + ) P `ͯͥ (̓ͯ(͖ − ͨ) )(ͬ) Γ() `ͮPͯͥ = (͖ − ͬ) (͙͌ ( ) > 0) (1.2.6) Γ( + ) and P `ͯͥ (̾ͮ(ͨ − ͕) )(ͬ) Γ() `ͯPͯͥ = (ͬ − ͕) (͙͌ ( ) ≥ 0) (1.2.7) Γ( − ) P `ͯͥ (̾ͯ(͖ − ͨ) )(ͬ) Γ() `ͯPͯͥ = (͖ − ͬ) (͙͌ ( ) ≥ 0).