Conformable Fractional Differintegral Method for Solving Fractional Equations

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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 9, ISSUE 03, MARCH 2020 ISSN 2277-8616 Conformable Fractional Differintegral Method For Solving Fractional Equations

Saber T.R. Syouri, Mustafa Mamat, Ibrahim M.M. Alghrouz, Ibrahim Mohammed Sulaiman

Abstract: The standard approaches to the problem of conformable fractional calculus has been studied extensively. Many researchers have shown that  the obtained conditions for the theorem describing the general solution of; y a( x ) y b ( x ) are generally weaker than those derived by using the classical norm-type expansion and compression theorem. In this paper, we propose conformable method for the fractional differential transform and established the prove for basic properties of differintegrals. Some solved examples have been reported to illustrate the possible application of the obtained results.

Index Terms: Conformable Fractional Derivatives, Fractional Calculus, α- differintegrals, the integrating factor, Caputo, Riemann– Liouville, positive solution ————————————————————

1 INTRODUCTION 2.1 Fractional Integral: The fractional Calculus (FC) is a branch of mathematics that The fractional Integral is an extension of fractional derivatives; investigate the properties of integrals and derivatives of non- there are some accepted and common definitions in many integer orders called differintegrals which represents either researches as we stated a number of important ones above. fractional (FD) derivatives or fractional integrals (FI). The The conformable fractional Integral is defined as follows. Definition 1.[2] following definitions are very important in this study; I. conformable fractional derivative;[1] Suppose f is differentiable on aa, , 0 . Then,

1   x  f()() x x f x a 11  ft() fx () =lim , I(())(), f x I x f d t   0  .  1   a t

x  0 , 0   1.[1] a   1 where I() x f is the general form of Riemann improper II. Riemann improper integral;[8] 1 x integral, and   (0 , 1) . nn1 1 D f()(-)(). x x t f t d t   (n - 1) ! a Theorem 1 These definitions are a natural extension of the usual Let   0,1 and fg , be -differentiable funct ions at a point derivatives and integral, and it achieves the general characteristics of the fractional integration. Lately, scholars x  0, and if f is differentiable ,then: have given new definitions of fractional calculus which is 1- d D ( f ( x )) x f ( x ) conspicuously compatible with the known definitions of  fractional derivative and integral [1]. Unlike previous dx definitions, this definition gratifies formularies the quotient of Proof: two functions with application to derivative of product. For By applying the fractional derivative definition we get; 1   more results on fractional derivatives and fractional differential f()() x x f x D(()) f x  lim . equations (FDE), see the references therein: [1], [2], [3], [4],    0  [5], [6]. Then,

1   S u p p o se that, h x, hence h  0 when  0 2 PRELIMINARIES   1 In this section, we define some preliminary results which and   xh , would help researchers to understand the basic concept behind this work. f()() x h f x D ()fx ( ) = lim    1 h  0 ______xh

1   f()() x h f x • Saber T.R. Syouri is currently pursuing Ph.D degree program in D ()f ( x ) = x lim  Mathematics at Sultan Zainal Abidin University, Malaysia, E-mail: h  0 h [email protected] H e n c e • Mustafa Mamat is currently a Professor of Mathematics Sultan Zainal Abidin University, Malaysia. E-mail: [email protected]. 1   d • Ibrahim M.M. Alghrouz is currently an Associate Professor of D ()f ( x ) = x ( f ( x )) Mathematics in Al-Quds University Al-Quds, Jerusalem, Palestine E-  dx mail: [email protected] • Ibrahim Muhammed Sulaiman is currently a Postdoctoral fellow in Sultan Zainal Abidin University, Malaysia. E-mail: [email protected]

Theorem 2 Hence, Supposse f is continuous on  aa, , 0 . Then The relation between fractional derivative and fractional a D ( I ( f ( x ))) f ( x ), for x a .  integration is satisfies. Proof: aa a IDfx( (())) DIfx ( (()))  fx () ,for xa   0. I(()) f x     Since f is smooth, by difinition1; and,  is differentiable. Then, by applying Theorem1, we get; From the above results, we derived the proposed scheme as

aa1   d D((()))(()) I f x x I f x follows.    dt 3 METHOD FORMULATION The derived formula for the solution of FDE with variable x 1   d f() t  x d t coefficients is defined as follows. The following Theorem is  1   d xa t very useful in the derivation process.

Theorem 4 (Variable Coefficients)

1   fx() Suppose y f( x ) is differentiable and a( x ) and b ( x ) are  x 1   x continuous functions, then the FDE,   fx ( ) y a( x ) y b ( x ) .

Theorem 3 Has infinitely many solutions given by Let f be a differentiable function on  a ,   , with f (a)=0, then A()()() x A x A x a y( x ) ce + e I e b ( x ) (1) I( D ( f ( x ))) f ( x ), for x  a  0.     Where A ( x ) I a ( x ),   (0,1) and c    . Proof: Let f be a differentiable function, then by the definition 1, Remarks: 1  

aaf()() x x f x I( D (())) f x I (lim )    a. The expression in Eq. (1) is called the solution of   0  fractional differential equation.

x  0,   0,1.  Ax() b. The function  ()xe is called the integrating factor 1   Let hx ; so clearly as 0 , of the equation.  -1 then h 0 and   x  h . Proof: T h e re fo re , Writing the fractional differential equation with y on one side only, as aaf()() x h f x  I( D (())) f x I (lim ) y a( x ) y b ( x ) .     1 h  0 x  h Then multiply the above by  ()x , we need to solve for  ()x  such that,  (x ) a ( x ) ( x ) aa1- f()() x h f x I((())) D f x  I ( x lim )    h  0 h Suppose there is a function  ()x satisfying the equation (4),

then df  a 1- ()()()()x y  x y  x b x (5) Ix ( )  dx By definition 1, we have  Using Definition 1 (())()()x y x b x (6)

x 1 df a 1- Using Theorem 1 to integrating both sides of equation (6), then I ( D ( f ( x )))  t d t   1- a t d t ()x y I (()()) x b x  c c   (7) x  11 a df I ( D ( f ( x )))  d t   a dt And so, 1 c y I(()()) x b x 1 (8)  f ( x ) - f ( a ) .  ()()xx But fa ( ) 0 that implies, To find  ()x , we go back to equation (4), then

 Ax()  (x ) c e , where A ( x ) I (( a x )). 23 1  a( x )  1 , b ( x )  x  2 x

Then, As in the above theorem the integrating factor;  Ax()  ().()x c e A x 1 .

 But, by Theorem 1 we get A ( x ) a ( x ) , then

  Ax()  ()()x c e a x 1 which implies

  (x ) a (x). ( x ) But Thus  ()x satisfies equation (4), back to equation (8)

So,

Let By Theorem 4

4 RESULTS AND DISCUSSIONS Since, In this section, we present the solutions of solved examples to support our theoretical analysis. We begin with a given From Theorem 1 remark.

Remark: The relation between fractional derivatives and fractional integral are solved using example 1.

Example 1 Let . If If f( x ) sin x , then show that a I( D ()) x sin x ,for x  .  Also, Solution: Since f (x) is differentiable function, then by applying definition 1on Substituting and solving the equation gives: Dx(sin )  we get

Applying definition 1.1 on , we have

Hence .

Problem 2. Consider the following initial fractional equation.

Solution Rewrite the equation with y on only one side,

1 3   2   y22  y  x  2 x Example 3. Consider the following initial fractional equation Using Theorem 4 that given general solution  1  y2  y 0, y (0 )  0.  y a( x ) y b ( x ) , where 294 IJSTR©2020 www.ijstr.org INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 9, ISSUE 03, MARCH 2020 ISSN 2277-8616

Solution: [4] V. Daftardar-Gejji. Fractional Calculus Theory and Rewrite the equation with y on only one side, Applications, India: Norosa Publishing House Pvt. Ltd,  1  yy2  2014.  Given the general solution y a( x ) y b ( x ) then by [5] S. Das. Functional Fractional Calculus, India: Theorem 4, we have, Scientific Publishing Services Pvt. Ltd. 2011. [6] M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El- a( x ) 1 , b ( x ) 0 Saka. On the Fractional-Order Logistic Equation, Applied Mathematics Letters, 20(7): pp. 817-823.

and the integrating factor;  I  1  2007. 1  (x ) ce 2 [7] M. Abu Hammad and R. Khalil. Abel's formula and Wronskian for Conformable Fractional Differential Equations, International Journal of Differential Equations and Applications, 13(3). 2014. [8] T. Harko, F. Lobo, and M.K. Mak. Analytical Solutions of the Riccati Equation with Coefficients Satisfying Integral Or Differential Conditions With Arbitrary Functions, Universal Journal of Applied Mathematics, 2: pp. 109-118, 2013.

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[11] M. Ishteva. Properties and Applications of Caputo Fractional Operator, Msc. Thesis, Universität Karlsruhe (TH), Sofia, Bulgaria. 2005. [12] Communications in Nonlinear Science and Numerical Simulation, 16: pp. 2535-2542. [13] M. M. Khader, T. S. EL Danaf, and A. S. Hendy. Efficient Spectral Collocation Method for Solving Multi-Term Fractional Differential Equations Based on the Generalized Laguerre Polynomials, Journal of Fractional Calculus and Applications, 3(13), pp. 1-14. 2012. 5 CONCLUSIONS In this study, some properties of fractional derivative and [14] R. Khalil, M. Horani, Y. Abdelrahman and M. integrals were established, and the fractional integral for Sababheh. A New Definition of Fractional Derivative, constant function is zero was proved. We also obtained a Computational and Applied Mathematics, 246: pp. 65- relationship between the fractional calculus and calculus that 70. 2014. satisfies, and a general solution was obtained to fractional [15] N. Khan, A, Asmat and M. Jamil. An Efficient  Approach for Solving the Riccatii Equation with differential equations y a( x ) y b ( x ) . Fractional Orders, Computers and Mathematics with Applications, 61(9): pp. 2683-2689, 2011. 6 ACKNOWLEDGMENT [16] J. Leszczynski. An Introduction of Fractional The authors wish to thank A, B, C. This work was supported in Mechanics, The Publishing Office of Czestochowa part by a grant from XYZ. University of Technology. 2011. [17] Mathai and H. Haubold. Special Functions for Applied 7 REFERENCES Scientists: Springer Science and Business Media, [1] R. Khalil, M. Horani, Y. Abdelrahman and M. LLC. New York 2008. Sababheh, A new Definition of Fractional Derivative, Computational and Applied Mathematics, 246: pp. 65- 70, 2014. [2] T. Abdeljawad, T. On Conformable Fractional Calculus, Journal of Computational and Applied Mathematics, 279: pp. 57-66, 2015. [3] B. William and R. Di/Prima. Elementary Differential Equations and Boundary Value Problems, (9th ed), Asia: John Willey & Sons, Inc. 2010. 295 IJSTR©2020 www.ijstr.org