Solution of Nonlinear Fractional Differential. Equations Using the Homotopy Perturbation. Sumudu Transform Method

Transcription

1 Applied Mathematical Sciences, Vol. 8, 2014, no. 44, HIKARI Ltd, Solution of Nonlinear Fractional Differential Equations Using the Homotopy Perturbation Sumudu Transform Method Eltayeb A. Yousif Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Khartoum, Khartoum 11111, Sudan Department of Mathematics, Faculty of Science, Northern Border University, Arar 91431, Saudi Arabia Sara H. M. Hamed Department of Mathematics, Faculty of Mathematical Sciences and Statistics, Alneelain University, Khartoum 11121, Sudan Copyright 2014 Eltayeb A. Yousif and Sara H. M. Hamed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we obtain exact analytical solutions of nonlinear fractional differential equations using a combined form of the Homotopy perturbation method with the Sumudu transform. The solutions are given in closed forms in terms of Mittage-Leffler functions. The fractional derivatives are considered in Caputo sense. The method is illustrated through a number of test examples. Keywords: Homotopy Perturbation Method, Sumudu Transform, Nonlinear Fractional Differential Equations, Mittage-Leffler functions

2 2196 Eltayeb A. Yousif and Sara H. M. Hamed 1. Introduction Fractional calculus is a generalization of differentiation and integration to non-integer orders. Many problems in physics and engineering are modulated in terms of fractional differential and integral equations, such as acoustics, diffusion, signal processing, electrochemistry, and may other physical phenomena [14,26]. During last decades, a great deal of interest appears in fractional differential equations. The solutions of fractional equations are investigated by many authors using powerful methods in obtaining exact and approximate solutions [1,25,30-32,35,36]. The Homotopy perturbation method (HPM) is proposed by He in 1999 [17]. This method is a coupling of traditional perturbation method and homotopy in topology. Later on He himself drawn many modifications and developments of the method [17-22]. In recent years Homotopy perturbation method has been extensively introduced by numerous authors, and implemented to obtain exact and approximate analytical solutions to a wide range of both linear and nonlinear problems in science and engineering [4,6,15,16,20,22,23,30,32]. Watugala in 1993 [10] introduced a new integral transform and named it as Sumudu transform, used it in obtaining the solution of ordinary differential equations in control engineering problems. Asiru [28] implemented the Sumudu transform for solving integral equations of convolution type. Belgacem et al [18,19] presented the fundamental properties of Sumudu transform. Kilicman and Eltayeb [2,3,12] investigated various types of problems via Sumudu transform, including ordinary and partial differential equations. Gupta and Bhavna [37] used Sumudu transform in determining the solution of reaction-diffusion equation. Rana et al [29] applied He's homotopy perturbation method to compute Sumudu transform. Several authors [1,31,34-36] have discussed many fractional partial differential equations using Sumudu transform. The Homotopy perturbation Sumudu transform method [7,11,24,25,29,33] is applied to solve many problems, for example, nonlinear equations, heat and wave-like equations.

3 Solution of nonlinear fractional differential equations 2197 In this paper the authors implemented the Homotopy perturbation Sumudu transform method (HPSTM) to evaluate the exact analytical solution of nonlinear fractional partial differential equations. This work is organized as follows: In section 2 we provide some preliminaries. Section 3 introduces the concept of Homotopy perturbation method, while section 4 gives the Sumudu transform. The Homotopy perturbation Sumudu transform method (HPSTM) is analyzed in section 5. Numerical examples are provided in section 6. The conclusions are given in section Preliminaries Definition (2.1): The Caputo fractional derivative of order is defined by [14,26] of a function { Where is called the Caputo derivative operator. Definition (2.2): The Mittag-Leffler function with two parameters, is defined by [13,14,26,27] The following results are obtained directly from definition : [ ] } Note (2.3): The special type of Mittag-Leffler function ( ) is given by [13] ( ) ( ) By using (4), we have to drive

4 2198 Eltayeb A. Yousif and Sara H. M. Hamed ( ) ( ) where is a complementary error function. These two functions are used further in this paper. The derivation of formula (5) based on definition of Mittage-Leffler function and formula (4), we have ( ) ( ) Replacing with in the RHS of, we get ( ) ( ) ( ( ) ) ( ( ) ) Substituting ( ) ( ), then we get the result. 3. Homotopy Perturbation Method To illustrate the concept of Homotopy perturbation method, we consider the nonlinear differential equation: with the boundary conditions: Where is a linear operator, is nonlinear operator, is boundary operator, is the boundary of the domain and is a known analytic function. The He s homotopy perturbation technique [8-10] defines the Homotopy [ ] which satisfies: [ ] [ ] (9) or [ ]

5 Solution of nonlinear fractional differential equations 2199 Where and [ ] is an impending parameter, is an initial approximation which satisfies the boundary condition. The basic assumption is that the solution of equation (9) and equation (10) can be expressed as power series in as follows: The approximate solution of equation (7) is given by 4. Sumudu Transform Consider functions in the set A, that defined by: { [ )} where is a constant must be finite, need not simultaneously exist, and each may be finite. The Sumudu transform is defined by [10] ( ) Definition (4.1): The Sumudu transform of fractional order derivative, is defined by [25,34] [ ] [ ] [ ] 5. Analysis of the method In this section we need to illustrate the concept and construction of Homotopy perturbation Sumudu transforms method (HPSTM) for fractional equations, that by considering the general nonlinear nonhomogenous time-fractional partial differential equation with the initial conditions to by: }

6 2200 Eltayeb A. Yousif and Sara H. M. Hamed Applying Sumudu transform on both sides of (15) with respect to t, we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) Taking the inverse Sumudu transform to the above result, we have ( ( ( ) ( ))) Application of the Homotopy perturbation method to (16), yields ( ) ( ( ( ( ) ( )))) Let: ( ( ( ) ( ))) ( )

7 Solution of nonlinear fractional differential equations 2201 ( ) ( ) Substituting (18) into equation (17), we get ( ( ( ) ( ( )))) ( ( ( ))) ( ( ( ))) ( ( ( ))) The solution of equation (15) is given by 6. Numerical examples Example (1): Consider the nonlinear nonhomogenous time-fractional invicid Burgers equation } Solution: By applying (HPSTM) to equation (21), then from (19), we have

8 2202 Eltayeb A. Yousif and Sara H. M. Hamed ( ( ( ( )))) ( ( ( ( ( )))) ) ( ( ( ( )))) ( ( ( ))) ( ( ( ))) ( ( )) ( ) ( ( ( ))) ( ( ))

9 Solution of nonlinear fractional differential equations 2203 ( ) ( ) Remark (1): As special case if we take then from (4) and (5), we have ( ). Remark (2 ): If then,. This agrees with the solution obtained by Wazwaz [5]. Example (2): Consider the following nonlinear time-fractional equation Solution: By applying (19) in (22), we get } ( ( ( ( ))))

10 2204 Eltayeb A. Yousif and Sara H. M. Hamed ( ( ( ( )))) ( ( ( ))) ( ( ( ))) ( ( )) ( ( ( ))) ( ( )) ( ) ( ) Remark (3): If then:. Example (3): Consider the time-fractional fifth order KdV equation

11 Solution of nonlinear fractional differential equations 2205 } Solution: Application of (19) into (23), yields ( ( ( ( )))) ( ( ( ( )))) ( ( ( ( )))) ( ( ( ))) ( ( ( ( )))) ( ( ( ( ) ( )))) ( ( ( ( ))))

12 2206 Eltayeb A. Yousif and Sara H. M. Hamed ( ( ( ( ) ( ) ( )))) Remark (4): If, then from, we have: ( ) ( ) Remark (5): If. then we get the solution of the classical equation as 7. Conclusion In the present paper, we applied the Homotopy perturbation Sumudu transform method (HPSTM) for solving fractional nonlinear partial differential equations. The time derivatives are considered in Caputo sense. Solutions are determined in a compact form in terms of Mittag-Leffler functions. The terms are obtained in a simplified way and straightforward. The method was tested on three different problems. This method is powerful, reliable and effective, easy to implement. Thus, this technique can be applied to solve many nonlinear problems in applied science. References [1] A. Kilicman, V. G. Gupta and B. Sharma, On the solution of fractional Maxwell equation by Sumudu transform, Journal of Mathematics Research, 2(4) 2010,

13 Solution of nonlinear fractional differential equations 2207 [2] A. Kilcman and H. Eltayeb. A note on classification of hyperbolic and elliptic equations with polynomial coefficients, Applied Mathematics Letters, 21(11) (2008), [3] A. Kilcman and H. Eltayeb, A note on integral transforms and partial differential equations, Applied Mathematical Sciences, 4(3) 2010, [4] A. M. Siddiqui, R. Mahmood, and Q. K Ghori, Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Physics Letters A, 352 (2006), [5] A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press Beijing and Springer-Verlag Heidelberg, [6] D. D. Ganji and M. Rafei, Solitary wave solutions for a generalized Hirota Satsuma coupled KdV equation by Homotopy perturbation method, Physics Letters A, 356 (2006), [7] D. Kumar, J. Singh, and Sushila, Sumudu Homotopy perturbation technique, Global Journal of Science Frontier Research, 11(6), Version 1.0, (Sept 2011). [8] F. M. Belgacem, A. A. Karaballi, and S. L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Mathematical Problems in Engineering, 3 (2003), [9] F. M. Belgacem and A. A. Karaballi, Sumudu transform fundamental properties investigations and applications, Journal of Applied Mathematics and Stochastic Analysis, 2006 (2006), Article ID 91083, [10] G. K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, International Journal of Mathematical Education in Science and Technology, 24(1) (1993), [11] H. Bulut, H. Baskonus, and S. Tulue, Homotopy perturbation Sumudu transform method for heat equations, Mathematics in Engineering, Science and Aerospace, 4(1) [12] H. Eltayeb and A. Kılıcman, On Some Applications of a new integral transform, International Journal of Mathematical Analysis, 4(3) (2010),

14 2208 Eltayeb A. Yousif and Sara H. M. Hamed [13] H. J. Haubold, A. M. Mathal, and R. K. Saxena, Mittage-Leffler functions and their applications, Journal of Applied Mathematics, 2011 (2011), Article ID , 51 pages. [14] I. Podlubny, Fractional Differential Equations, Academic Press, [15] J. Biazar, and H. Ghazvini, Exact solutions for nonlinear Schrödinger equations by He s Homotopy perturbation method, Physics Letters A, 366 (2007), [16] J. Biazar and H. Ghazvini, He s Homotopy perturbation method for solving system of Volterra integral equations of the second kind, Chaos, Solitons and Fractals, 39(3) (2009), [17] J. H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178 (1999), [18] J. H. He, Recent development of the Homotopy perturbation method, Topological Methods in Nonlinear Analysis, 31(2) (2008), [19] J. H. He, A modified perturbation technique depending upon an artificial parameter, Mechanical, 35 (2000), [20] J. H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A, 350 (2006), [21] J. H. He, New interpretation of Homotopy perturbation method, International Journal of Modern Physics B, 20 (2006), [22] J. H. He, Application of Homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (2005), [23] J. H. He, A coupling method of a Homotopy technique and a perturbation technique for nonlinear problems, International Journal of Nonlinear Mechanics, 35 (2000), [24] J. Singh, D. Kumar, and A. Kilicman, Application of Homotopy perturbation Sumudu transform method for solving heat and wave-like equations, Malaysian Journal of Mathematical Sciences, 7(1) (2013),

15 Solution of nonlinear fractional differential equations 2209 [25] J. Singh, D. Kumar, and A. Kilicman, Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform, Abstract and Applied Analysis, 2013 (2013), Article ID , 8 pages. [26] L. Debnath, Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, 54 (2003), [27] L. Debnath, D. Bhatta, Integral Transforms and their Applications, 2nd edition, Taylor & Francis Group, LLC, [28] M. A. Asiru, Sumudu transform and the solution of integral equations of convolution type, International Journal of Mathematical Education in Science and Technology, 32(6) (2001), [29] M.A. Rana, A.M. Siddiquib, Q.K. Ghoric, and R. Qamar, Application of He's Homotopy perturbation method to Sumudu transform, International Journal of Nonlinear Sciences and Numerical Simulation, 8(2) (2007), [30] Q. Wang, Homotopy perturbation method for fractional KdV Burgers equation, Chaos, Solitons and Fractals, 35 (2008), [31] R. Darzi, B. Mohammadzade, S. Mousavi, and R. Beheshti, Sumudu transform method for solving fractional differential equations and fractional diffusion-wave equation, Journal of Mathematics and Computer Science, 6 (2013) [32] S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A, 365 (2007), [33] S. Rathore, D. Kumer, J. Singh, and S. Gupta, Homotopy analysis Sumudu transform method for nonlinear equations, International Journal of Industrial Mathematics, 4(4) (2012), 13 pages. [34] V. B. Chaurasia and J. Singh, Application of Sumudu transform in fractional kinetic equations, General Mathematics Notes, 2(1) (2011),

16 2210 Eltayeb A. Yousif and Sara H. M. Hamed [35] V. G. Gupta, B. Sharma, and A. Kilicman, A note on fractional Sumudu transform method, Journal of Applied Mathematics, 2010 (2010), Article ID 15489, 9 pages. [36] V. G. Gupta, B. Sharma and A. Kilcman, A note on fractional Sumudu transform, Journal of Applied Mathematics, 2010 (2010), Article ID , 9 pages. [37] V. G. Gupta and B. Sharma, Application of Sumudu transform in reaction-diffusion systems and nonlinear waves, Applied Mathematical Sciences, 4(9) 2010, Received: February 11, 2014