# American Institute of Mathematical Sciences

December  2015, 8(6): 1065-1077. doi: 10.3934/dcdss.2015.8.1065

## A new solution method for nonlinear fractional integro-differential equations

 1 Department of Mathematics, Mustafa Kemal University, 31000, Hatay, Turkey

Received  April 2015 Revised  August 2015 Published  December 2015

The aim of this paper is to obtain approximate solution of a class of nonlinear fractional Fredholm integro-differential equations by means of sinc-collocation method which is not used for solving them in the literature before. The fractional derivatives are defined in the Caputo sense often used in fractional calculus. The important feature of the present study is that obtained results are stated as two new theorems. The introduced method is tested on some nonlinear problems and it seems that the method is a very efficient and powerful tool to obtain numerical solutions of nonlinear fractional integro-differential equations.
Citation: Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
##### References:
 [1] A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method,, Chaos, 40 (2009), 521. doi: 10.1016/j.chaos.2007.08.001. [2] R. L. Bagley and J. P. Torvik, On the fractional calculus model of viscoelastic behavior,, Journal of Rheology, 30 (1986), 133. doi: 10.1122/1.549887. [3] M. El-Gamel and A. Zayed, Sinc-Galerkin method for solving nonlinear boundary-value problems,, Comput. Math. Appl., 48 (2004), 1285. doi: 10.1016/j.camwa.2004.10.021. [4] E. Hesameddini and E. Asadolahifard, Solving Systems of Linear Volterra Integro-Differential Equations by Using Sinc-Collocation Method,, International Journal of Mathematical Engineering and Science, 2 (2013), 1. [5] L. Huang, X. F. Li, Y. Zhao and X. Y. Duan, Approximate solution of fractional integro-differential equations by Taylor expansion method,, Computers & Mathematics with Applications, 62 (2011), 1127. doi: 10.1016/j.camwa.2011.03.037. [6] J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Differential Equations,, SIAM, (1992). doi: 10.1137/1.9781611971637. [7] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation,, Applied Mathematics Letters, 9 (1996), 23. doi: 10.1016/0893-9659(96)00089-4. [8] A. Mohsen and M. El-Gamel, On the Galerkin and collocation methods for two-point boundary value problems using sinc bases,, Computers & Mathematics with Applications, 56 (2008), 930. doi: 10.1016/j.camwa.2008.01.023. [9] A. Mohsen and M. El-Gamel, Sinc-collocation algorithm for solving nonlinear fredholm integro-differential equations,, British Journal of Mathematics & Computer Science, 4 (2014), 1693. doi: 10.9734/BJMCS/2014/8247. [10] A. Mohsen and M. El-Gamel, A Sinc-Collocation method for the linear Fredholm integro-differential equations,, Zeitschrift für angewandte Mathematik und Physik, 58 (2007), 380. doi: 10.1007/s00033-006-5124-5. [11] S. Momani and M. A. Noor, Numerical methods for fourth-order fractional integro-differential equations,, Applied Mathematics and Computation, 182 (2006), 754. doi: 10.1016/j.amc.2006.04.041. [12] S. Momani and R. Qaralleh, An efficient method for solving systems of fractional integro-differentialequations,, Computers & Mathematics with Applications, 52 (2006), 459. doi: 10.1016/j.camwa.2006.02.011. [13] Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations,, Computers & Mathematics with Applications, 61 (2011), 2330. doi: 10.1016/j.camwa.2010.10.004. [14] D. Nazari and S. Shahmorad, Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions,, Journal of Computational and Applied Mathematics, 234 (2010), 883. doi: 10.1016/j.cam.2010.01.053. [15] I. Podlubny, Fractional Differential Equations,, Academic Press, (1999). [16] J. Rashidinia and M. Nabati, Sinc-Galerkin and Sinc-Collocation methods in the solution of nonlinear two-point boundary value problems,, Computational and Applied Mathematics, 32 (2013), 315. doi: 10.1007/s40314-013-0021-y. [17] F. Riewe, Mechanics with fractional derivatives,, Physical Review E, 55 (1997), 3581. doi: 10.1103/PhysRevE.55.3581. [18] J. Sabatier, O. P. Agrawal and J. T. Machado, Advances in Fractional Calculus,, Springer, (2007). doi: 10.1007/978-1-4020-6042-7. [19] R. K. Saedd and H. M. Sdeq, Solving a system of linear fredholm fractional integro-differential equations using homotopy perturbation method,, Australian Journal of Basic and Applied Sciences, 4 (2010), 633. [20] A. Secer, S. Alkan, M. A. Akinlar and M. Bayram, Sinc-Galerkin method for approximate solutions of fractional order boundary value problems,, Boundary Value Problems, 2013 (2013), 281. [21] M. Zarebnia and N. Zeinab, Solution of linear Volterra integro-differential equations via Sinc functions,, International Journal of Applied Mathematics and Computation, 2 (2009), 1. [22] M. Zarebnia and M. G. A. Abadi, Numerical solution of system of nonlinear second-order integro-differential equations,, Computers & Mathematics with Applications, 60 (2010), 591. doi: 10.1016/j.camwa.2010.05.005.

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##### References:
 [1] A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method,, Chaos, 40 (2009), 521. doi: 10.1016/j.chaos.2007.08.001. [2] R. L. Bagley and J. P. Torvik, On the fractional calculus model of viscoelastic behavior,, Journal of Rheology, 30 (1986), 133. doi: 10.1122/1.549887. [3] M. El-Gamel and A. Zayed, Sinc-Galerkin method for solving nonlinear boundary-value problems,, Comput. Math. Appl., 48 (2004), 1285. doi: 10.1016/j.camwa.2004.10.021. [4] E. Hesameddini and E. Asadolahifard, Solving Systems of Linear Volterra Integro-Differential Equations by Using Sinc-Collocation Method,, International Journal of Mathematical Engineering and Science, 2 (2013), 1. [5] L. Huang, X. F. Li, Y. Zhao and X. Y. Duan, Approximate solution of fractional integro-differential equations by Taylor expansion method,, Computers & Mathematics with Applications, 62 (2011), 1127. doi: 10.1016/j.camwa.2011.03.037. [6] J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Differential Equations,, SIAM, (1992). doi: 10.1137/1.9781611971637. [7] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation,, Applied Mathematics Letters, 9 (1996), 23. doi: 10.1016/0893-9659(96)00089-4. [8] A. Mohsen and M. El-Gamel, On the Galerkin and collocation methods for two-point boundary value problems using sinc bases,, Computers & Mathematics with Applications, 56 (2008), 930. doi: 10.1016/j.camwa.2008.01.023. [9] A. Mohsen and M. El-Gamel, Sinc-collocation algorithm for solving nonlinear fredholm integro-differential equations,, British Journal of Mathematics & Computer Science, 4 (2014), 1693. doi: 10.9734/BJMCS/2014/8247. [10] A. Mohsen and M. El-Gamel, A Sinc-Collocation method for the linear Fredholm integro-differential equations,, Zeitschrift für angewandte Mathematik und Physik, 58 (2007), 380. doi: 10.1007/s00033-006-5124-5. [11] S. Momani and M. A. Noor, Numerical methods for fourth-order fractional integro-differential equations,, Applied Mathematics and Computation, 182 (2006), 754. doi: 10.1016/j.amc.2006.04.041. [12] S. Momani and R. Qaralleh, An efficient method for solving systems of fractional integro-differentialequations,, Computers & Mathematics with Applications, 52 (2006), 459. doi: 10.1016/j.camwa.2006.02.011. [13] Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations,, Computers & Mathematics with Applications, 61 (2011), 2330. doi: 10.1016/j.camwa.2010.10.004. [14] D. Nazari and S. Shahmorad, Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions,, Journal of Computational and Applied Mathematics, 234 (2010), 883. doi: 10.1016/j.cam.2010.01.053. [15] I. Podlubny, Fractional Differential Equations,, Academic Press, (1999). [16] J. Rashidinia and M. Nabati, Sinc-Galerkin and Sinc-Collocation methods in the solution of nonlinear two-point boundary value problems,, Computational and Applied Mathematics, 32 (2013), 315. doi: 10.1007/s40314-013-0021-y. [17] F. Riewe, Mechanics with fractional derivatives,, Physical Review E, 55 (1997), 3581. doi: 10.1103/PhysRevE.55.3581. [18] J. Sabatier, O. P. Agrawal and J. T. Machado, Advances in Fractional Calculus,, Springer, (2007). doi: 10.1007/978-1-4020-6042-7. [19] R. K. Saedd and H. M. Sdeq, Solving a system of linear fredholm fractional integro-differential equations using homotopy perturbation method,, Australian Journal of Basic and Applied Sciences, 4 (2010), 633. [20] A. Secer, S. Alkan, M. A. Akinlar and M. Bayram, Sinc-Galerkin method for approximate solutions of fractional order boundary value problems,, Boundary Value Problems, 2013 (2013), 281. [21] M. Zarebnia and N. Zeinab, Solution of linear Volterra integro-differential equations via Sinc functions,, International Journal of Applied Mathematics and Computation, 2 (2009), 1. [22] M. Zarebnia and M. G. A. Abadi, Numerical solution of system of nonlinear second-order integro-differential equations,, Computers & Mathematics with Applications, 60 (2010), 591. doi: 10.1016/j.camwa.2010.05.005.
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