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

A new solution method for nonlinear fractional integro-differential equations

Sertan Alkan 1,

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.
Keywords: Nonlinear fractional Fredholm integro-differential equation, Caputo derivative, sinc-collocation method, quadrature rule, mathematica..
Mathematics Subject Classification: Primary: 41A55, 41A30; Secondary: 65D15, 65D3.
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, Solitons & Fractals, 40 (2009), 521-529. doi: 10.1016/j.chaos.2007.08.001.  Google Scholar

[2]

R. L. Bagley and J. P. Torvik, On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 30 (1986), 133-155. doi: 10.1122/1.549887.  Google Scholar

[3]

M. El-Gamel and A. Zayed, Sinc-Galerkin method for solving nonlinear boundary-value problems, Comput. Math. Appl., 48 (2004), 1285-1298. doi: 10.1016/j.camwa.2004.10.021.  Google Scholar

[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), 1-9. Google Scholar

[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), 1127-1134. doi: 10.1016/j.camwa.2011.03.037.  Google Scholar

[6]

J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Englewood Cliffs, 1992. doi: 10.1137/1.9781611971637.  Google Scholar

[7]

F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Applied Mathematics Letters, 9 (1996), 23-28. doi: 10.1016/0893-9659(96)00089-4.  Google Scholar

[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), 930-941. doi: 10.1016/j.camwa.2008.01.023.  Google Scholar

[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), 1693-1700. doi: 10.9734/BJMCS/2014/8247.  Google Scholar

[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), 380-390. doi: 10.1007/s00033-006-5124-5.  Google Scholar

[11]

S. Momani and M. A. Noor, Numerical methods for fourth-order fractional integro-differential equations, Applied Mathematics and Computation, 182 (2006), 754-760. doi: 10.1016/j.amc.2006.04.041.  Google Scholar

[12]

S. Momani and R. Qaralleh, An efficient method for solving systems of fractional integro-differentialequations, Computers & Mathematics with Applications, 52 (2006), 459-470. doi: 10.1016/j.camwa.2006.02.011.  Google Scholar

[13]

Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations, Computers & Mathematics with Applications, 61 (2011), 2330-2341. doi: 10.1016/j.camwa.2010.10.004.  Google Scholar

[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), 883-891. doi: 10.1016/j.cam.2010.01.053.  Google Scholar

[15]

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.  Google Scholar

[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), 315-330. doi: 10.1007/s40314-013-0021-y.  Google Scholar

[17]

F. Riewe, Mechanics with fractional derivatives, Physical Review E, 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581.  Google Scholar

[18]

J. Sabatier, O. P. Agrawal and J. T. Machado, Advances in Fractional Calculus, Springer, Dordrecht, 2007. doi: 10.1007/978-1-4020-6042-7.  Google Scholar

[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), 633-638. Google Scholar

[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), 281-294.  Google Scholar

[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), 1-10. Google Scholar

[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), 591-601. doi: 10.1016/j.camwa.2010.05.005.  Google Scholar

show all references

References:
[1]

A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos, Solitons & Fractals, 40 (2009), 521-529. doi: 10.1016/j.chaos.2007.08.001.  Google Scholar

[2]

R. L. Bagley and J. P. Torvik, On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 30 (1986), 133-155. doi: 10.1122/1.549887.  Google Scholar

[3]

M. El-Gamel and A. Zayed, Sinc-Galerkin method for solving nonlinear boundary-value problems, Comput. Math. Appl., 48 (2004), 1285-1298. doi: 10.1016/j.camwa.2004.10.021.  Google Scholar

[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), 1-9. Google Scholar

[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), 1127-1134. doi: 10.1016/j.camwa.2011.03.037.  Google Scholar

[6]

J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Englewood Cliffs, 1992. doi: 10.1137/1.9781611971637.  Google Scholar

[7]

F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Applied Mathematics Letters, 9 (1996), 23-28. doi: 10.1016/0893-9659(96)00089-4.  Google Scholar

[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), 930-941. doi: 10.1016/j.camwa.2008.01.023.  Google Scholar

[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), 1693-1700. doi: 10.9734/BJMCS/2014/8247.  Google Scholar

[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), 380-390. doi: 10.1007/s00033-006-5124-5.  Google Scholar

[11]

S. Momani and M. A. Noor, Numerical methods for fourth-order fractional integro-differential equations, Applied Mathematics and Computation, 182 (2006), 754-760. doi: 10.1016/j.amc.2006.04.041.  Google Scholar

[12]

S. Momani and R. Qaralleh, An efficient method for solving systems of fractional integro-differentialequations, Computers & Mathematics with Applications, 52 (2006), 459-470. doi: 10.1016/j.camwa.2006.02.011.  Google Scholar

[13]

Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations, Computers & Mathematics with Applications, 61 (2011), 2330-2341. doi: 10.1016/j.camwa.2010.10.004.  Google Scholar

[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), 883-891. doi: 10.1016/j.cam.2010.01.053.  Google Scholar

[15]

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.  Google Scholar

[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), 315-330. doi: 10.1007/s40314-013-0021-y.  Google Scholar

[17]

F. Riewe, Mechanics with fractional derivatives, Physical Review E, 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581.  Google Scholar

[18]

J. Sabatier, O. P. Agrawal and J. T. Machado, Advances in Fractional Calculus, Springer, Dordrecht, 2007. doi: 10.1007/978-1-4020-6042-7.  Google Scholar

[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), 633-638. Google Scholar

[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), 281-294.  Google Scholar

[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), 1-10. Google Scholar

[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), 591-601. doi: 10.1016/j.camwa.2010.05.005.  Google Scholar

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