An efficient Chebyshev wavelets method for solving a class of nonlinear fractional integrodifferential equations in a large interval is developed, and a new technique for computing nonlinear terms in such equations is proposed. Existence of a unique solution for such equations is proved. Convergence and error analysis of the proposed method are investigated. Moreover in order to show efficiency of the proposed method, the new approach is compared with some numerical methods. 1. Introduction Fractional integrodifferential equations (FIDEs) arise in modelling processes in applied sciences such as physics, engineering, and biology. The nonlinear fractional integro-differential equation (NFIDE) of the type where is Caputo fractional derivative, is a parameter describing the order of the fractional derivative, , , and are fixed constants, and is a nonlinear continuous function, arise in the mathematical modelling of various physical phenomena, such as heat conduction in materials with memory. Moreover, these equations are encountered in combined conduction, convection, and radiation problems [1–3]. Therefore in recent years, numerous works have been focusing on the solution of these problems. Some of these methods are Adomian decomposition method (ADM) [4], fractional differential transform method (FDTM) [5], and the collocation method [6]. Most of these methods have been utilized in linear problems, and a few number of works have considered nonlinear problems. In [7] Rawashdeh applied Legendre wavelet method for solving fractional Voltera integro-differential equations in the form Also in [8] Awawdeh et al. applied homotopy analysis method (HAM) for solution of (2). Mittal and Nigam [9] applied the ADM for (1) in the form In view of successful application of wavelets in approximation theory [10–17], we will use the Chebyshev wavelets for solving a generalized form of the previous described equations of the form with high nonlinearity in a large interval under the initial condition . Here, for simplicity, we assume that and are continuous functions on , is continuous on , and also and are analytic functions. The method is based on reducing the equation to a system of nonlinear algebraic equations by expanding the solution as Chebyshev wavelet bases with unknown coefficients. We note that, for , (4) is an ordinary integro-differential equation and the method can be easily applied to it. Also the method is fast and mathematically simple and guarantees the necessary accuracy for a small number of basic functions. Moreover in order to show the efficiency of the
References
[1]
M. Caputo, “Linear models of dissipation whose q is almost frequency independent-ii,” Geophysical Journal of the Royal Astronomical Society, vol. 13, pp. 529–539, 1967.
[2]
W. E. Olmstead and R. A. Handelsman, “Diffusion in a semi-infinite region with nonlinear surface dissipation,” SIAM Review, vol. 18, no. 2, pp. 275–291, 1976.
[3]
F. Mainardi, “Fractional calcauls: some basic problems in continument and statistical mechanics,” Fractals and Fractional Calulus in Continuum Mechanics, pp. 291–348, 1976.
[4]
S. Momani and R. Qaralleh, “An efficient method for solving systems of fractional integro-differential equations,” Computers and Mathematics with Applications, vol. 52, no. 3-4, pp. 459–470, 2006.
[5]
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, vol. 234, no. 3, pp. 883–891, 2010.
[6]
E. A. Rawashdeh, “Numerical solution of fractional integro-differential equations by collocation method,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 1–6, 2006.
[7]
E. A. Rawashdeh, “Legendre wavelets method for fractional integro-differential equations,” Applied Mathematical Sciences, vol. 5, no. 49-52, pp. 2467–2474, 2011.
[8]
F. Awawdeh, E. A. Rawashdeh, and H. M. Jaradat, “Analytic solution of fractional integro-differential equations,” Annals of the University of Craiova, Mathematics and Computer Science Series, vol. 38, no. 1, pp. 1–10, 2011.
[9]
R. C. Mittal and R. Nigam, “Solution of fractional integro-differential equations by adomian decomposition method,” Appl Math Comput, vol. 4, no. 2, pp. 87–94, 2008.
[10]
C. Cattani, “Fractional calculus and shannon wavelets,” Mathematical Problems in Engineering, vol. 2012, Article ID 502812, 25 pages, 2012.
[11]
C. Cattani, “Shannon wavelets for the solution of integro-differential equations,” Mathematical Problems in Engineering, vol. 2010, Article ID 408418, 22 pages, 2010.
[12]
C. Cattani and A. Kudreyko, “Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4164–4171, 2010.
[13]
C. Cattani, “Shannon wavelets theory,” Mathematical Problems in Engineering, vol. 2008, Article ID 164808, 24 pages, 2008.
[14]
M. H. Heydari, M. R. Hooshmandasl, F. M. M. Ghaini, and F. Fereidouni, “Two-dimensionallegendre wavelets forsolving fractional poisson equation with dirichlet boundary conditions,” Engineering Analysis with Boundary Elements, vol. 37, pp. 1331–1338, 2013.
[15]
M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and F. Mohammadi, “Wavelet collocation method for solving multiorder fractional differential equations,” Journal of Applied Mathematics, vol. 2012, Article ID 542401, 19 pages, 2012.
[16]
M. H. Heydari, M. R. Hooshmandasl, F. Mohammadi, and C. Cattani, “Wavelets method for solving systems of nonlinear singular fractional volterra integro-differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 1, pp. 37–48, 2014.
[17]
M. R. Hooshmandasl, M. H. Heydari, and F. M. M. Ghaini, “Numerical solution of the one dimensional heat equation by using chebyshev wavelets method,” Applied & Computational Mathematics, vol. 1, no. 6, pp. 1–7, 2012.
[18]
J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.
[19]
J. He, “A new approach to nonlinear partial differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 4, pp. 230–235, 1997.
[20]
S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons and Fractals, vol. 31, no. 5, pp. 1248–1255, 2007.
[21]
Z. Odibat and S. Momani, “Numerical solution of Fokker-Planck equation with space- and time-fractional derivatives,” Physics Letters A, vol. 369, no. 5-6, pp. 349–358, 2007.
[22]
V. Daftardar-Gejji and H. Jafari, “An iterative method for solving nonlinear functional equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 753–763, 2006.
[23]
S. Bhalekar and V. Daftardar-Gejji, “New iterative method: application to partial differential equations,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 778–783, 2008.
[24]
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
[25]
Y. LI, “Solving a nonlinear fractional differential equation using Chebyshev wavelets,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2284–2292, 2010.
[26]
A. Kilicman and Z. A. A. Al Zhour, “Kronecker operational matrices for fractional calculus and some applications,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 250–265, 2007.
[27]
S. Sohrabi, “Comparison Chebyshev wavelets method with BPFs method for solving Abel's integral equation,” Ain Shams Engineering Journal, vol. 2, no. 3-4, pp. 249–254, 2011.
[28]
H. Saeedi, N. Mollahasani, M. Moghadam, and G. Chuev, “An operational haar wavelet method for solving fractional volterra integral equations,” International Journal of Applied Mathematics and Computer Science, vol. 21, no. 3, pp. 535–547, 2011.
