Mojahedfar, M., Tari Marzabad, A. (2017). Solving two-dimensional fractional integro-differential equations by Legendre wavelets. Bulletin of the Iranian Mathematical Society, 43(7), 2419-2435.

M. Mojahedfar; A. Tari Marzabad. "Solving two-dimensional fractional integro-differential equations by Legendre wavelets". Bulletin of the Iranian Mathematical Society, 43, 7, 2017, 2419-2435.

Mojahedfar, M., Tari Marzabad, A. (2017). 'Solving two-dimensional fractional integro-differential equations by Legendre wavelets', Bulletin of the Iranian Mathematical Society, 43(7), pp. 2419-2435.

Mojahedfar, M., Tari Marzabad, A. Solving two-dimensional fractional integro-differential equations by Legendre wavelets. Bulletin of the Iranian Mathematical Society, 2017; 43(7): 2419-2435.

Solving two-dimensional fractional integro-differential equations by Legendre wavelets

^{}Department of Mathematics, Shahed University, Tehran, Iran.

Receive Date: 09 December 2015,
Revise Date: 10 May 2017,
Accept Date: 11 May 2017

Abstract

In this paper, we introduce the two-dimensional Legendre wavelets (2D-LWs), and develop them for solving a class of two-dimensional integro-differential equations (2D-IDEs) of fractional order. We also investigate convergence of the method. Finally, we give some illustrative examples to demonstrate the validity and efficiency of the method.

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas and Mathematical Tables, Dover, NewYork, 1965.

RT. Baillie, Long memory processes and fractional integration in econometrics, J. Econometrics 73 (1996) 5--59.

E. Banifatemi, M. Razzaghi and S. Yousefi, Two-dimensional Legendre wavelets method for the Mixed Volterra-Fredholm integral equations, J. Vib. Control 13 (2007), no. 11, 1667--1675.

M.S. EL-Azab, I.L. EL-Kalla and S.A. EL-Morsy, Solution of KDVB equation via Block-Pulse functions method, Electron. J. Math. Anal. Appl. 1 (2013), no. 2, 361-- 367.

R. Goreno and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: Fractals and Fractional Calculus in Cintinuum Mechanics, pp. 223--276, Springer Verlag, Wien-New York, 1997.

R. Hilfer, Application of Fractional Calculus in Phyisics, World Scientific, Singapore, 2000.

S.A. Hosseini, S. Shahmorad and A.Tari, Existence of an Lp-solution for two dimensional integral equations of the Hammerstein type, Bull. Iranian Math. Soc. 40 (2014), no. 4, 851--862.

A.A. Kilbas, H.M. Srivastava and J.J. Trujiilo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

A. Kilicman and ZAA. Al Zhour, Kronecker operational matrices for fractional calculus and some applications, Appl. Math. Comput. 187 (2007) 250--265.

Z. Menga, L. Wanga, H. Lib and W. Zhangb, Legendre wavelets method for solving fractional integro-differential equations, Int. J. Comput. Math. 92 (2015), no. 6, 1275--1291.

M.A. Mohammed and F.S. Fadhel, Solution of two-dimensional fractional order volterra integro-differential equations, J. Al-Nahrain Univ. 12 (2009), no. 4, 185--189.

S. Momani, An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simulation 70 (2005) 110--118.

S. Momani and M. Noor, Numerical methods for fourth order fractional integro differential equations, Appl. Math. Comput. 182 (2006) 754--760.

S. Momani and Z. Odibat, Analytical approach to linear fractional partial differential equations arising in uid mechanics, Phys. Lett. A 355 (2006) 271--279.

Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth order fractional integro-differential equations, Comput. Math. Appl. 61 (2011) 2330--2341.

A. Neamaty, B. Agheli and R. Darzi, Solving fractional partial differential equation by using wavelet operational method, J. Math. Comput. Sci. 7 (2013) 230--240.

K. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw. 41 (2010) 9--17.

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, 1999.

M.Y. Rahimi, S. Shahmorad, F. Talati and A.Tari, An operational method for the numerical solution of two dimensional linear Fredholm integral equations with an error estimation, Bull. Iranian Math. Soc. 36 (2010), no. 2, 119--132.

S.S. Ray, Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1295--1306.

M. Tavassoli Kajani and A. Hadi Vencheh, Solving linear integro-differential equation with Legendre wavelets, Int. J. Comput. Math. 81 (2004), no. 6, 719--726

V. Turut and N. Guzel, On solving partial differential equations of fractional order by using the variational iteration method and multivariate pade approximations, Eur. J. Pure App. Math. 6 (2013), no. 2, 147--171.

Y. Xu and V. Suat Erturk, A finite difference technique for solving variable-order fractional integro-differential equations, Bull. Iranian Math. Soc. 40 (2014), no. 3, 699--712.

L. Zhu and Q. Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelets, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 2333--2341.