Application of Legendre wavelet method coupled with the Gauss quadrature rule for solving fractional integro-differential equations

Document Type : Original Paper


Department of Mathematics, Higher Education Complex of Saravan, Saravan, Iran


In this work, we propose a novel technique for solving the nonlinear fractional Volterra-Fredholm integro-differential equations (FVFIDEs). This method approximates the unknown function with the Legendre wavelets. To do this, the Legendre wavelets are used in conjunction with the quadrature rule for converting the problem into a linear or nonlinear system of algebraic equations which can be easily solved by applying the mathematical programming techniques. Furthermore, the existence and uniqueness of the solution are proved by preparing some theorems and lemmas. Also, the error estimate and convergence analysis of the method will be shown. Moreover, some examples are presented and their results are compared to the results of Chebyshev wavelet, modification of hat functions, NystrÖm and Newton-Kantorovitch methods to show the capability and accuracy of this scheme.


Main Subjects

[1] Abbasbandy S, Hashemi M. and Hashim I., On convergence of homotopy analysis method and its application to fractional integro-differential equations, Quaest. Math. 36 (2013) 93–105.
[2] Alkan S. and Hatipoglu V. F., Approximate solutions of volterra-fredholm integro-differential equations of fractional order, Tbil. Math. J. 10 (2017) 1–13.
[3] Amin R, Shah K, Asif M, Khan I. and Ullah F., An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet, J. Comput. Appl. Math. 381 (2021) 113–128.
[4] Bhrawy A, Zaky M. and Van Gorder R. A., A space-time legendre spectral tau method for the two-sided space-time caputo fractional diffusion-wave equation, Numer. Algorithms. 71 (2016) 151–180.
[5] Erfanian M, Gachpazan M. and Beiglo H., A new sequential approach for solving the integrodifferential equation via haar wavelet bases, Comput. Math. & Math. Phys. 57 (2017) 297–305.
[6] Guner O. and Bekir A., Exp-function method for nonlinear fractional differential equations, Nonlinear. Sci. Lett. A. 8 (2017) 41–49.
[7] Hamoud A. and Ghadle K., The reliable modified of laplace adomian decomposition method to solve nonlinear interval volterra-fredholm integral equations, Korean. J. Math. 25 (2017) 323–334.
[8] Heris J. M., Solving the integro-differential equations using the modified laplace adomian decomposition method, J. Math. Ext. 6 (2012) 1–15.
[9] Hesameddini E, Rahimi A. and Asadollahifard E., On the convergence of a new reliable algorithm for solving multi-order fractional differential equations, Commun. Nonlinear. Sci. Numer. Simul. 34 (2016) 154–164.
[10] Hesameddini E. and Riahi M., Bernoulli galerkin matrix method and its convergence analysis for solving system of volterra-fredholm integro-differential equations, Iran. J. Sci. Technol. A. 43 (2018) 1203–1214.
[11] Hesameddini E, Riahi M. and Latifzadeh H., A coupling method of homotopy technique and laplace transform for nonlinear fractional differential equations, Int. J. Adv. Appl. Sci. 1 (2012) 159-170.
[12] Hesameddini E. and Shahbazi M., Hybrid bernstein block-pulse functions for solving system of fractional integro-differential equations, Int. J. Comput. Math. 95 (2018) 2287–2307.
[13] He S, Sun K. and Wang H., Dynamics of the fractional-order lorenz system based on adomian decomposition method and its DSP implementation, IEEE/CAA Journal of Automatica Sinica (2016) 1–6.
[14] Jahanshahi M., numerical solution of nonlinear fractional volterra-fredholm integro-differential equations with mixed boundary conditions, Int. J. Ind. Math. 7 (2015) 63–69.
[15] Liu Z, Cheng A. and Li X., A second-order finite difference scheme for quasi-linear time fractional parabolic equation based on new fractional derivative, Int. J. Comput. Math. 95 (2017) 396–411.
[16] Mahdy A. M. and Mohamed E. M., Numerical studies for solving system of linear fractional integrodifferential equations by using least squares method and shifted chebyshev polynomials, J. Abstr. Comput. Math. 1 (2016) 24–32.
[17] Modanli M. and Akgül A., On Solutions of Fractional order Telegraph partial differential equation by Crank-Nicholson finite difference method, Appl. Math. Non. Sci. 31 (2020) 163–170.
[18] Mohyud-Din S. T, Khan H, Arif M. and Rafiq M., Chebyshev wavelet method to nonlinear fractional volterra-fredholm integro-differential equations with mixed boundary conditions, Adv. Mech. Eng. 9 (2017) 1–8.
[19] Nazari D. and Shahmorad S., Application of the fractional differential transform method to fractionalorder integro-differential equations with nonlocal boundary conditions, J. Comput. Appl. Math. 234 (2010) 883–891.
[20] Nemati S. and Lima P. M., Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions, Appl. Math. Comput, 327 (2018) 79–92.
[21] Ordokhani Y. and Rahimi N., Numerical solution of fractional volterra integro-differential equations via the rationalized haar functions, J. Sci. Kharazmi. Univ. 14 (2014) 211-224.
[22] Pirim N. A. and Ayaz F., A new technique for solving fractional order systems: Hermite collocation method, Appl. Math. 7 (2016) 2307.
[23] Singh B. K., Homotopy perturbation new integral transform method for numeric study of space-and time-fractional (n+ 1)-dimensional heat-and wave-like equations, Waves, Wavelets and Fractals, 4 (2018) 19–36.
[24] Sun H, Zhao X. and Sun Z. Z., The temporal second order difference schemes based on the interpolation approximation for the time multi-term fractional wave equation, J. Sci. Comput. 78 (2019) 467–498.
[25] Sweilam N, Nagy A, Youssef I. K. and Mokhtar M. M., New spectral second kind chebyshev wavelets scheme for solving systems of integro-differential equations, Int. J. Appl. Comput. 3 (2017) 333–345.
[26] Wang Y. and Zhu L., Solving nonlinear volterra integro-differential equations of fractional order by using euler wavelet method, Adv. Differ. Equ. 2017 (2017) 1–16.
[27] Yang X, Zhang H. and Tang Q., A spline collocation method for a fractional mobile–immobile equation with variable coefficients, Comput. Appl. Math. 39 (2020) 1–20.
[28] Yin X. B, Kumar S. and Kumar D., A modified homotopy analysis method for solution of fractional wave equations, Adv. Mech. Eng. 7 (2015) 1–8.
[29] Zhu L. and Fan Q., Solving fractional nonlinear fredholm integro-differential equations by the second kind chebyshev wavelet, Commun. Nonlinear. Sci. Numer. Simul. 17 (2012) 2333–2341.
Volume 11, Issue 3
September 2021
Pages 463-480
  • Receive Date: 10 March 2021
  • Revise Date: 19 June 2021
  • Accept Date: 23 July 2021
  • First Publish Date: 13 August 2021