2D-fractional Muntz–Legendre polynomials for solving the fractional partial differential equations

Document Type : Research Article


Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.


We present a numerical method for solving linear and nonlinear fractional partial differential equations (FPDEs) with variable coefficients. The main aim of the proposed method is to introduce an orthogonal basis of twodimensional fractional Muntz–Legendre polynomials. By using these polynomials, we approximate the unknown functions. Furthermore, an operational matrix of fractional derivative in the Caputo sense is provided for computing the fractional derivatives. The proposed approximation together with the Tau method reduces the solution of the FPDEs to the solution of a system of algebraic equations. Finally, to show the validity and accuracy of the presented method, we give some numerical examples.


1. Abbasbandy, S. Kazem, S. Alhuthali, M.S. and Alsulami, H.H. Application of the operational matrix of fractional-order Legendre functions for solving the time-fractional convection-diffusion equation. Appl. Math. Comput. 266 (2015), 31–40.
2. Ahmadi Darani, M. and Nasiri, M. A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations. Computational Methods for Differential Equations, 1(2) (2013), 96–107.
3. Chen, Y. Sun, Y. and Liu, L. Numerical solution of fractional partial differential equations with variable coefficients using generalized fractional order Legendre functions. Appl. Math. Comput. 244 (2014), 847–858.
4. Esmaeili, S. Shamsi, M. and Luchko, Y. Numerical solution of fractional differential equations with a collocation method based on Mntz polynomials. Comput. Math. Appl. 62(3) (2011), 918–929.
5. Hilfer, R. Applications of Fractional calculus in Physics, World Scientific Publishing Company, Singapore, 2000.
6. Kazem, S. Abbasbandy, S. and Kumar, S. Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 37 (2013) (7), 5498–5510.
7. Kumar, S. and Rashidi, M.M. New analytical method for gas dynamics equation arising in shock fronts, Comput. Phys. Commun. 185 (7) (2014) 1947–1954.
8. Liu, N. and Lin, E.B. Legendre wavelet method for numerical solutions of partial differential equations. Numer. Methods Partial Differential Equations, 26(1) (2010), 81–94.
9. Nagy, A.M. Numerical solution of time fractional nonlinear Klein–Gordon equation using Sinc-Chebyshev collocation method, Appl. Math. Comput. , 310 (2017) 139–148.
10. Odibat, Z. and Momani, S. An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. inform. 26 (2008) 15–27.
11. Podlubny, I. Fractional Differential Equations, Academic Press, San Diego, 1999.
12. Podlubny, I. and Kacenak, M. MATLAB implementation of the Mittag Leffler function, 2005–2009. http://www.mathworks.com.
13. Saadatmandi, A. and Dehghan, M. A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 59(3) (2010), 1326–1336.
14. Saadatmandi, A. and Dehghan, M. A tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 62(3) (2011), 1135–1142.
15. Shen, J. and Tang, T. High order numerical methods and algorithms. Abstract and Applied Analysis, Chinese Science Press, 2005.
16. Szeg, G. Orthogonal Polynomials, 4th edn (Providence, RI: American Mathematical Society), 1975.
17. Tamsir, M. and Srivastava, V.K. Revisiting the analytical solution of gas dynamics equation, Alex. Eng. J. 55 (2016), 867–874.
18. Yin, F. Song, J. Wu Y. and Zhang, L. Numerical solution of the fractional partial differential equations by the two-dimensional fractional-order Legendre functions, Abstr. Appl. Anal. 2013, Art. ID 562140, 13 pp.