Numerical treatment for the solution of fractional fifth-order Sawada–Kotera equation using second kind Chebyshev wavelet method

Abstract In this paper, a new method based on the Chebyshev wavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is proposed to solve time-fractional fifth-order Sawada–Kotera (SK) equation. Two-dimensional Chebyshev wavelet method is applied to compute the numerical solution of nonlinear time-fractional Sawada–Kotera equation. The approximate solutions of nonlinear time fractional Sawada–Kotera equation thus obtained by Chebyshev wavelet method are compared with the exact solutions as well as homotopy analysis method (HAM). The present scheme is very simple, effective and convenient for obtaining numerical solution of fractional Sawada–Kotera equation.

[1]  M. Dehghan,et al.  Solving nonlinear fractional partial differential equations using the homotopy analysis method , 2010 .

[2]  A. Wazwaz.  Partial Differential Equations and Solitary Waves Theory , 2009 .

[3]  L. Debnath.  Nonlinear Partial Differential Equations for Scientists and Engineers , 1997 .

[4]  A. K. Gupta,et al.  On the Solutions of Fractional Burgers-Fisher and Generalized Fisher's Equations Using Two Reliable Methods , 2014, Int. J. Math. Math. Sci..

[5]  Qibin Fan,et al.  Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet , 2012 .

[6]  B. Zheng.  Exact Solutions for Some Fractional Partial Differential Equations by the Method , 2013 .

[7]  Santanu Saha Ray,et al.  Two-dimensional Legendre wavelet method for the numerical solutions of fuzzy integro-differential equations , 2015, J. Intell. Fuzzy Syst..

[8]  Fawang Liu,et al.  Numerical methods and analysis for a class of fractional advection-dispersion models , 2012, Comput. Math. Appl..

[9]  Santanu Saha Ray,et al.  Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system , 2015, Appl. Math. Comput..

[10]  O. Iyiola.  A NUMERICAL STUDY OF ITO EQUATION AND SAWADA-KOTERA EQUATION BOTH OF TIME-FRACTIONAL TYPE , 2013 .

[11]  S. Mohyud-Din,et al.  Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation , 2013 .

[12]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[13]  A. K. Gupta,et al.  Comparison between homotopy perturbation method and optimal homotopy asymptotic method for the soliton solutions of Boussinesq–Burger equations , 2014 .

[14]  A. Gupta,et al.  Traveling wave solution of fractional KdV-Burger-Kuramoto equation describing nonlinear physical phenomena , 2014 .

[15]  Mohammad Mehdi Rashidi,et al.  The Homotopy Analysis Method for Solving the Sawada-Kotera and Lax's Fifth-Order KdV Equations , 2008 .

[16]  Santanu Saha Ray,et al.  On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation , 2012, Appl. Math. Comput..

[17]  A. Gupta,et al.  A two-dimensional Haar wavelet approach for the numerical simulations of time and space fractional Fokker–Planck equations in modelling of anomalous diffusion systems , 2014, Journal of Mathematical Chemistry.

[18]  S. Das.  Functional Fractional Calculus , 2011 .

[19]  O. Agrawal.  A General Formulation and Solution Scheme for Fractional Optimal Control Problems , 2004 .

[20]  Yanxin Wang,et al.  The second kind Chebyshev wavelet method for solving fractional differential equations , 2012, Appl. Math. Comput..

[21]  A. K. Gupta,et al.  Numerical Solution of Fractional Partial Differential Equation of Parabolic Type With Dirichlet Boundary Conditions Using Two-Dimensional Legendre Wavelets Method , 2016 .

[22]  Mehdi Dehghan,et al.  A new operational matrix for solving fractional-order differential equations , 2010, Comput. Math. Appl..

[23]  K. Burrage,et al.  Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain , 2012 .

[24]  I. Podlubny.  Fractional differential equations , 1998 .

[25]  M. Dehghan,et al.  THE SOLUTION OF THE LINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING THE HOMOTOPY ANALYSIS METHOD , 2010 .

[26]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .