Non-polynomial B-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers’ equations numerically

Abstract

Abstract This paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval [ 0 , L ] $[0, L]$ . The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

Keywords

• Liouville–Caputo fractional derivative
• Non-polynomial B-spline functions
• Fractional coupled Burgers’ equation
• Shifted Jacobi polynomial
• Jacobi–Gauss quadrature
• Von Neumann stability