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

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.

Numerical treatments of the nonlinear coupled time‐fractional Schrödinger equations

“…Lemma (Hadhoud et al 28 ) Let and 0,1,…, then ··· , as .…”

“…Lemma 1. (Hadhoud et al 28 ) Let 0 < 𝛼 < 1 and 𝜑 q = (q + 1) 1−𝛼 − q 1−𝛼 , q = 0,1, … , then 1 = 𝜑 𝛼 0 > 𝜑 𝛼 1 > •••> 𝜑 𝛼 q → 0, as q → ∞.…”

“…In this section, we use the Von Neumann method to analyze the stability of the scheme (28) to (31). First, we linearize the nonlinear terms R, S, W, and Z as local constants d 1 , d 2 , d 3 , and d 4 , respectively, as is done in the Von Neumann method.…”

Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method

“…Fractional calculus is a powerful and versatile tool for modeling a wide range of scientific phenomena, including image processing, earthquake engineering, biomedical engineering, computational fluid mechanics, and physics. In recent decades, the conventional Schrödinger equation has been generalized to a fractional order partial differential equation that takes into consideration the Riemann-Liouville, Caputo, and Riesz derivatives instead of the classical Laplacian [4][5][6][7]. The Caputo fractional derivative is considered here because it allows traditional initial and boundary conditions to be included in the formulation of the problem [8].…”

Numerical solution of system of Fredholm-Volterra integro-differential equations using Legendre polynomials

“…Nemati [34] presented a numerical solution of Volterra-Fredholm integral equations using Legendre collocation method. Hadhoud et al [35] proposed the non-polynomial B-spline method and the shifted Jacobi spectral collocation method to solve time-fractional nonlinear coupled Burgers' equations numerically. Some dynamical models involving fractional-order derivatives with the Mittag-Leffler type kernels and their applications based upon the Legendre spectral collocation method have been investigated by Srivastava et al [36].…”