Solving time fractional Burgers’ and Fisher’s equations using cubic B-spline approximation method

Abstract

This article presents a numerical algorithm for solving time fractional Burgers' and Fisher's equations using cubic B-spline finite element method. The L1 formula with Caputo derivative is used to discretized the time fractional derivative, whereas the Crank-Nicolson scheme based on cubic B-spline functions is used to interpolate the solution curve along the spatial grid. The numerical scheme has been implemented on three test problems. The obtained results indicate that the proposed method is a good option for solving nonlinear fractional Burgers' and Fisher's equations. The error norms L 2 and L ∞ have been calculated to validate the efficiency and accuracy of the presented algorithm.

Numerical Simulation of Time Fractional BBM-Burger Equation Using Cubic B-Spline Functions

“…Lemma 3. The coefficients ℵ ℘ = ð℘+1Þ 1−α − ℘ 1−α and ϖ = ðt n+1 − sÞ in Equation ( 6) satisfy the following properties [16]:…”

“…where Ψ j ðt n+1 Þ′s are unknowns to be determined and Λ j ðxÞ are spline function basis of third degree defined as follows [16]:…”

“…So, several investigations have been done in recent years to determine the numerical solution of water wave model (see, for example, [14,15] and the references therein). Researchers introduced many different methods to develop an approximate analytical solution for the fractional differential equation and systems, such as the cubic B-spline method (CBS) by [16,17] and homotopy analysis method (HAM) by [18], unified method (UM) [19], Runge-Kutta method [20], Hermite wavelet technique together with Newton Raphson iteration method [21], and Lie symmetry method [22]. The time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation was proposed to discuss the dynamic physical system.…”

An accurate numerical method and its analysis for time-fractional Fisher’s equation

“…Khader and Saad [14] introduced a numerical scheme for solving the space-fractional Fisher's equation using spectral collocation method which is based upon Chebyshev approximations. Majeed et al [15] focused on developing a numerical technique based on cubic B-spline (CS) basis functions for TFF -and Burgers' equations. This method uses the L1 formula to approximate the Caputo fractional derivative and the third degree basis spline functions based on Crank-Nicolson scheme for the space derivatives.…”

Effective numerical technique applied for Burgers' equation of (1 + 1)‐, (2 + 1)‐dimensional, and coupled forms

“…Moreover, miscellaneous numerical techniques have been employed either for Burgers' equation or other engineering applications such as boundary element techniques for cavitation of hydrofoils, 36,37 step cubic polynomial, 38 technique of modified diffusion coefficient for studying convection diffusion equation, 39 and differential quadrature for functionally graded nanobeams 40,41 . Majeed et al, 42 presented a technique for solving time fractional Burgers' and Fisher's equations via cubic B‐spline approximation. The results indicated the accuracy of the presented scheme.…”