On a weakly L‐stable time integration formula coupled with nonstandard finite difference scheme with application to nonlinear parabolic partial differential equations

Abstract

In the present paper, we establish an efficient numerical scheme based on weakly L‐stable time integration convergent formula and nonstandard finite difference (NSFD) scheme. We solve Burgers' equation with Dirichlet boundary conditions as well as Neumann boundary conditions. We also solve the Fisher equation. We use Hermite approximation polynomial of order five and backward explicit Taylor's series approximation of order six to derive the numerical integration formula for the initial value problem (IVP) y′false(tfalse)=gfalse(t,yfalse),0.1emyfalse(t0false)=ρ0. We combine this method with the NSFD scheme and convert the problem into the system of algebraic equations. We discuss the convergence, truncation error, and stability of the developed method. To demonstrate the efficiency of the developed method, we compare the numerical results with some existing numerical results and exact solutions.

Half-sweep Modified SOR Approximation of A Two-dimensional Nonlinear Parabolic Partial Differential Equation

“…For instance, [8] studied a hybrid spectral collocation method to precisely find the approximate solutions of NPPDE in studying gene propagation and transmission of nerve impulses. Then, [9] proposed a numerical method based on a weak L-stable time integration with a nonstandard finite difference method. Next, [10] introduced the combined quintic trigonometric B-spline collocation, finite difference, and Rubin-Graves linearization.…”

On numerical solutions of Telegraph, viscous, and modified Burgers equations via Bernoulli collocation method

“…Zang et al [4] investigated a novel approach for solving high-dimensional PDE, which has a wide range of applications in science. Other models involving PDE can be found in [5,6] with di erent numerical and analytical methods for solving these types of problems.…”