Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

Abstract

This article deals with a numerical approach based on the symmetric space-time Chebyshev spectral collocation method for solving different types of Burgers equations with Dirichlet boundary conditions. In this method, the variables of the equation are first approximated by interpolating polynomials and then discretized at the Chebyshev–Gauss–Lobatto points. Thus, we get a system of algebraic equations whose solution is the set of unknown coefficients of the approximate solution of the main problem. We investigate the convergence of the suggested numerical scheme and compare the proposed method with several recent approaches through examining some test problems.

A Novel Spatial–Temporal Radial Trefftz Collocation Method for 3D Transient Wave Propagation Analysis with Specified Sound Source Excitation

“…This is why in practical calculations often small artificial damping is added to the model. (3) Spatial-temporal semianalytical basis function methods [37][38][39] employ the spatial-temporal semi-analytical basis function a priori to satisfy the transient wave equation and then solve it directly. Among these three time-discretization schemes, the first two have been widely used for transient wave propagation analysis; the last one has not been widely used because the time-dependent semi-analytical basis functions are not easy to construct, in particular, the transient wave equation, including the source excitations.…”

Analyzing Similarity Solution of Modified Fisher Equation

“…Especially, partial di erential equations can be used to describe a wide variety of phenomena in nature such as acoustics, electrodynamics, uid ow, heat, and sound. Nonlinear partial di erential equations are mostly renowned for describing the underlying behavior of nonlinear phenomena related to the nature of the real world [1][2][3][4][5][6][7][8].…”

“…It is well-known that obtaining analytical solutions of nonlinear partial di erential equations has a signi cant role in de ning physical phenomena that are rising in several areas such as physics, biology, chemistry, and engineering, and there is a tremendous amount of work on the theory and techniques for solving partial di erential equations both analytically [6][7][8] and numerically [9][10][11][12][13][14].…”

Numerical Solution of Reaction–Diffusion Equations with Convergence Analysis

“…We fully show that the approximate solutions are convergent to the exact solution when the number of collocation points tends to infinity. Note that spectral collocation methods have high accuracy and exponential convergence and, up to now many researchers utilized them to solve different continuous-time problems involving the ordinary and partial differential equations [12,13,18].…”