The Laplace transform method for Burgers' equation

Abstract

SUMMARYThe Laplace transform method (LTM) is introduced to solve Burgers' equation. Because of the nonlinear term in Burgers' equation, one cannot directly apply the LTM. Increment linearization technique is introduced to deal with the situation. This is a key idea in this paper. The increment linearization technique is the following: In time level t, we divide the solution u(x, t) into two parts: u(x, t k ) and w(x, t), t k t t k+1 , and obtain a time-dependent linear partial differential equation (PDE) for w(x, t). For this PDE, the LTM is applied to eliminate time dependency. The subsequent boundary value problem is solved by rational collocation method on transformed Chebyshev points. To face the well-known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that the present method is effective and competitive.

Efficient structure‐preserving schemes for good Boussinesq equation

“…Our goal to construct the new schemes is to reduce the scale of corresponding matrices from order 2 N to order N when we choose the spatial step size be ; thereby, the computational cost will be reduced. () It cannot neglect the fact that the leading terms of local truncation error of the existing schemes and with spatial step size are and ; hence, the leading terms of the local truncation error of the new schemes (denoted by and ) with spatial step size must be . If , the effect of new schemes is not improved, which means the new schemes do not realize our goal; if , we obtain the optimal error in theory under positive integer 2 N .…”

An Efficient Parallel Laplace Transform Spectral Element Method for the Two-Dimensional Schrödinger Equation

“…It removes the time dependency and converts the original problem into a series of boundary value problems (BVPs). Chen and Wu et al combined LT with the rational spectral collocation method to solve the advectiondi usion equation [17] and one-dimensional Burgers' equation (18). A coupled method of LT and Legendre wavelets was presented in [19] for one-dimensional Klein-Gordon equations.…”

“…e increment linearization method was first proposed by Wu et al [26] and then integrated with LT to solve Burgers' equation (18). Its key idea is to define an increment function at each time step: Δu(x, t) � u(x, t) − u(x, t n ), t n ≤ t ≤ t n+1 , linearize the equation by omitting the nonlinear terms of the order O(τ c ), c ≥ 2, and finally get the approximation by u(x, t n+1 ) ≈ u(x, t n ) + Δu(x, t n+1 ).…”

The representation on solutions of Burger's equation by Laplace transform

“…We call it viscous Burger's equation, and in case of ν = 0, we call it inviscid Burger's equation. Normally, the solution of Burger's equation can be obtained by Hopf-Cole transformation which transforms Burgers equation into a linear parabolic equation [4], and the numerical solution has been still pursued because of comparing it with the exact solution. In this article, we would like to approach the topic by using Laplace transform.…”

“…With relation to this topic, several researches have been pursued for integral transforms related to differential equations [3,[5][6][7][10][11][12][13][14][15][16], and for nonlinear equation [2,[8][9]. Benton researched exact solutions of the one-dimensional Burger's equation [1], Elzaki has proposed Elzaki homotopy perturbation method to find the solution of Burger's equation [9], and Chen applied the increment linerarization technique(ILT) and Laplace transform to solve the equation [4]. Chen employed Laplace transform to control linear terms, and ILT to control nonlinear term.…”