用差分格式法求解电报偏微分方程

Bawar Mohammed Faraj, mahmut mondali
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引用次数: 8

摘要

在这项工作中,我们提出了以下电报双曲偏微分方程{utt (t, x) + ut (t, x) + u (t, x) = uxx (t, x) +用户体验(t, x) + f (t, x), 0≤t≤t u (t, 0) = u (t、L) = 0, u (0, x) =φ(x),但(0,x) =Ψ(x) 0≤≤x L(1)虽然这个偏微分方程的精确解是已知testreliability差分格式的方法是很重要的。给出了该电报偏微分方程的稳定性估计。利用初始条件,给出了上述方程抽象形式的一阶和二阶差分格式。对这些差分格式建立了矩阵稳定性定理。给出了近似求解该问题的一阶和二阶精度差分格式。对于该初边值问题的近似解,我们考虑了依赖于小参数τ =TN(N > 0)和h =LN(N > 0)的网格点族的集合w(τ,h) = [0, T]τ × [0, L]h。在电报偏微分方程的情况下,应用高斯消去法求解了这种差分格式。将拉普拉斯变换法得到的精确解与近似解进行了比较。数值实验结果支持了这些差分格式解的理论项。用matlab程序求得的数值解在精度上取得了良好的效果。举例说明了所提出技术的有效性和适用性。因此,差分格式法对于上述方程的求解是十分重要的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Using Difference Scheme Method for the Numerical Solution of Telegraph Partial Differential Equation
In this work, we presented the following hyperbolic telegraph partial differential equation{utt (t, x) + ut (t, x) + u(t, x) = uxx (t, x) + ux (t, x) + f(t, x), 0 ≤ t ≤ T u(t, 0) = u(t, L) = 0 , u(0, x) = φ(x) , ut (0, x) = Ψ(x), 0 ≤ x ≤ L (1)Although exact solution of this partial differential equation is known it is important to testreliability of difference scheme method. The Stability estimates for this telegraph partialdifferential equation are given. The first and second order difference schemes are formed for theabstract form of the above given equation by using initial conditions. Theorem on matrix stabilityis established for these difference schemes. The first and second order of accuracy differenceschemes to approximate solution of this problem are stated. For the approximate solution of thisinitial-boundary value problem, we consider the set w(τ,h) = [0, T]τ × [0, L]h of a family of gridpoints depending on the small parameters τ =TN(N > 0) and h =LN(N > 0). Gauss eliminationmethod is applied for solving this difference schemes in the case of telegraph partial differentialequations. Exact solutions obtained by Laplace transform method is compared with obtainedapproximation solutions. The theoretical terms for the solution of these difference schemes aresupported by the results of numerical experiments. The numerical solutions which found by Matlabprogram has good results in terms of accuracy. Illustrative examples are included to demonstratethe validity and applicability of the presented technique. As a result, difference scheme method isimportant for above mentioned equation.
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