Using matrix stability for variable telegraph partial differential equation

Abstract

The variable telegraph partial differential equation depend on initial boundary value problem has been studied. The coefficient constant time-space telegraph partial differential equation is obtained from the variable telegraph partial differential equation throughout using Cauchy-Euler formula. The first and second order difference schemes were constructed for both of coefficient constant time-space and variable time-space telegraph partial differential equation. Matrix stability method is used to prove stability of difference schemes for the variable and coefficient telegraph partial differential equation. The variable telegraph partial differential equation and the constant coefficient time-space telegraph partial differential equation are compared with the exact solution. Finally, approximation solution has been found for both equations. The error analysis table presents the obtained numerical results.

On the Numerical Solution for Two Dimensional Laplace Equation with Initial Boundary Conditions by using Finite Difference Methods

“…Some theoretical and numerical treatments of this problem are found in [20,34]. Finite difference methods are the oldest numerical methods for solving this equation [21]. In the other hand, Al-Haq and Mohammad in [28] investigated the method of fundamental solutions for solving the Laplace equation with the Dirichlet boundary condition in a disk [35].…”

“…In the case of Partial differential equations, much attention has been given in the works to test the reliability and accuracy of the approximation and numerical techniques [17]. Many researchers applied finite difference approximations for numerical solution of different kinds of PDE's in works [7,[18][19][20][21][22][23][24][25].…”

“…Akgul and Faraj have applied Finitie difference methods for Wave equations under IBV conditions [18]. Moreover, various studies on PDE stability estimates have been done, scholars presented several methods for proving the stability of the proposed methods [7,[19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. In the other hand, various numerical methods presented for solving IBVP elliptic partial differential equations [26,[28][29][30].…”

On the numerical solution of the fractional telegraph partial differential equation with variational iteration method

“…He'nin ufuk açıcı çalışmasından sonra, Cantor kümeleri üzerinde difüzyon ve dalga denklemleri [28], Riccati diferansiyel denklemi [29], iki boyutlu Burger's denklemlerinin kesirli modeli [30], Fisher's denklemleri, lineer olmayan osilatörler, Evrim denklemleri, tekil problemler, gözenekli bir ortam boyunca kararsız akım, sızıntı akımı, bağımsız adi diferansiyel denklemler, Korteweg-de Vries denklemleri, parabolik denklemler, integral-diferansiyel denklemleri, kimyasal problemler, Boussinesq denklemleri, Schrödinger denklemleri, Helmholtz denklemleri, Sine-Gordon denklemleri ve zamana bağlı kesirli Fornberg-Whitham denklemleri [31] bunların fiziksel doğasıyla uyum içerisinde varyasyonel iterasyon metodunun önerileri uyarınca başarılı bir şekilde çözülmeye çalışılmıştır [32][33]. Sonlu fark şeması metodu kullanılarak üçüncü mertebeden kısmi diferansiyel denklemler ve telegraf denklemlerin yaklaşık çözümleri verildi [34][35][36]. Bunlarla beraber, Kesirli mertebeden kısmi diferansiyel denklemlerin yaklaşık çözümleri ile ilgili pek çok çalışma yapılmıştır [37][38][39][40][41][42][43][44][45].…”

Difference Scheme Method for the Numerical Solution of Partial Differential Equations

“…Moreover, several different approximation methods were presented for solving wave equations [7]. Finite difference approximations have been used for solving different types of partial differential equations in the works [8][9][10][11][12][13][14][15]. Qian and Weiss worked on Wavelets and the numerical solution of partial differential equations [11].…”