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.