زانکۆی گەرمیانJournal of Garmian University231000874ICBS Conference20170701Using Difference Scheme Method for the Numerical Solution of Telegraph Partial Differential Equation1571636565810.24271/garmian.133KUBawarFarajDepartment of Mathematics ,Science and Art Faculty, Harran University ,Sanliurfa, Turkey.MahmutMondaliDepartment of Mathematics ,Science and Art Faculty, Harran University ,Sanliurfa, Turkey.Journal Article20180713In this work, we presented the following hyperbolic telegraph partial differential equation<br />{<br />utt (t, x) + ut<br /> (t, x) + u(t, x) = uxx (t, x) + ux<br /> (t, x) + f(t, x), 0 ≤ t ≤ T<br /> u(t, 0) = u(t, L) = 0 , u(0, x) = φ(x) , ut<br /> (0, x) = Ψ(x), 0 ≤ x ≤ L<br /> (1)<br />Although exact solution of this partial differential equation is known it is important to test<br />reliability of difference scheme method. The Stability estimates for this telegraph partial<br />differential equation are given. The first and second order difference schemes are formed for the<br />abstract form of the above given equation by using initial conditions. Theorem on matrix stability<br />is established for these difference schemes. The first and second order of accuracy difference<br />schemes to approximate solution of this problem are stated. For the approximate solution of this<br />initial-boundary value problem, we consider the set w(τ,h) = [0, T]τ × [0, L]h of a family of grid<br />points depending on the small parameters τ =<br />T<br />N<br />(N > 0) and h =<br />L<br />N<br />(N > 0). Gauss elimination<br />method is applied for solving this difference schemes in the case of telegraph partial differential<br />equations. Exact solutions obtained by Laplace transform method is compared with obtained<br />approximation solutions. The theoretical terms for the solution of these difference schemes are<br />supported by the results of numerical experiments. The numerical solutions which found by Matlab<br />program has good results in terms of accuracy. Illustrative examples are included to demonstrate<br />the validity and applicability of the presented technique. As a result, difference scheme method is<br />important for above mentioned equation.https://jgu.garmian.edu.krd/article_65658_d0699173fd1e41d4885bc43add4a4e02.pdf