جۆری توێژینه‌وه‌: Original Article

نوسه‌ران

Department of Mathematics ,Science and Art Faculty, Harran University ,Sanliurfa, Turkey.

پوخته‌

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 test
reliability of difference scheme method. The Stability estimates for this telegraph partial
differential equation are given. The first and second order difference schemes are formed for the
abstract form of the above given equation by using initial conditions. Theorem on matrix stability
is established for these difference schemes. The first and second order of accuracy difference
schemes to approximate solution of this problem are stated. For the approximate solution of this
initial-boundary value problem, we consider the set w(τ,h) = [0, T]τ × [0, L]h of a family of grid
points depending on the small parameters τ =
T
N
(N > 0) and h =
L
N
(N > 0). Gauss elimination
method is applied for solving this difference schemes in the case of telegraph partial differential
equations. Exact solutions obtained by Laplace transform method is compared with obtained
approximation solutions. The theoretical terms for the solution of these difference schemes are
supported by the results of numerical experiments. The numerical solutions which found by Matlab
program has good results in terms of accuracy. Illustrative examples are included to demonstrate
the validity and applicability of the presented technique. As a result, difference scheme method is
important for above mentioned equation.

وشه‌ بنچینه‌ییه‌كان

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