An operator method for telegraph partial differential and difference equations

被引:0
作者
Allaberen Ashyralyev
Mahmut Modanli
机构
[1] Fatih University,Department of Mathematics
[2] Siirt University,Department of Mathematics
来源
Boundary Value Problems | / 2015卷
关键词
telegraph equations; Cauchy problem; Hilbert space; difference schemes; stability;
D O I
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摘要
The Cauchy problem for abstract telegraph equations d2u(t)dt2+αdu(t)dt+Au(t)+βu(t)=f(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\frac{d^{2}u(t)}{dt^{2}}}+\alpha{\frac{du(t)}{dt}}+Au(t)+\beta u(t)= f(t)$\end{document} (0≤t≤T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\leq t\leq T$\end{document}), u(0)=φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u(0)=\varphi$\end{document}, u′(0)=ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u^{\prime}(0)=\psi $\end{document} in a Hilbert space H with the self-adjoint positive definite operator A is studied. Stability estimates for the solution of this problem are established. The first and second order of accuracy difference schemes for the approximate solution of this problem are presented. Stability estimates for the solution of these difference schemes are established. In applications, two mixed problems for telegraph partial differential equations are investigated. The methods are illustrated by numerical examples.
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