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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 1, Pages 78–92 (Mi vuu467)  

MATHEMATICS

Convergence of the difference method of solving the two-dimensional wave equation with heredity

E. E. Tashirova

Ural Federal University, ul. Turgeneva, 4, Yekaterinburg, 620000, Russia

Abstract: The paper presents the consideration of the wave equation with two space variables and one time variable and with heredity effect
$$ \frac{\partial^2 u}{\partial t^2}=a^2(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}) + f(x,y,t,u(x,y,t),u_t(x,y,\cdot)),\quad u_t(x,y,\cdot)=\{u(x,y,t+\xi),-\tau\le\xi\le0\}. $$
A family of grid methods is constructed for the numerical solution of this equation; the methods are based on the idea of separating the current state and the history function. A complete analog of the factorization method which is known for an equation without delay is constructed according to the current state. Influence of prehistory is taken into consideration by interpolation constructions. The local error order of the algorithm is investigated. A theorem on the convergence and on the order of convergence of methods is obtained by means of embedding into a general difference scheme with aftereffect. The results of calculating a test example with variable delay are presented.

Keywords: difference methods, two-dimensional wave equation, time delay, interpolation, factorization, order of convergence.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 02.А03.21.0006
Russian Foundation for Basic Research 13--01--00089


Full text: PDF file (367 kB)
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UDC: 519.633
MSC: 35L20
Received: 17.12.2014

Citation: E. E. Tashirova, “Convergence of the difference method of solving the two-dimensional wave equation with heredity”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:1 (2015), 78–92

Citation in format AMSBIB
\Bibitem{Tas15}
\by E.~E.~Tashirova
\paper Convergence of the difference method of solving the two-dimensional wave equation with heredity
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2015
\vol 25
\issue 1
\pages 78--92
\mathnet{http://mi.mathnet.ru/vuu467}
\elib{http://elibrary.ru/item.asp?id=23142055}


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  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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