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## Archive of Issues

Russia Yekaterinburg
Year
2015
Volume
25
Issue
1
Pages
78-92
 Section Mathematics Title Convergence of the difference method of solving the two-dimensional wave equation with heredity Author(-s) Tashirova E.E.a Affiliations Ural Federal Universitya Abstract The paper presents the consideration of the wave equation with two space variables and one time variable and with heredity effect $$\frac{\partial^2 u}{\partial t^2}=a^2\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) + f\big(x,y,t,u(x,y,t),u_t(x,y,\cdot)\big), \quad u_t(x,y,\cdot)=\big\{u(x,y,t+\xi),-\tau \leqslant \xi\leqslant 0\big\}.$$ 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 UDC 519.633 MSC 35L20 DOI 10.20537/vm150109 Received 17 December 2014 Language Russian Citation Tashirova E.E. Convergence of the difference method of solving the two-dimensional wave equation with heredity, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 1, pp. 78-92. References Wu J. Theory and application of partial functional differential equations, New York: Springer-Verlag, 1996, 438 p. Pimenov V.G., Tashirova E.E. Numerical methods for solving a hereditary equation of hyperbolic type, Proceedings of the Steklov Institute of Mathematics, 2013, vol. 281, issue 1 supplement, pp. 126-136. Pimenov V.G., Lozhnikov A.B. Difference schemes for the numerical solution of the heat conduction equation with aftereffect, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 275, issue 1 supplement, pp. 137-148. Samarskii А.А. Teoriya raznostnykh skhem (Theory of difference schemes), Moscow: Nauka, 1989, 656 p. Pimenov V.G. General linear methods for the numerical solution of functional-differential equations, Differential Equations, 2001, vol. 37, no. 1, pp. 116-127. Kim A.V., Pimenov V.G. i-gladkii analiz i chislennye metody resheniya funktsional'no-differentsial'nykh uravnenii (i-smooth calculus and numerical methods for functional differential equations), Moscow-Izhevsk: Regular and Chaotic Dynamics, 2004, 256 p. Lekomtsev A.V., Pimenov V.G. Convergence of the alternating direction method for the numerical solution of a heat conduction equation with delay, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 272, issue 1 supplement, pp. 101-118. Kalitkin N.N. Chislennye metody (Numerical methods), St. Petersburg: BHV-Petersburg, 2011, 586 p. Tashirova E.E. Numerical methods for solving two-dimensional wave equation with aftereffect, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 2013, vol. 18, no. 5-2, pp. 2704-2706 (in Russian). Full text