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Solution of the first boundary-value problem for a system of autonomous second-order linear partial differential equations of parabolic type with a single delay. (English) Zbl 1250.35117

Summary: The first boundary-value problem for an autonomous second-order system of linear partial differential equations of parabolic type with a single delay is considered. Assuming that a decomposition of the given system into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical solution to the problem is given in the form of formal series and the character of their convergence is discussed. A delayed exponential function is used in order to analytically solve auxiliary initial problems (arising when Fourier method is applied) for ordinary linear differential equations of the first order with a single delay.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35R10 Partial functional-differential equations
35C10 Series solutions to PDEs
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[1] I. G. Petrovsky, Lectures on Partial Differential Equations, Dover Publications, New York, NY, USA, 1991, Translated from the Russian by A. Shenitzer, reprint of the 1964 English translation. · Zbl 0059.08402
[2] E. Kamke, Spravochnik po obyknovennym differentsial’nym uravneniyam, Izdat. Nauka, Moscow, Russia, 1971, Supplemented by a translation by N. H. Rozov of excerpts of articles by D. S. Mitrinović and J. Zbornik. Fourth revised edition. Translated from the sixth German edition and revised in accordance with the eighth German edition by S.V. Fomin. · Zbl 0096.28204
[3] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2002. · Zbl 1127.35320
[4] A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2nd edition, 2003. · Zbl 1054.80010
[5] B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics, vol. 55 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1983, Translated from the Russian by J. R. Schulenberger. · Zbl 0513.35002
[6] F. R. Gantmacher, The Theory of Matrices, vol. 1, AMS Chelsea Publishing, Providence, RI, USA, 2000, Translated from the Russian by K. A. Hirsch, reprint of the 1959 translation. · Zbl 0927.15002
[7] A. Boichuk, J. Diblík, D. Khusainov, and M. Rů\vzi, “Fredholm’s boundary-value problems for differential systems with a single delay,” Nonlinear Analysis, vol. 72, no. 5, pp. 2251-2258, 2010. · Zbl 1190.34073
[8] A. Boichuk, J. Diblík, D. Khusainov, and M. Rů\vzi, “Boundary value problems for delay differential systems,” Advances in Difference Equations, vol. 210, Article ID 593834, 2010. · Zbl 1204.34087
[9] A. Boichuk, J. Diblík, D. Khusainov, and M. Rů\vzi, “Boundary-value problems for weakly nonlinear delay differential systems,” Abstract and Applied Analysis, vol. 2011, Article ID 631412, 19 pages, 2011. · Zbl 1222.34075
[10] J. Diblík, D. Khusainov, Ya. Luká, and M. Rů\vzi, “Representation of the solution of the Cauchy problem for an oscillatory system with pure delay,” Nelinijni Kolyvannya, vol. 11, no. 2, pp. 261-270, 2008 (Russian), English translation: J. Diblík, D. Khusainov, J. Luká and M. Rů\vzi, “Representation of a solution of the Cauchy problem for an oscillating system with pure delay”, Nonlinear Oscillations, vol. 11, no. 2, pp. 276-285, 2008. · Zbl 1276.34055
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