Vector Transformation Operators for a Systems Partial Differential Equation

O. Yaremko^1, , N. Yaremko¹ and T. Eliseeva¹

¹Penza State University, str. Lermontov, 37, 440038, Penza, Russia

Cite this paper:
O. Yaremko, N. Yaremko and T. Eliseeva. Vector Transformation Operators for a Systems Partial Differential Equation. International Journal of Partial Differential Equations and Applications. 2015; 3(1):7-11. doi: 10.12691/ijpdea-3-1-2

Abstract

Vector transformation operators method is applied to vector heat equation in one space dimension, to vector wave equation in one space dimension and to vector Laplace equation in two dimensions. The transformation operators method which allows us to interpret piecewise-homogeneous physical processes as a perturbing of a homogeneous ones. The analytical description of mathematical models of thermal conductivity and wave processes for piecewise homogeneous media with flat symmetry is obtained by the developed in this paper vector transformation operators method.

Keywords:
poisson formula wave equation heat equation dirichlet problem laplace equation

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References:

[1]	Marchenko, V. A. (2011), Sturm-Liouville operators and applications (2 ed.), Providence: American Mathematical Society.
[2]	A. Kipriyanov,Lyakhov L.N., Raykhelgauz L.B. Singular Heat Equation with -Bessel Operator. Fundamental Solutions Journal of Mathematical Sciences V. 188, No. 3, 2013. pp. 283-293.
[3]	Samko, S.G., Kilbas, A.A. and Marichev, O.I., Fractional Integralsand Derivatives: Theory and Applications, Gordon and Breach, New York (1993).
[4]	Carslaw, H. S.; Jaeger, J. C., Conduction of Heat in Solids (2nd ed.), 1959, Oxford University Press,
[5]	Evans, L.C., Partial Differential Equations, American Mathematical Society, 1998.
[6]	R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II. Interscience (Wiley) New York, 1962.
[7]	Polyanin, A. D. and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman and Hall. CRC Press, 2007.
[8]	E. Mogileva, O. Yaremko, Hermite functions with discontinuous coefficients for the solution of fractal diffusion retrospective problems, International journal of applied mathematics and informatics, Issue 3, Volume 7, 2013, p. 78-86.
[9]	O.E. Yaremko, Matrix integral Fourier transforms for problems with discontinuous coefficients and transformation operators, Reports Of Academy Of Sciences, Volume. 417, Issue 3, 2007, p. 323-325.
[10]	Bavrin I.I., Matrosov V.L., Jaremko O. E.(2006) Operators of transformation in the analysis, mathematical physics and Pattern recognition. Moscow, Prometheus, p 292.
[11]	O. Yaremko, V. Selutin, N. Yaremko, The Fourier Transform with Piecewise Trigonometric Kernels and its Applications, WSEAS transactions on mathematics, Volume 13, 2014, pp. 615-625.
[12]	O.E. Yaremko, Transformation operator and boundary value problems, Differential Equation. Vol. 40, No. 8, 2004, pp. 1149-1160
[13]	O. Yaremko, O. Nikitina, E. Zuravleva Matrix Fourier Transforms and Application, International Journal of Partial Differential Equations and Applications , Vol. 2, No. 5, 2014, pp. 91-95.