Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

Author： Ivanov D.

Publisher： MAIK Nauka/Interperiodica

ISSN： 0012-2661

Source： Differential Equations, Vol.46, Iss.8, 2010-08, pp. : 1104-1113

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Abstract

We consider boundary value problems for the differential equations ∆₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation ∆₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.