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On explicit and numerical solvability of parabolic initial-boundary value problems

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Abstract

A homogeneous boundary condition is constructed for the parabolic equation in an arbitrary cylindrical domain ( being a bounded domain, and being the identity operator and the Laplacian) which generates an initial-boundary value problem with an explicit formula of the solution. In the paper, the result is obtained not just for the operator, but also for an arbitrary parabolic differential operator, where is an elliptic operator in of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation in is reduced to an integral equation in a thin lateral boundary layer. An approximate solution to the integral equation generates a rather simple numerical algorithm called boundary layer element method which solves the 3D Cauchy-Dirichlet problem (with three spatial variables).

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Correspondence to Alexander Kozhevnikov.

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Keywords

  • Boundary Layer
  • Integral Equation
  • Bounded Domain
  • Parabolic Equation
  • Explicit Formula