## Abstract

A homogeneous boundary condition is constructed for the parabolic equation (∂_{t} + I - Δ)u = f in an arbitrary cylindrical domain Ω × Rdbl; (Ω ⊂ ℝ^{n} being a bounded domain, I and Δ being the identity operator and the Laplacian) which generates aninitial-boundary value problem with an explicit formula of the solution u. In the paper, the result is obtained not just for the operator ∂_{t} + I - Δ, but also for an arbitrary parabolic differential operator ∂_{t} + A, where A is an elliptic operator in ℝ^{n} of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation (∂_{t} + I - Δ)u = 0 in Ω × ℝ 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).

Original language | English |
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Article number | 75458 |

Journal | Boundary Value Problems |

Volume | 2006 |

DOIs | |

State | Published - 2006 |

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory