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Boundary integral solution of the two‐dimensional heat equation
Author(s) -
Hsiao George C.,
Saranen J.
Publication year - 1993
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670160203
Subject(s) - mathematics , heat equation , sobolev space , mathematical analysis , heat kernel , boundary value problem , robin boundary condition , integral equation , compact space , operator (biology) , free boundary problem , chemistry , biochemistry , repressor , transcription factor , gene
Here we consider initial boundary value problems for the heat equation by using the heat potential representation for the solution. Depending on the choice of the representation we are led to a solution of the various boundary integral equations. We discuss the solvability of these equations in anisotropic Sobolev spaces. It turns out that the double‐layer heat potential D and its spatial adjoint D ′ have smoothing properties similar to the single‐layer heat operator. This yields compactness of the operators D and D ′. In addition, for any constant c ≠ 0, cI + D ′ and cI + D ′ are isomorphisms. Based on the coercivity of the single‐layer heat operator and the above compactness we establish the coerciveness of the hypersingular heat operator. Moreover, we show an equivalence between the weak solution and the various boundary integral solutions. As a further application we describe a coupling procedure for an exterior initial boundary value problem for the non‐homogeneous heat equation.