On explicit and numerical solvability of parabolic initial-boundary value problems
<p/> <p>A homogeneous boundary condition is constructed for the parabolic equation <inline-formula><graphic file="1687-2770-2006-75458-i1.gif"/></inline-formula> in an arbitrary cylindrical domain <inline-formula><graphic file="1687-2770-2006-754...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2006-01-01
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| Series: | Boundary Value Problems |
| Online Access: | http://www.boundaryvalueproblems.com/content/2006/75458 |
| Summary: | <p/> <p>A homogeneous boundary condition is constructed for the parabolic equation <inline-formula><graphic file="1687-2770-2006-75458-i1.gif"/></inline-formula> in an arbitrary cylindrical domain <inline-formula><graphic file="1687-2770-2006-75458-i2.gif"/></inline-formula> ( <inline-formula><graphic file="1687-2770-2006-75458-i3.gif"/></inline-formula> being a bounded domain, <inline-formula><graphic file="1687-2770-2006-75458-i4.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-75458-i5.gif"/></inline-formula> being the identity operator and the Laplacian) which generates an initial-boundary value problem with an <it>explicit</it> formula of the solution <inline-formula><graphic file="1687-2770-2006-75458-i6.gif"/></inline-formula>. In the paper, the result is obtained not just for the operator <inline-formula><graphic file="1687-2770-2006-75458-i7.gif"/></inline-formula>, but also for an arbitrary parabolic differential operator <inline-formula><graphic file="1687-2770-2006-75458-i8.gif"/></inline-formula>, where <inline-formula><graphic file="1687-2770-2006-75458-i9.gif"/></inline-formula> is an elliptic operator in <inline-formula><graphic file="1687-2770-2006-75458-i10.gif"/></inline-formula> of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation <inline-formula><graphic file="1687-2770-2006-75458-i11.gif"/></inline-formula> in <inline-formula><graphic file="1687-2770-2006-75458-i12.gif"/></inline-formula> 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).</p> |
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| ISSN: | 1687-2762 1687-2770 |