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On explicit and numerical solvability of parabolic initial-boundary value problems
Boundary Value Problems volume 2006, Article number: 75458 (2006)
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).
References
- 1.
Agranovič MS, Višik MI: Elliptic problems with a parameter and parabolic problems of general type. Uspekhi Matematicheskikh Nauk 1964,19(3 (117)):53–161. Russian Mathematical Surveys 19, 53–159
- 2.
Eidelman SD, Zhitarashu NV: Parabolic Boundary Value Problems, Operator Theory: Advances and Applications. Volume 101. Birkhäuser, Basel; 1998:xii+298.
- 3.
Grubb G: Parameter-elliptic and parabolic pseudodifferential boundary problems in globalSobolev spaces. Mathematische Zeitschrift 1995,218(1):43–90. 10.1007/BF02571889
- 4.
Kozhevnikov A: On explicit solvability of an elliptic boundary value problem and its application. Applicable Analysis 2005,84(8):789–805. 10.1080/00036810500137542
- 5.
Ladyzhenskaya OA, Solonnikov VA, Uralzeva NN: Linear and Quasi-Linear Equations of Parabolic Type. American Mathematical Society, Rhode Island; 1968.
- 6.
Lions J-L, Magenes E: Non-Homogeneous Boundary Value Problems and Applications. Vol. II, Die Grundlehren der mathematischen Wissenschaften. Volume 182. Springer, New York; 1972:xi+242.
- 7.
Press WH, Flannery BP, Teukolsky SA, Vetterling WT: Numerical Recipes in C. The Art of Scientific Computing. Cambridge University Press, Cambridge; 1988:xxii+735.
- 8.
Ryaben'kii VS: Method of Difference Potentials and Its Applications, Springer Series in Computational Mathematics. Volume 30. Springer, Berlin; 2002:xviii+538.
- 9.
Slobodeckiĭ LN: Generalized Sobolev spaces and their application to boundary problems for partial differential equations. Leningradskii Gosudarstvennyi Pedagogičeskii Institute Učenye Zapiski 1958, 197: 54–112.
- 10.
Volevich LR, Shirikyan AR: Stable and unstable manifolds for nonlinear elliptic equations with a parameter. Transactions of the Moscow Mathematical Society 2000, 2000: 97–138.
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Kozhevnikov, A., Lepsky, O. On explicit and numerical solvability of parabolic initial-boundary value problems. Bound Value Probl 2006, 75458 (2006). https://doi.org/10.1155/BVP/2006/75458
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Keywords
- Boundary Layer
- Integral Equation
- Bounded Domain
- Parabolic Equation
- Explicit Formula