- Research Article
- Open Access
On explicit and numerical solvability of parabolic initial-boundary value problems
Boundary Value Problems volume 2006, Article number: 75458 (2006)
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).
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
Eidelman SD, Zhitarashu NV: Parabolic Boundary Value Problems, Operator Theory: Advances and Applications. Volume 101. Birkhäuser, Basel; 1998:xii+298.
Grubb G: Parameter-elliptic and parabolic pseudodifferential boundary problems in globalSobolev spaces. Mathematische Zeitschrift 1995,218(1):43–90. 10.1007/BF02571889
Kozhevnikov A: On explicit solvability of an elliptic boundary value problem and its application. Applicable Analysis 2005,84(8):789–805. 10.1080/00036810500137542
Ladyzhenskaya OA, Solonnikov VA, Uralzeva NN: Linear and Quasi-Linear Equations of Parabolic Type. American Mathematical Society, Rhode Island; 1968.
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.
Press WH, Flannery BP, Teukolsky SA, Vetterling WT: Numerical Recipes in C. The Art of Scientific Computing. Cambridge University Press, Cambridge; 1988:xxii+735.
Ryaben'kii VS: Method of Difference Potentials and Its Applications, Springer Series in Computational Mathematics. Volume 30. Springer, Berlin; 2002:xviii+538.
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.
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.