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Keywords:

  • Feller classification of boundary points;
  • Feller theory;
  • finite-difference schemes;
  • parabolic equations singular and degenerate at the boundary;
  • preconditioning of degenerate parabolic problems

Abstract

A numerical method is devised to solve a class of linear boundary-value problems for one-dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite-difference scheme, grid, and treatment of the boundary data. Second-order accuracy, unconditional stability, and unconditional convergence of solutions of the finite-difference scheme to a constant as the time-step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011