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Paper   IPM / M / 7375
School of Mathematics
  Title:   Saulyev's techniques for solving a parabolic equation with a non linear boundary specification
  Author(s):  M. Dehghan
  Status:   Published
  Journal: Int. J. Comput. Math.
  No.:  2
  Vol.:  80
  Year:  2003
  Pages:   257-265
  Supported by:  IPM
In this paper a two-dimensional heat equation is considered. The problem has both Numann and Dirichlet boundary conditions and one non-local condition in which an integral of the unknown solution u occurs. The Dirichlet boundary condition contains an additional unknown function μ(t). In this paper the numerical solution of this equation is treated. Due to the structure of the boundary conditions a reduced one-dimensional heat equation for the new unknown v(y,t)=∫u(x,y,t)dx can be formulated. The resulting problem has a non-local boundary condition. This one-dimensional heat equation is solved by Saulyev's formula. From the solution of this one-dimensional problem an approximation of the function μ(t) is obtained. Once this approximation is known, the given two-dimensional problem reduces to a standard heat equation with the usual Neumann's boundary conditions. This equation is solved by an extension of the Saulyev's techniques. Results of numerical experiments are presented.

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