Boundary Value Problem For Heat Equation Research Articles

Determining the internal boundary of a region in the two-dimensional initial–boundary-value problem for the heat equation

The article investigates the reconstruction of the internal boundary of a two-dimensional region in the two-dimensional initial---boundary-value problem for the homogeneous heat equation. The initial values for the determination of the internal boundary are provided by a boundary condition of second kind on the external boundary and the solution of the initial---boundary-value problem at finitely many points inside the region. The inverse problem is reduced to solving a system of integral equations nonlinear in the function describing the sought boundary. An iterative numerical procedure is proposed involving linearization of integral equations.

Determining the boundary of a two-dimensional region from the solution of the external initial boundary-value problem for the heat equation

We determine the boundary of a two-dimensional region using the solution of the external initial boundary-value problem for the nonhomogeneous heat equation. The initial values for the boundary determination include the right-hand side of the equation and the solution of the initial boundary-value problem given for finitely many points outside the region. The inverse problem is reduced to solving a system of two integral equations nonlinear in the function defining the sought boundary. An iterative procedure is proposed for numerical solution of the problem involving linearization of integral equations. The efficiency of the proposed procedure is investigated by a computer experiment.