The boundary element method for the solution of the backward heat conduction equation
In an earlier paper the same authors introduced the boundary element method modified with a minimal energy scheme to solve the Laplace equation with insufficient boundary data. Here this method is extended to solve the backward unsteady heat conduction problem. This problem is improperly posed, therefore neither the direct solution of the linear system obtained by discretisation nor the least squares solution is possible. It is necessary to add restrictions. The problem is reformulated and an energy functional \(J(u)\) is minimized additionally. Three choices for \(J(u)\) are considered, but for different reasons only the simplest possibility \(J(u) = \int^ 1_ 0 u^ 2 (x,0) dx\) is used for the presented results. These results show that the method is very robust, and that a good initial guess is not important. An alternative method using a time- marching scheme is being investigated by the authors to reduce the large amount of computer storage.
backward unsteady heat conduction problem
scientific article; zbMATH DE number 740631