Development of a numerical method for solving the inverse Cauchy problem for the heat equation

In this work, the initial temperature has been investigated in the Cauchy inverse problem for linear heat conduction equation that it depends on the given temperature at specification time. In this problem, the initial temperature distribution is unknown, but instead, there is a known temperature at the time, $t = T > 0$. The heat conduction problem can be formulated as Fredholm integral first kind equation. It is well known that this problem is an ill-posed problem and direct solution to this problem is unacceptable. An algorithm has been used to define a finite-dimensional operator for this problem also used the generalized discrepancy method to reduce the conditional extremum variation problem to unconditional extremum variation problem for the integral equation. The discretization of the integral equation has made it possible to reduce this problem to a system of linear algebraic equations. Then, Tikhonov’s regularization inversion method has been used to find an approximation solution. Finally, the numerical computation example has been presented to verify the accuracy of the estimated solution.