An approximate method for solving the inverse coefficient problem for the heat equation

A numerical method is constructed for recovering a variable coefficient in the Cauchy problem and also in the initial boundary value problem for the one-dimensional heat equation. The desired coefficient is assumed to be time-dependent, but not space-dependent. Our approach is based on the construction of an auxiliary ordinary differential equation for the unknown coefficient and the subsequent solving it by some numerical method of solving ordinary differential equations. The apparatus of Lyapunov stability theory is used as well. The main advantages of the proposed method are its simplicity and stability with respect to the initial data perturbations. For the implementation, the method requires some additional information on the solution of the original heat equation at no more than finitely many points. The efficiency of the proposed approach is illustrated by solving several model examples.