A Numerical Method of Solving the Coefficient Inverse Problem for the Nonlinear Equation of Diffusion-Reaction

Abstract

We consider two inverse problems for determining the coecients for a one-dimensional nonlinear diffusion-reaction equation of the FisherKolmogorovPetrovskyPiskunov type. The rst problem consists in determining the kinetic coeffcient for a nonlinear lower term, depending only on the time variable, according to a given integral condition. And the second problem consists in determining the time-dependent diusion coeffcient, again according to a given integral condition. To solve both problems, the time derivative of the derivative is rst sampled. In the rst problem, the diusion term is approximated in time according to the implicit scheme, and the nonlinear minor term in the semi-explicit scheme. And in the second problem, the diusion term is approximated in time in an explicitly implicit scheme, and the nonlinear minor term is again in a semi-explicit scheme. As a result, both problems reduce to dierential-dierence problems with respect to functions depending on the spatial variable. For numerical solution of the problems obtained, a non-iterative computational algorithm is proposed, based on reduction of the dierential-dierence problem to two direct boundary-value problems and a linear equation with respect to the unknown coeffcient. On the basis of the proposed numerical method, numerical experiments were performed for model problems.