A Numerical Method to Solve Identification Problem for the Lower Coefficient and the Source in the Convection–Reaction Equation

Abstract

We consider the problem of identifying simultaneously the kinetic reaction coefficient and source function depending only on a spatial variable in one-dimensional linear convection–reaction equation. As additional conditions, a non-local integral condition for the solution of the equation and condition of final overdetermination are given. This problem belongs to the class of combined inverse problems. By integrating the equation with the use of additional integral condition, the problem is transformed to a coefficient inverse problem with local conditions. The derivative with respect to the spatial variable is discretized and a special representation is proposed to solve the resultant semi-discrete problem. As a result, for each discrete value of the spatial variable, the semi-discrete problem splits into two parts: a Cauchy problem and a linear equation with respect to the approximate value of the unknown kinetic coefficient. To determine the source function, an explicit formula is also obtained. The numerical solution of the Cauchy problem uses the implicit Euler method. Numerical experiments are carried out on the basis of the proposed method.