Computational identification of the right-hand side of a parabolic equation

P. N. Vabishchevich ; V. l. Vasil'ev ; M. V. Vasil'eva

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 6, pp. 1020-1027

Résumé

Among inverse problems for partial differential equations, a task of interest is to study coefficient inverse problems related to identifying the right-hand side of an equation with the use of additional information. In the case of nonstationary problems, finding the dependence of the right-hand side on time and the dependence of the right-hand side on spatial variables can be treated as independent tasks. These inverse problems are linear, which considerably simplifies their study. The time dependence of the right-hand side of a multidimensional parabolic equation is determined using an additional solution value at a point of the computational domain. The inverse problem for a model equation in a rectangle is solved numerically using standard spatial difference approximations. The numerical algorithm relies on a special decomposition of the solution whereby the transition to a new time level is implemented by solving two standard grid elliptic problems. Numerical results are presented.