Determination of a control function in three‐dimensional parabolic equations by Legendre pseudospectral method

A Legendre pseudospectral method is proposed for solving approximately an inverse problem of determining an unknown control parameter p(t) which is the coefficient of the solution u(x, y, z, t) in a diffusion equation in a three‐dimensional region. The diffusion equation is to be solved subject to suitably prescribed initial‐boundary conditions. The presence of the unknown coefficient p(t) requires an extra condition. This extra condition considered as the integral overspecification over the spacial domain. For discretizing the problem, after homogenization of the boundary conditions, we apply the Legendre pseudospectral method in a matrix based manner. As a results a system of nonlinear differential algebraic equations is generated. Then by using suitable transformation, the problem will be converted to a homogeneous time varying system of linear ordinary differential equations. Also a pseudospectral method for efficient solving of the resulted system of ordinary differential equations is proposed. The solution of this system gives the approximation to values of u and p. The matrix based structure of the present method makes it easy to implement. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 74‐93, 2012