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Determination of a control function in three-dimensional parabolic equations by Legendre pseudospectral method. (English) Zbl 1252.65161
Summary: 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 result, a system of nonlinear differential algebraic equations is generated. Then, by using a suitable transformation, the problem is converted to a homogeneous time varying system of linear ordinary differential equations. Also, a pseudospectral method for the efficient solution of the resulting system of ordinary differential equations is proposed. The solution of this system gives an 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.

MSC:
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R30 Inverse problems for PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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