Abstract
A numerical procedure for an inverse problem concerning diffusion equation with source control parameter is considered. The proposed method is based on shifted Legendre‐tau technique. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution as a shifted Legendre function with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre functions. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.
Highlights
Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution u and p as a shifted Legendre function with unknown coefficients
The operational matrices of integral and derivative are given. These matrices together with the tau method are utilized to evaluate the unknown coefficients of shifted Legendre functions
A function u(x,t) of two independent variables defined for 0 ≤ x ≤ and 0 ≤ t ≤ τ may be expanded in terms of double shifted Legendre polynomials as nm u(x, t) =
Summary
We want to identify the function p(t) that will yield a desired energy prescribed in a portion of the spatial domain This kind of problem has many important applications [13, 17, 20, 22]. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution u and p as a shifted Legendre function with unknown coefficients. The operational matrices of integral and derivative are given These matrices together with the tau method are utilized to evaluate the unknown coefficients of shifted Legendre functions.
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