A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems
In this paperý, two inverse problems of determining an unknown source term in a parabolicý equation are consideredý. ýFirstý, ýthe unknown source term is ýestimated in the form of a combination of Chebyshev functionsý. ýThený, ýa numerical algorithm based on Chebyshev polynomials is presented for obtaining the solution of the problemý. ýFor solving the problemý, ýthe operational matrices of integration and derivation are introduced and utilized to reduce the mentioned problem into the matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficientsý. Dueý to ill-posedness of these inverse problemsý, ýthe Tikhonov regularization method with generalized cross validation (GCV) criterion is applied to find stableý solutions. ýFinallyý, some examples are presented to illustrate the efficiency of this numerical methodý. The numerical results show that the proposed method is a reliable method and can give high accuracy approximate solutions.
Journal of Mathematical Researches, Volume:2 Issue:1, 2016
47 - 68
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