Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications

Abstract

In this paper, we consider the problem of the simultaneous determination of time-dependent coefficients in a one-dimensional partial differential equation. The main aim is to apply the tau technique to determine unknown coefficients in a time-dependent partial differential equation. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of shifted Legendre polynomials 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 polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

Keywords:

• tau method
• operational matrix
• inverse problem
• parabolic partial differential equations
• simultaneous parameter determination

2000 AMS Subject Classification::

• 35K15
• 35Q99
• 39A10

Acknowledgements

The authors are very grateful to one of the reviewers for careful reading of this paper and for the comments and suggestions that have greatly improved this paper.