The modified polynomial expansion method for identifying the time dependent heat source in two-dimensional heat conduction problems

• A modified polynomial expansion method to identify the unknown time dependent heat source.
• By using the characteristic length, we obtain a well-posed linear system and compute the weighting coefficients accurately.
• The method is robust enough against noise perturbation on the measurement data.
• The polynomial expansion method with the concept of characteristic length can solve multi-dimensional problems straightforward.

In this paper, the modified polynomial expansion method is developed to solve problems of identifying the time dependent heat source, in which an inverse problem is encountered. Aimed at this problem, the variation of variables is adopted to eliminate the unknown heat source and obtain a six-line boundary value problem. As compared with the conventional four-line boundary value problem, the six-line boundary value problem is quite hard to be dealt with. After the unknown non-homogeneous term being eliminated, the polynomial expansion method is introduced to discretize the time and space fields, respectively. Then, the distribution of temperature is expressed as a linear superposition of polynomial functions. After that, a characteristic length concept is adopted to resolve the ill-posed matrix problems arising in those conventional polynomial expansion methods. The desired heat source function can be obtained by putting the solution of the six-line boundary value problem into differential operations. Several numerical experiments with designed examples are included to validate the accuracy and effectiveness of the proposed approach.