Modulating functions-based method for parameters and source estimation in one-dimensional partial differential equations

In this paper, modulating functions-based method is proposed for estimating space-time-dependent unknowns in one-dimensional partial differential equations. The proposed method simplifies the problem into a system of algebraic equations linear in unknown parameters. The well-posedness of the modulating functions-based solution is proved. The wave and the fifth-order KdV equations are used as examples to show the effectiveness of the proposed method in both noise-free and noisy cases.