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 simplified the problem into a system of algebraic equations linear in unknown parameters. The well-posedness of modulating functions-based solution is proven. 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.