The idea of the modulating function-based method (MFBM) is to multiply the considered differential equation by a set of modulating functions to transform the differential equation into a set of algebraic integral equations by applying integration by parts, where the unknown initial conditions are eliminated by the properties of the modulating functions. The method does not require solving the direct problem by which the computational cost is reduced. Further, approximating the derivatives of the measurements, which are usually noisy, is avoided with this method.
The main contributions of this paper are the following. First, we extend the MFBM to estimate space-time dependent parameters and source, separately and simultaneously, using a finite number of measurements in both noisy and non-noisy cases. Secondly, the well-posedness of the modulating functions-based solution is proved. Then, a mathematical analysis of the estimation error is performed. Finally, the influence of the number of modulating functions is investigated and discussed independently on the choice of the type of the modulating function. The figure shows that the estimated space varying velocity c(x) in the first order wave equation is in quite good agreement with the exact one; therefore, the MFBM is an efficient and a robust method for solving inverse problems.