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.