No abstract

No abstract

No abstract

No abstract

No abstract

In this work, we discuss the inverse problem which consists of the determination of an unknown time-dependent force function from one time-dependent measurement collected in any space point for the one-dimensional wave equation. This problem is motivated by the question of estimating the timedependent body force which needs to be exerted on a given string to reach a desired shape at the final time. We prove its unique solvability using as data a linear combination of displacement and flux measured at one arbitrary fixed point of the string. We also derive a conditional Hölder stability estimate of this inverse problem. The numerical solution of the problem is investigated by means of the Ritz-Galerkin technique along with the application of the satisfier function to obtain cost-effective and stable results. Some numerical examples are provided to show the performance of the proposed scheme.

The inverse problem of identifying the diffusion coefficient in the one‐dimensional parabolic heat equation is studied. We assume that the information of Dirichlet boundary conditions along with an integral overdetermination condition is available. By applying the given assumptions, the problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. Then we employ the direct technique based on the operational matrices for integration, differentiation, and the product of the orthonormal polynomials together with the Ritz–Galerkin technique to reduce the main problem to the solution of a system of nonlinear algebraic equations. Numerical simulations are presented to show the applicability of the proposed method.

No abstract