Uniqueness and CB-Spline Approach for Identifying the Time-Varying Coefficient of a Second-Order Hyperbolic PDE with Non-Classical Boundary Condition

Abstract

This work is concerned with an inverse problem for the second-order hyperbolic PDE, in which extra data are imposed. The time-varying coefficient is not known a priori. We begin by introducing the notion of a classical solution for the problem at hand. The primary aim is to recover both the solution θ(κ,τ) and the unknown coefficient s(τ). To analyze the well-posedness of the problem, we introduce an auxiliary inverse formulation. We establish its parity with the primary one in an appropriate sense. Employing the Fourier method, we prove that the auxiliary problem admits a unique solution. This equivalence then allows us to establish uniqueness and existence for the classical solution of the original inverse problem. The numerical solution of the inverse problem is obtained using a CB-spline method in conjunction with Tikhonov regularization. The CB-spline approach reduces the PDE to a system of ODEs, discretized in time by a Crank–Nicolson scheme. The regularization stabilizes the ill-posed reconstruction. Numerical experiments with two test problems, one smooth and one non-smooth in s(τ), confirm the accuracy and stability of the suggested procedure. These results are cross-validated against an independently implemented finite-difference discretization. The unconditional stability and second-order convergence of the numerical method are both established theoretically and confirmed numerically.