Approximation of a timewise dependent function in the inverseone-dimensional telegraph equation

In this article, we study the linear inverse problem for approximating a timewise-dependentfunction in the second-order hyperbolic equation. To solve the problem, information such as Neumannboundary conditions along with an integral condition and initial conditions at the initial moment and thefinal instant have been provided. In the first step, we show that this problem has a unique solution. Then,we change the main problem into a new one and then we present the spectral approximation based onthe Ritz-collocation method to recover the unknown functions. Discretization of the problem by usingthe presented technique leads to a linear system of algebraic equations, which Tikhonov’s regularizationmethod is used to obtain stable solutions. The results of the numerical simulation confirm the high accuracyand stability of the approximate solution.