An Inverse Problem for a Semilinear Integro-Differential Pseudoparabolic Equation

This paper investigates a semilinear integro-differential pseudoparabolic equation in a bounded domain. In the inverse problem, the unknown source function is reconstructed using an additional integral condition. First, the problem is discretized in time and is reduced to an elliptic problem at each time level. Then, by applying the Lax–Milgram theorem, the existence and uniqueness of the solution at each time step are established. Second, fundamental energy estimates for the unknown functions are derived. Based on these estimates, the existence and uniqueness of the solution over the entire time interval are proved using Rothe’s method. After that, the elliptic problem is discretized in space using the Legendre pseudospectral method. The fully discretized scheme leads to a symmetric linear algebraic system, which is advantageous from a computational point of view. Finally, the effectiveness of the constructed algorithm is verified using a numerical method. Mathematics Subject Classification: 35K90 , 34M50