The Numerical Solution of an Inverse Pseudoparabolic Problem with a Boundary Integral Observation

Direct and inverse problems for a pseudoparabolic equation are considered. The direct (forward) problem is to find the solution of the corresponding initial–boundary-value problem for known model parameters, as well as the initial and boundary conditions. The well-posedness of the direct problem is shown and a priori estimates of the solution are obtained. We study the inverse problem for identifying the flux on a part of the boundary of a rectangle, using integral measurement on the same part of the boundary. We first reduce the inverse problem to a direct one. The initial–boundary-value direct problem is with nonclassical (integrodifferential) boundary conditions. We develop a finite-difference scheme for numerically solving this problem. Numerical test examples demonstrate the effectiveness of the proposed method. It successfully handles the nonclassical integrodifferential boundary conditions and provides accurate numerical solutions.