Numerical Determination of a Time-Dependent Boundary Condition for a Pseudoparabolic Equation from Integral Observation

The third-order pseudoparabolic equations represent models of filtration, the movement of moisture and salts in soils, heat and mass transfer, etc. Such non-classical equations are often referred to as Sobolev-type equations. We consider an inverse problem for identifying an unknown time-dependent boundary condition in a two-dimensional linear pseudoparabolic equation from integral-type measured output data. Using the integral measurements, we reduce the two-dimensional inverse problem to a one-dimensional problem. Then, we apply appropriate substitution to overcome the non-local nature of the problem. The inverse ill-posed problem is reformulated as a direct well-posed problem. The well-posedness of the direct and inverse problems is established. We develop a computational approach for recovering the solution and unknown boundary function. The results from numerical experiments are presented and discussed.