Numerical solution of the Cauchy–Wentzell problem for the dzekzer model in a bounded domain

In terms of the theory of the $p$-sectorial operator, the Cauchy problem for the Dzekzer equation describing the evolution of the free surface of a filtered liquid with pure Wentzell boundary conditions are investigated. In particular, we consider the relative spectrum in the Dzekzer equation and construct a resolving holomorphic semigroup of the operator in the Cauchy–Wentzell problem. In the article, these problems are solved under the assumption that the initial space in which the Laplace operator operates on the bounded domain is a Lebesgue space $L^{2}(\Omega)$. The purpose of this work is to show new approach for resolvability of this problem with pure Wentzell boundary conditions. Namely, according to the modified Galerkin method, describe the solution of the Cauchy–Wentzell problem.