Abstract
Sufficient conditions of existence and uniqueness of weak generalized solution to the Dirichlet--Cauchy problem for equation modeling a quasi-steady process in conducting nondispersive medium with relaxation are obtained. The main equation of the model is considered as a representative of the class of quasi-linear equations of Sobolev type. It enables to prove a solvability of the Dirichlet--Cauchy problem in a weak generalized meaning by methods developed for this class of equations. In suitable functional spaces the Dirichlet--Cauchy problem is reduced to the Cauchy problem for abstract quasi-linear operator differential equation of the special form. Algorithm of numerical solution to the Dirichlet--Cauchy problem based on the Galerkin method is developed. Results of computational experiment are provided.
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