Approximation of Solutions to the Boundary Value Problems for the Generalized Boussinesq Equation

Abstract

The paper is devoted to one of the Sobolev type mathematical models of uid ltration in a porous layer. Results that allow to obtain numerical solutions are signicant for applied problems. We propose the following algorithm to solve the initial-boundary value problems describing the motion of a free surface ltered in a uid layer having nite depth. First, the boundary value problems are reduced to the Cauchy problems for integrodierential equations, and then the problems are numerically integrated. However, numerous computational experiments show that the algorithm can be simplied by replacing the integro-dierential equations with the corresponding approximating Riccati dierential equations, whose solutions can also be found explicitly. In this case, the numerical values of the solution to the integro-dierential equation are concluded between successive values of approximating solutions. Therefore, we can pointwise estimate the approximation errors. Examples of results of numerical integration and corresponding approximations are given.