Approximation of solutions to the boundary value problems for the generalized Boussinesq equation

The paper is devoted to one of the Sobolev type mathematical models of fluid filtration in a porous layer. Results that allow to obtain numerical solutions are significant for applied problems. We propose the following algorithm to solve the initial-boundary value problems describing the motion of a free surface filtered in a fluid layer having finite depth. First, the boundary value problems are reduced to the Cauchy problems for integro-differential equations, and then the problems are numerically integrated. However, numerous computational experiments show that the algorithm can be simplified by replacing the integro-differential equations with the corresponding approximating Riccati differential equations, whose solutions can also be found explicitly. In this case, the numerical values of the solution to the integro-differential 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.