At present, investigations of Sobolev-type models are actively developing. In the solution of applied problems the results allowing to get their numerical solutions are very signicant. In the article the algorithm for numerical solving of the initial boundary value problem is developed. The problem describes the pressure distribution of the homogeneous uid in the horizontal layer in the circle. The layer is opened by a vertical well of a small radius. In our research we suppose that random disturbing loads have an inuence on the uid. The problem was solved under two assumptions. Firstly, we suppose that an unstable uid ow is axially symmetric, and secondly, that in initial moment the pressure in the layer is constant. After the process of the discretization we modify the original model to the Cauchy problem for the system of ordinary dierential equations. For the numerical solution we use algorithms based on explicit one-step formulas of the Runge Kutta type with the seventh-order accuracy and with the selection of the integration step. We also use the scheme of the eighth-order accuracy to evaluate the calculation accuracy on each steps of time. According to the results of this control, we choose the time-step. A lot of numerical experiments have shown high numerical eciency of the algorithm that we use to solve the investigated initial-boundary problem.

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The authors developed a numerical non-iterative method of nding of the value of eigenfunctions of perturbed self-adjoint operators, which was called the method of regularized traces. It allows to nd the value of eigenfunctions of perturbed discrete operators, using the spectral characteristics of the unperturbed operator and the eigenvalues of the perturbed operator. In contrast to the known methods, in the method of regularized traces the value of eigenfunctions are found by the linear equations. It signicantly increases the computational eciency. The diculty of the method is to nd sums of functional series of "suspended" corrections of perturbation theory, which can be found only numerically. The formulas, which are convenient to nd "suspended" corrections such that one can approximate the amount of these functional series by summing up of them, are presented in the paper. However, if a norm of the perturbing operator is large, then the summation of "suspended" corrections can be not eective. We obtain analytical formulas, which allow to nd the values of sums of functional series of "suspended" corrections of perturbation theory in the discrete nodes without direct summation of its terms. Computational experiments are performed. These experiments allowed to nd the values of the eigenfunctions of the perturbed one-dimensional Laplace operator. The experimental results showed the accuracy and computational eciency of the developed method.