The paper is devoted to the development of a numerical algorithm for solving nonlinear integro-differential equations based on the use of quadrature formulas. The Koltunov-Rzhanitsyn kernel with weakly singular features of the Abel type is used as a kernel. To conduct a computational experiment, a computer program was developed; the results obtained by this program are reflected in the form of tables and graphs. A test example was solved, and the obtained approximate numerical results were compared with exact solutions. The influence of nonlinearity and integral parts on the nature of oscillatory process of a viscoelastic body was investigated.

The paper presents a solution to the problem of the gas-dynamic state of an elementary section of a gas pipeline with variable input and output mass gas outflows, which is obtained by the method of separation of variables. The quasi-one-dimensional mathematical model of pipeline transport of real gas takes into account factors such as friction, gravity and the local component of the inertia of the gas. By introducing the mass flow rate and averaging the velocity in the quadratic law of resistance, the autonomous equations of the telegraph equation with respect to pressure and mass flow are compiled from a system of equations. Taking into account possible abrupt changes of the unknowns in time and distance, the solution is sought in the form of functional series. The demonstrativeness of this method lies in the fact that the perturbation frequencies relevant to the parabolic and hyperbolic types of equations and the intermediate variant are distinguished. A description is given of a general method for solving the problem when temporary changes in the gas mass flow rate are set at the ends of the section. The exact solutions of mass flow rate and hydrostatic pressure are obtained for the case of a spasmodic change in mass flow rate at the boundaries of the site with a constant slope from the horizon. Numerical results related to the case of instantaneous closure of the output section of a linear section are presented. The results of the work are useful for assessing the energy intensity and reliability of gas pipelines in conditions of large diameters, high working pressures, and also when the gas pipeline is laid along a relief line.