Solution of Integral Equations Arising in Mathematical Problems of Construction Science, using the Bogolyubov – Krylov method

The possibility of a numerical solution of construction problems leading to Fredholm integral equations of the first and second kinds is considered. The following problems of building sciences, for example, lead to such equations: in the theory of elasticity – elastoplastic torsion of bodies, stress concentration in a body during its bending; in the theory of vibrations, in the mechanics of structural failure – in the formation of cracks; when studying non-stationary phenomena in solids – non-stationary heat transfer, quasistatic viscoelasticity, propagation of longitudinal and transverse waves and other processes. According to the formulas of numerical integration, the problem can be reduced to a system of linear algebraic equations with a matrix of coefficients, which can be ill-conditioned, and the task is classically incorrect. This circumstance complicates its decision by traditional methods. To solve ill-posed problems similar to the problem in question, there are classical regularization methods. However, the numerical implementation of regularizing algorithms is not always easy to implement. In this regard, the goal of our study was to develop a numerical method for solving the Fredholm integral equations of the first and second kinds, which allow solving such problems without using regularization methods. The authors propose to solve problems of this type on the basis of the Bogolyubov – Krylov formula for representing certain integrals, in the form of finite-dimensional sums. At the same time, it becomes possible to use a non-uniform grid of nodes for splitting the integration interval. To efficiently define the set of nodes for splitting the integration interval, it is proposed to use a priori information about the properties of the solution of the problem. For example, the number of split points of the integration interval increases where the intended solution undergoes the most rapid changes, and decreases in areas where the intended solution is close to linear. The optimal arrangement of the splitting nodes of the integration interval makes it possible to increase the conditionality of the system of linear equations corresponding to the difference analogue of the problem and, thereby, to prevent the divergence of the iterative process. The article provides an example of calculating the optimal arrangement of the nodes splitting the integration interval when solving a test problem.