Методика наближеного розв’язання рівняння Фредгольма першого роду в деяких осесиметричних задачах механіки деформівного твердого тіла

Abstract

Проведено представлення наближеного розв’язку рівняння Фредгольма першого роду у вигляді полінома за ортогональними функціями. Проаналізовано можливість застосування варіаційної задачі з нерухомими кінцями та задачі поточкового зведення розв’язку до системи лінійних алгебраїчних рівнянь для визначення коефіцієнтів полінома. Отримано умову для вибору оптимальної кількості членів полінома-розв’язку.

While solving many applied tasks, axisymmetric contact problems of elasticity and thermoelasticity in particular, it arises a need to build an approximate solution of the Fredholm first kind equations. However, the problem of finding the solution of these equations is incorrect [1], since even small errors in the calculation of the right side of the equation or the core can result in distortion of the solution. For a long time reasonability of solving problems of this type was under question. However, with the appearance of various regularization methods it became possible to search approximate solutions of the Fredholm first kind equations [2-8]. Classical the Lavrentiev and Tikhonov methods are adapted to be used for finding approximate solutions of the Fredholm first kind equations that arise in solving of axisymmetric problems of fracture mechanics. Desired function as a linear combination of polynom by cylindrical functions with unknown coefficients was presented. Then the system for finding these unknown coefficients, and hence the approximation of the desired function was obtained. Besides, the conditions for choosing the optimal number of polynom solution members are obtained. However, the direct use of classical methods for solving of the Fredholm first kind equations in the applied problems of fracture mechanics becomes more complicated problem because of the necessity of finding the regularization parameter. In many cases, while solving axisymmetric contact problems of elasticity theory in particular [9], simpler approach can be used. After the unknown functions in a generalized Fourier series with unknown coefficients for orthogonal functions of a special type are presented, integrating of the left and right side of the equation are made. The systems of linear algebraic equations for finding of these coefficients are obtained. The characteristic feature of the obtained system is that the increase of the number of equations results in the increase of the accuracy of the approximation solution. It makes possible to build solutions with arbitrary predetermined accuracy.