Method of Solving a Singular Dynamic Problem in the Integral Equation Form

• Видавець: Кам'янець-Подільський національний університет імені Івана Огієнка
• Дата: 2018
• Автори: Верлань, Анатолий Федорович, Фуртат, Юрий Олегович
• Формат: Стаття
• Мова: rus
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2018
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/159381

Volterra integral equations of the second kind are a universal mathematical model used in problems of identification and computer simulation. At the same time, the singularity of these equations makes it difficult to solve these problems. To solve this problem, regularization algorithms for ill-posed problems are used. In this case, the regularization parameter can be determined in various ways, in particular, by the method of model examples. The article also shows how to solve the obtained approximate expression from the regularization algorithm using quadrature formulas.The problem of determining the error of solving the second-kind Volterra integral equations based on the quadrature formula method is also considered. The evaluation of the error is carried out by proving the corresponding theorem and its consequences. One of the consequences of the theorem on limitness of error states that, for an infinitely small value of the regularization parameter, the error of the solution also tends to zero. This statement is also proved in the article with the demonstration of computations and calculations.A final expression is given for evaluating the errors in solving the Volterra in­tegral equations of the second kind using regularization methods and quadratu­re formulas, and it is concluded that the proposed methods allow one to overco­me the problem of singularity in Volterra integral equations of the second kind.