A numerical method for solving ill-posed problems for two-dimensional integral equations of the first kind

Abstract

Propose a method of numerical solution integral equations of the first kind. This assumes that the source data in the equation can be set approximately with errors, and the integral equation kernel may have features that leads to the formulation of so-called ill-posed problems. This in turn requires the construction of regularizing algorithms in accordance with the theory of solving ill-posed problems of computational mathematics. The solution to this problem is of primary importance in computational mathematics, hence its relevance. The paper describes the formulation of the problem, highlighting the existing problems that need to be resolved, developed and validated a suitable numerical method. It uses the theoretical bases and methods of functional analysis, computational mathematics, theory of solution of incorrectly formulated problems of the theory of calculus of variations and optimization methods. The work also examines the possibility of applying this method to the solution of the corresponding three-dimensional problem. When building a computing method performed, we state the corresponding variational problem, which is then solved by the method of steepest descent, the construction of the regularizing algorithm. In the end, the developed numerical method and algorithm allow to obtain a stable solution of the original problem taking into account the errors in the original data, which corresponds to a practical situation simulation of processes in specific applications.

Keywords

• интегральное уравнение
• сингулярный интеграл
• метод регуляризации
• обобщенный обратный оператор
• численные методы
• вычислительный алгоритм
• integral equation
• singular integral regularization method
• generalized inverse of an operator
• numerical methods
• computational algorithm