The Method of the Integral Equations to Construct the Green's Function
Keywords:
linear boundary problem, Green's function, integral equationsAbstract
The Method of the Integral Equations to Construct the Green's Function
Linear differential operator and the system of the boundary conditions were considered. The boundary conditions are linear and linear independent functionals. The Green's function for the defined by this operator and functionals boundary problem was build as solution of the Fredholm's integral equation of the second kind. Characteristics of the Fredholm's equation was defined by the Green's function of the auxiliary problem. Resulting Green's function makes it possible to solve both direct (the problem of finding solutions) and inverse (the problem of finding the right part of the equation from the experimentally obtained solution) problems. The numerical algorithm to solve boundary problem and inverse problem was build on the basis of the proposed method and tested .
References
Shestakov, A. L. Dynamic Error Correction Method /A.L. Shestakov // IEEE Transactions on Instrumentation and Measurement. - 1996. - V. 45, № 1. - P. 250-255.
Сансоне, Дж. Обыкновенные дифференциальные уравнения, т.1 / Сансоне Дж. - М.: ИЛ, 1953.
Ильин, В.А. Нелокальная краевая задача для оператора Штурма-Лиувилля в дифференциальной и разностной форме /Ильин В.А., Моисеев Е.И. // ДАН СССР. - 1986. - Т. 291, № 3. - С. 534-539.
Тихомиров, В.М. Некоторые вопросы теории приближений / Тихомиров В.М. - М.: МГУ, 1976. - 305 с.
Zalyapin, V. Inverse problem of the measurements theory / Zalyapin V.I., Kharitonova H.V., Yermakov S.V. // Inverse problems, Design and Optimization Symposium, Miami, Florida, USA. - 2007. - P. 91-96.
Асфандиярова, Ю.С. Функция Грина линейной краевой задачи с нелокальными данными / Асфандиярова Ю.С., Заляпин В.И. // Тр. Мат. центра им. Н.И. Лобачевского / Казан. матем. общ-во, Казань. - 2009. - Т. 39. - С. 128.
Наймарк, М.А. Линейные дифференциальные операторы / М.А. Наймарк. - М.: Наука, 1969. - 528 с.
