TY - JOUR

T1 - Метод моментов в задаче обращения преобразования Лапласа и его регуляризация

AU - Рябов, Виктор Михайлович

AU - Лебедева, Анастасия Владимировна

PY - 2022

Y1 - 2022

N2 - Integral equations of the first kind are considered, which belong to the class of ill-posed problems. This also includes the problem of inverting the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to illconditioned systems of linear algebraic equations, in which the unknowns are the coefficients of the expansion in a series in the shifted Legendre polynomials of some function that simply expresses in terms of the sought original. This function is found as a solution to a certain finite moment problem in a Hilbert space. To obtain a reliable solution to the system, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications. A specific type of stabilizer in the regularization method is indicated, focused on a priori low degree of smoothness of the desired original. The results of numerical experiments are presented, confirming the effectiveness of the proposed inversion algorithm.

AB - Integral equations of the first kind are considered, which belong to the class of ill-posed problems. This also includes the problem of inverting the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to illconditioned systems of linear algebraic equations, in which the unknowns are the coefficients of the expansion in a series in the shifted Legendre polynomials of some function that simply expresses in terms of the sought original. This function is found as a solution to a certain finite moment problem in a Hilbert space. To obtain a reliable solution to the system, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications. A specific type of stabilizer in the regularization method is indicated, focused on a priori low degree of smoothness of the desired original. The results of numerical experiments are presented, confirming the effectiveness of the proposed inversion algorithm.

UR - https://www.mendeley.com/catalogue/3febb0f6-8ec5-3072-aba4-8236f167b987/

U2 - 10.21638/spbu01.2022.105

DO - 10.21638/spbu01.2022.105

M3 - статья

VL - 9(67)

SP - 46

EP - 52

JO - ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. МАТЕМАТИКА. МЕХАНИКА. АСТРОНОМИЯ

JF - ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. МАТЕМАТИКА. МЕХАНИКА. АСТРОНОМИЯ

SN - 1025-3106

IS - 1

ER -