Abstract:
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.
Keywords:system of linear algebraic equations, integral equations of the first kind, ill-posed problems, ill-conditioned problems, condition number, regularization method.
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