An approximate solution of a hypersingular integral equation using Chebyshev series on a class of function bounded at one end and unbounded at the other end of the integration interval
Abstract:
Approximate methods for solving hypersingular integral equations are an actively developing field of computational mathematics. This is due to the numerous applications of hypersingular integral equations in various fields of mathematics and the fact that analytical solutions of such equations are possible only in exceptional cases. A method for finding a solution, limited at one end and unlimited at the other end of the integration interval [-1,1], of hypersingular integral equations of the first kind using Chebyshev series is proposed and justified. The core and the right side of the equation are also decomposed into Chebyshev series, the decomposition coefficients of which are calculated approximately using quadrature Gauss formulas with the highest degree of accuracy. The coefficients of decomposition of an unknown function into a Chebyshev series are found by solving systems of linear algebraic equations. Methods of functional analysis and the theory of orthogonal polynomials are used to substantiate the computational scheme. The Helder space of functions with corresponding norms is introduced. Given hypersingular and corresponding approximate operators are considered in this space. When the condition for the existence of derivatives up to a certain order belonging to the Helder class for given functions is fulfilled, the calculation error is estimated and the order of its tendency to zero is given.
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