An approximate solution of singular integral equations using the Chebyshev series on the class of functions vanishing at one end and turning into infinity at the other end of the integration interval

Sh. S. Khubezhty

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 3, pp. 331-341

Résumé

Computational schemes for an approximate solution of a singular integral equation of the first kind, bounded at one end and not bounded at the other end of the integration interval are constructed $[-1,1]$. The solution of the equation is sought for in the form of a series in the Chebyshev polynomials of the third and the fourth kinds. The kernel and the right-hand side of the equation decompose into series with the use of the Chebyshev polynomials of the third and the fourth kinds, whose coefficients are approximately calculated by the Gaussian quadrature formulas, i.e. the highest algebraic degree of accuracy. For the coefficients of the expansion of the Chebyshev polynomials, exact values in the series are found. The coefficients of the expansion of the unknown function, i.e. solutions of the equation, are found from the solution to systems of linear algebraic equations. To justify the constructed computing schemes, the methods of functional analysis and the theory of orthogonal polynomials are used. When the existence condition for the given functions of the derivatives up to some order belonging to the Holder class is satisfied, the calculation error is estimated and the order of its turning into zero is given.