"A P-version of the collocation method for solving the Fredholm integral equations of the second kind in the Mathematica environment"
Shapeev V.P. and Vorozhtsov E.V.

A p-version of the collocation method for the numerical solution of Fredholm integral equations of the second kind is proposed and implemented. In the considered implementation, the possibilities are realized for the variation of the polynomial degree in the polynomial representation of the approximate solution of equations and the variation of the number of nodes of the employed Gauss quadrature formula to affect the solution accuracy. The influence of the number of collocation points used for the solution approximation and of the number of nodes of the Gauss quadrature formula on the condition number of the system of linear algebraic equations to the solution of which the construction of the approximate solution is reduced and on its accuracy are studied by the numerical solution of examples, including some examples presented in well-known publications. The proposed algorithm is implemented in the language of the program package Mathematica. In all considered examples, the proposed version of the collocation method has enabled us to reach the accuracy of the solution of equations, which is close to the level of the machine rounding errors. The program product implementing the proposed p-version has proved to be compact and the method turned out to be economical: the machine time required for the solution of problems considered in the paper did not exceed 3 seconds of the CPU time of a personal computer. We describe an algorithm allowing us to estimate the accuracy of the approximate solution obtained by the proposed p-version of the method in the cases where the exact solution of the integral equation is unknown.

Keywords: Fredholm integral equation of the second kind, collocation method, condition number, Gauss quadrature.

  • Shapeev V.P. – Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of Russian Academy of Sciences; Institutskaya ulitsa 4/1, Novosibirsk, 630090, Russia; Dr. Sci., Professor, Principal Scientist, e-mail: Shapeev@itam.nsc.ru
    Novosibirsk National Research University, Faculty of Mechanics and Mathematics; ulitsa Pirogova 2, Novosibirsk, 630090, Russia; Dr. Sci., Professor, vshapeev@ngs.ru
  • Vorozhtsov E.V. – Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of Russian Academy of Sciences; Institutskaya ulitsa 4/1, Novosibirsk, 630090, Russia; Dr. Sci., Professor, Leading Scientist, e-mail: vorozh@itam.nsc.ru