On the Collocation Method in Constructing a Solution to the Volterra Integral Equation of the Second Kind Using Chebyshev and Legendre Polynomials

Abstract

The paper proposes a matrix implementation of the collocation method for constructing a solution to Volterra integral equations of the second kind using systems of orthogonal Chebyshev polynomials of the first kind and Legendre polynomials. The integrand in the equations considered in this work is represented as a partial sum of a series for these polynomials. The roots of the Chebyshev and Legendre polynomials are chosen as collocation points. Using matrix and integral transformations, properties of finite sums of products of these polynomials and weight functions at the zeros of the corresponding polynomials with degree equal to the number of nodes, integral equations are reduced to systems of linear algebraic equations for unknown values of the sought functions at these points. As a result, solutions to Volterra integral equations of the second kind are found by polynomial interpolations of the obtained function values at collocation points using inverse matrices, the elements of which are written on the basis of orthogonal relations for these polynomials. In the presented work, the elements of integral matrices are also given in explicit form. Error estimates for the constructed solutions with respect to the infinite norm are obtained. The results of computational experiments are presented, which demonstrate the effectiveness of the collocation method used.

Keywords

• polynomial interpolation
• collocation method
• chebyshev polynomials
• legendre polynomials
• integral equations