Solving integral equations with logarithmic kernels by Chebyshev polynomials

Abstract

In this paper, a finite Chebyshev expansion is developed to solve Volterra integral equations with logarithmic singularities in their kernels. The error analysis is derived. Numerical results are given showing a marked improvement in comparison with the piecewise polynomial collocation method given in literature.Keywords Volterra integral equations · Integral equations with logarithmic kernels · Chebyshev polynomials · Error analysis and numerical approximation of solutions

A Product Integration Approach Based on New Orthogonal Polynomials for Nonlinear Weakly Singular Integral Equations

“…Reported results of them show that for N = 128 and 64 issued maximum errors of this problem are O(10 −6 ) and O(10 −9 ), respectively. Looking at Table 4, we can observe an improvement of the accuracy for N = 3 in the case of our method respect to methods in [5] and [10].…”

“…on the nodes coinciding with the zeroes of the mentioned orthogonal polynomials of the N th degree in addition to the endpoints x = 0 and 1) together with a product integration method described in section (3.1) has been implemented to the following test problems, taken from [5,10,24,25]:…”

“…1 and 2 the absolute errors of the test problems (5.2) and (5.3). The computational results have been reported in Tables 2 and 3, respectively. Brunner et al [5] and recently, Khater et al [10] have been solved the equation (5.4), by piecewise polynomial collocation method and Chebyshev polynomials expansion. Reported results of them show that for N = 128 and 64 issued maximum errors of this problem are O(10 −6 ) and O(10 −9 ), respectively.…”

“…We remark here that equations of this type have been the focus of many papers [5,7,10,17,22,24,25] in recent years.…”

“…Also, the piecewise polynomial collocation scheme has been proposed by Brunner et al in [5]. Recently, Khater et al [10] have been concerned with the method of one step ν-stage Chebyshev expansion for the logarithmic singularities of (1.1).…”