Nonpolynomial Jacobi Spectral-Collocation Method for Weakly Singular Fredholm Integral Equations of the Second Kind (Advances in Applied Mathematics and Mechanics, 2024)
In this paper a nonpolynomial Jacobi spectral-collocation (NJSC) methodfor the second kind Fredholm integral equations (FIEs) with weakly singular kernel $|s−t|^{−γ}$ is proposed. By dividing the integral interval symmetrically into two partsand applying the NJSC method symmetrically to the two weakly singular FIEs respectively, the mild singularities of the interval endpoints can be captured and the exponential convergence can be obtained. A detailed $L^∞$ convergence analysis of the numericalsolution is derived. The NJSC method is then extended to two dimensional case andsimilar exponential convergence results are obtained for low regular solutions. Numerical examples are presented to demonstrate the efficiency of the proposed method.