A remark on a special class of Volterra–Fredholm integral equations

We propose a global method based on Lagrange interpolation at Jacobi nodes to approximate the solution of second-kind Volterra–Fredholm integral equations on the interval $[-1,1]$. The considered equations involve kernels with a multiple zero at the origin and Jacobi-type weight functions, allowing for algebraic endpoint singularities in the data. Accordingly, the problem is studied in suitable weighted spaces of continuous functions.

We prove the stability and convergence of our numerical method and derive explicit a priori error estimates in the weighted uniform norm, showing that the approximation essentially achieves the convergence rate of the best weighted polynomial approximation, up to an additional logarithmic factor $logm$. Some numerical experiments are presented to illustrate the accuracy and efficiency of the proposed approach.