A M\"untz-collocation spectral method for weakly singular Volterra delay-integro-differential equations

TL;DR

This paper introduces a M"untz spectral collocation method for solving weakly singular Volterra delay-integro-differential equations, providing error estimates and demonstrating exponential decay of errors in certain cases.

Contribution

The paper develops a novel M"untz spectral collocation scheme specifically designed for weakly singular VDIEs with delays, including rigorous error analysis and convergence proofs.

Findings

Method effectively handles weak singularities at initial point

Numerical errors decay exponentially in specific scenarios

Convergence analysis supported by multiple examples

Abstract

A M\"untz spectral collocation method is implemented for solving weakly singular Volterra integro-differential equations (VDIEs) with proportional delays. After constructing the numerical scheme to seek an approximate solution, we derive error estimates in a weighted $L^2$ and $L^{\infty}$-norms. A rigorous proof reveals that the proposed method can handle the weak singularity of the exact solution at the initial point $t=0$, with the numerical errors decaying exponentially in certain cases. Moreover, several examples will illustrate our convergence analysis.