Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

Abstract

A spectral Jacobi-collocation approximation is proposed for Volterra delay integro-differential equations with weakly singular kernels. In this paper, we consider the special case that the underlying solutions of equations are sufficiently smooth. We provide a rigorous error analysis for the proposed method, which shows that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially in L ∞ norm and weighted L 2 norm. Finally, two numerical examples are presented to demonstrate our error analysis.

Spectral Collocation Methods for Fractional Integro‐Differential Equations with Weakly Singular Kernels

“…Lemma 3 (see [26]). Assume that v ∈ H m ω α,β (− 1, 1) and denote by I α,β N v(x) its interpolation polynomial associated with the Jacobi-Gauss points…”

“…For nonnegative integer r and κ ∈ (0, 1), we denote by C r,κ [− 1, 1] the set of continuous function v: [− 1, 1] ⟶ R whose r-th derivatives are H o .. lder continuous with exponent κ, equipped with norm ‖ • ‖ r,κ (see [26]). From [31,32], we know that there exists a constant C r,κ > 0 such that for any function v ∈ C r,κ [− 1, 1], there exists a polynomial function…”

Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

“…The Legendre spectral collocation method was proposed in for linear VIEs or VIDEs with smooth kernels. In , the Jacobi spectral collocation method was successfully applied to solve linear VIEs or VIDEs with weakly kernels. The authors in developed spectral methods to solve nonlinear VDEs.…”