Weak Singularity Modulation in Discrete Solvers for Volterra-Type Integral Equations

Abstract

Weakly singular kernels in Volterra-type integral equations appear in viscoelasticity applications and anomalous diffusion, and in fractional-order control. When the kernel has weak singularities, particularly near the initial time, numerical solvers usually become less accurate and less stable. The paper presents a unified framework known as Weak Singularity Modulation, which enhances classical methods such as convolution quadrature, product-integration, and collocation. The system design uses steady-state weights, compact start-up correctors, time mesh that grows in size to get convergence stable. Through thorough investigation, it is shown that WSM recovers the optimal accuracy predicted by local asymptotic while being robust to both linear and nonlinear problems. Examples with solutions that are known show how fast the method improves the solution. The presented methodology makes little to no impact on computational effort. Additionally, it easily integrates with developed or existing solvers. Thus, large-scale simulations are possible in a fractional modeling of practical applications.