Xiangcheng Zheng | Search arXiv

arXiv2026

High-order smoothness and high-order collocation approximation for Volterra integral equation with multiscale and nonlinear exponent kernel

We consider a Volterra integral equation with multiscale and nonlinear exponent kernel. We propose the high-order smoothing conditions, under which the initial singularity of the solutions can be eliminated up to any prescribed order. This indicates the application of the multiscale nature of the kernel on local modification of the solutions. Then a discontinuous high-order collocation method with arbitrary polynomial degree is developed and analyzed on uniform or graded meshes based on the solution regularity. This work serves as a comprehensive extension and complement to [Zheng, Qiu and Stynes, SIAM J. Numer. Anal., to appear] in both mathematical and numerical aspects.

math.NA↗

arXiv2026

Analysis and Approximation of Stochastic Multiscale Subdiffusion Driven by Fractional Gaussian Noise

This paper investigates a stochastic multiscale subdiffusion model driven by fractional Gaussian noise, where the multiscale Abel kernel with variable exponent $α(t)\in(0,1)$ is used to capture multiscale and crossover behavior in anomalous diffusion. The main difficulties of this model lie in the complexity of the multiscale Abel kernel (e.g. non-monotonicity and non-coercivity) and the low regularity caused by the noise. Concerning these issues, we prove the well-posedness and regularity of the mild solutions by means of solution operator approach and a perturbation technique for multiscale Abel kernel. Then both the semidiscrete-in-time and fully-discrete numerical schemes are proposed and analyzed under the low-regularity numerical analysis framework, with proved temporal and spatial convergence rates. Numerical experiments are presented to substantiate the theoretical results.

math.NA↗