Boundary behaviour of potential-type integrals for the multi-term time-fractional diffusion equation

TL;DR

This paper analyzes the boundary behavior of potential-type integrals for the multi-term time-fractional diffusion equation, establishing jump and continuity relations crucial for boundary integral equation analysis in time-dependent domains.

Contribution

It extends boundary behavior analysis to the complex multi-term fractional diffusion case, overcoming challenges due to the fundamental solution's structure.

Findings

Established jump relation for the integral operator.

Proved continuity of the integral operator.

Addressed complexities due to non-standard scaling.

Abstract

This paper investigates the boundary behaviour of potential-type integrals for the multi-term time-fractional diffusion equation (MTFDE) across the moving boundary. First, we establish the jump relation for the integral operator associated with the fundamental solution of the inhomogeneous MTFDE. Second, we prove the continuity of the integral operator generated by the kernel corresponding to the homogeneous MTFDE. Krasnoschok obtained similar results for the time-fractional diffusion equation. However, in the multi-term case, the fundamental solution has more complex structure and does not admit standard scaling properties, which requires a different approach. Our results are essential for the analysis of boundary integral equations related to the MTFDE in time-dependent domains.