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

Karolina Pawlak

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.

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

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.

Paper Structure

This paper contains 6 sections, 11 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimates for kernels E and Z
  4. Jump relation for integral operator corresponding to kernel E
  5. Continuity of the integral operator corresponding to the kernel Z
  6. Acknowledgments

Key Result

Theorem 1

Let $t^{1-\alpha_m}\varphi(t) \in C[0,T]$ and we assume that there exists $\beta>\frac{\alpha_m}{2}$ such that $|s(t)- s(\tau)|\leq |t- \tau|^{\beta}$ for $t, \tau \in [0, T]$. Then for each $t\in (0,T]$ there holds

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2: luchko, p. 3
  • Lemma 1: pskhu2, § 2, § 3, MFbook, equality (7.2.3)
  • Remark 1: pskhu4, Remark 1
  • Remark 2
  • Remark 3
  • Definition 3: pskhu3, § 2
  • Theorem 3: see pskhu3, Theorem 1
  • ...and 15 more