Time-discretization method for a multi-term time fractional differential equation with delay

TL;DR

The paper tackles the existence of strong solutions for a multi-term time-fractional delay differential equation in a real Hilbert space, of the form $\dfrac{d\vartheta(t)}{dt}+\sum_{q=1}^{k}a_q{\;^C{D}_t^{\alpha_q}}\vartheta(t)+A\vartheta(t)=f(t,\vartheta(t-\nu))$, with $0<\alpha_q<1$ and delay $\nu>0$. It extends Rothe's semi-discretization to this setting, using backward-Euler time stepping and a Caputo discretization, under assumptions (A1)-(A4)$ to obtain a priori estimates and convergence of the Rothe approximations. The main result proves the existence of a strong, Lipschitz continuous solution $\vartheta$ on $[-\nu,T_0]$ (with $T_0=\lfloor T/\nu\rfloor\nu$) that satisfies the original problem, plus an illustrative application to a multi-term fractional diffusion equation with delay. Overall, the work expands existing results from single-term to multi-term Caputo derivatives with delay in Hilbert spaces by providing a robust Rothe-based existence framework and establishing regularity of the solution.

Abstract

This paper discusses a multi-term time-fractional delay differential equation in a real Hilbert space. An iterative scheme for a multi-term time-fractional differential equation is established using Rothe's method. The method of semi-discretization is extended to this kind of time fractional problem with delay in the case that the time delay parameter $ν>0$ satisfies $ν\leq T$, where $T$ denotes the final time. We apply the accretivity of the operator $A$ in an iterative scheme to establish the existence and regularity of strong solutions to the considered problem. Finally, an example is provided to demonstrate the abstract result.

Theorems & Definitions (16)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Lemma 2.6
• Definition 2.7
• Lemma 3.2
• proof
• Lemma 3.3