An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data

First Published in SIAM Journal on Numerical Analysis (SINUM), 56(1), 2018, published by the Society of Industrial and Applied Mathematics (SIAM). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin \et (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197-221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has convergence rate $O(k^{2-\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results

Malware propagation model of fractional order, optimal control strategy and simulations

In this paper, an improved SEIR model of fractional order is investigated to describe the behavior of malware propagation in the wireless sensor network. Firstly, the malware propagation model of fractional order is established based on the classical SEIR epidemic theory, the basic reproductive number is obtained by the next-generation method and the stability condition of the model is also analyzed. Then, the inverse approach for the uncertainty propagation based on the discrete element method and least square algorithm is presented to determine the unknown parameters of the propagation process. Finally, the optimal control strategy is also discussed based on the adaptive model. Simulation results show the proposed model works better than the propagation model of integer order. The error of proposed model is smaller than integer order models

L1 scheme for solving an inverse problem subject to a fractional diffusion equation

This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle $< \pi/2$, that is the resolvent set of this operator contains $\{z\in\mathbb C\setminus\{0\}:\ |\operatorname{Arg} z|< \theta\}$ for some $\pi/2 < \theta < \pi$. The relationship between the time fractional order $\alpha \in (0, 1)$ and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as $\alpha$ approaches $1$. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results