In this paper, we study the analytical solutions of two-dimensional fractional-order linear system ${\mathcal{D}}_t^{\alpha }X(t)=AX(t)$ described by fractional differential equations, where $\mathcal{D}$ is the fractional derivative in the Caputo-Fabrizio sense and $A=(a_{ij}{)}_{2\times 2}$ is nonsingular coefficient matrix with $a_{ij}\in \mathbb{R}$. The analytical solutions of fractional-order linear system will be compared to the solution of classical linear system. Examples are provided to characterize the behavior of the solutions for fractional-order linear system.

This study develops a finite difference/local discontinuous Galerkin (LDG) framework for solving a fourth-order fractional diffusion-wave equation. The temporal fractional operator is approximated through a finite difference approach, achieving a truncation accuracy of $O((\mathrm{\Delta }t{)}^{3-\alpha })$, where $\mathrm{\Delta }t$ denotes the time increment and $\alpha$ represents the fractional order. For spatial discretization, LDG technique is employed, which leads to a fully implicit discrete formulation of the considered model. By applying mathematical induction, we establish the unconditional stability and convergence of the proposed algorithm. A series of computational experiments is presented to verify the theoretical error bounds.

In this paper, an effective numerical method for the variable-order(VO) fourth-order problem with Caputo-Fabrizio derivative will be constructed and analyzed. Based on generalized alternating numerical flux, appropriate spatial and temporal discretization, we get a fully discrete local discontinuous Galerkin(LDG) scheme. The theoretic properties of the fully discrete LDG scheme are proved in detail by mathematical induction, and the method is proved to be unconditionally stable and convergent with $\mathrm{O}(\tau +h^{k+1})$, where $h$ is the spatial step, $\tau$ is the temporal step and $k$ is the degree of the piecewise $P^k$ polynomial. In order to show the efficiency of our method, some numerical examples are carried out by Matlab.

In this paper, an implicit compact finite difference (CFD) scheme was constructed to get the numerical solution for time fractional diffusion-wave equation (TFDWE), in which the time fractional derivative was denoted by Caputo-Fabrizio (C-F) sense. We proved that the full discrete scheme is unconditionally stable. We also proved that the rate of convergence in time is near to $O({\tau }^2)$ and the rate of convergence in space is near to $O(h^4)$. Test problem was considered for regular domain with uniform points to validate the efficiency and accuracy of the method. The numerical results can support the theoretical claims.