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This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0,1). By providing an a priori estimate of the solution, we have established the existence and uniqueness of a numerical solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2 type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings.

The shifted fractional trapezoidal rule (SFTR) with a special shift is adopted to construct a finite difference scheme for the time-fractional Allen-Cahn (tFAC) equation. Some essential key properties of the weights of SFTR are explored for the first time. Based on these properties, we rigorously demonstrate the discrete energy decay property and maximum-principle preservation for the scheme. Numerical investigations show that the scheme can resolve the intrinsic initial singularity of such nonlinear fractional equations as tFAC equation on uniform meshes without any correction. Comparison with the classic fractional BDF2 and L2-1$_\sigma$ method further validates the superiority of SFTR in solving the tFAC equation. Experiments concerning both discrete energy decay and discrete maximum-principle also verify the correctness of the theoretical results.

In this paper, we study the mean-square stability of the solution and its stochastic theta scheme for the following stochastic differential equations drive by fractional Brownian motion with Hurst parameter $H\in (\frac 12,1)$: $$ dX(t)=f(t,X(t))dt+g(t,X(t))dB^{H}(t). $$ Firstly, we consider the special case when $f(t,X)=-\lambda\kappa t^{\kappa-1}X$ and $g(t,X)=\mu X$. The solution is explicit and is mean-square stable when $\kappa\geq 2H$. It is proved that if the parameter $2H\leq\kappa\le 3/2$ and $\frac{\sqrt{3/2}\cdot e}{\sqrt{3/2}\cdot e+1} (\approx 0.77)\leq \theta\leq 1$ or $\kappa>3/2$ and $1/2<\theta\le 1$, the stochastic theta method reproduces the mean-square stability; and that if $0<\theta<\frac 12$, the numerical method does not preserve this stability unconditionally. Secondly, we study the stability of the solution and its stochastic theta scheme for nonlinear equations. Due to the presence of long memory, even the problem of stability in the mean square sense of the solution has not been well studied since the conventional techniques powerful for stochastic differential equations driven by Brownian motion are no longer applicable. Let alone the stability of numerical schemes. We need to develop a completely new set of techniques to deal with this difficulty. Numerical examples are carried out to illustrate our theoretical results., Comment: 29 pages, 6 figures

In the current work we build a difference analog of the Caputo fractional derivative with generalized memory kernel ($_\lambda$L2-1$_\sigma$ formula). The fundamental features of this difference operator are studied and on its ground some difference schemes generating approximations of the second order in time for the generalized time-fractional diffusion equation with variable coefficients are worked out. We have proved stability and convergence of the given schemes in the grid $L_2$ - norm with the rate equal to the order of the approximation error. The achieved results are supported by the numerical computations performed for some test problems., Comment: arXiv admin note: text overlap with arXiv:1404.5221

In this paper, we consider a second-order scalar auxiliary variable (SAV) Fourier spectral method to solve the nonlinear fractional generalized wave equation. Unconditional energy conservation or dissipation properties of the fully discrete scheme are first established. Next, we utilize the temporal-spatial error splitting argument to obtain unconditional optimal error estimate of the fully discrete scheme, which overcomes time-step restrictions caused by strongly nonlinear system, or the restrictions that the nonlinear term needs to satisfy the assumption of global Lipschitz condition in all previous works for fractional undamped or damped wave equations. Finally, some numerical experiments are presented to confirm our theoretical analysis., Comment: 31pages, 9figures

The present paper is devoted to constructing L2 type difference analog of the Caputo fractional derivative. The fundamental features of this difference operator are studied and it is used to construct difference schemes generating approximations of the second and fourth order in space and the $(3-\alpha)$ th-order in time for the time fractional diffusion equation with variable coefficients. Stability of the schemes under consideration as well as their convergence with the rate equal to the order of the approximation error are proven. The received results are supported by the numerical computations performed for some test problems., Comment: 16 pages, 4 tables. arXiv admin note: substantial text overlap with arXiv:1404.5221

In this paper, we first establish the existence, uniqueness and H\"older continuity of the solution to stochastic Volterra integral equations with weakly singular kernels. Then, we propose a $\theta$-Euler-Maruyama scheme and a Milstein scheme to solve the equations numerically and we obtain the strong rates of convergence for both schemes in $L^{p}$ norm for any $p\geq 1$. For the $\theta$-Euler-Maruyama scheme the rate is $\min\{1-\alpha,\frac{1}{2}-\beta\}~ % (0<\alpha<1, 0< \beta<\frac{1}{2})$ and for the Milstein scheme the rate is $\min\{1-\alpha,1-2\beta\}$ when $\alpha\neq \frac 12$, where $(0<\alpha<1, 0< \beta<\frac{1}{2})$. These results on the rates of convergence are significantly different from that of the similar schemes for the stochastic Volterra integral equations with regular kernels. The difficulty to obtain our results is the lack of It\^o formula for the equations. To get around of this difficulty we use instead the Taylor formula and then carry a sophisticated analysis on the equation the solution satisfies.

This paper presents and analyzes the compensated projected Euler-Maruyama method for stochastic differential equations with jumps under a global monotonicity condition. Compared with existing conditions, this condition allows the jump-diffusion coefficient to be growth superlinearly. Moreover, the method is proved to be convergent with strongly order $\frac{1}{2}$ on the discrete time level. Finally, some numerical experiments are carried out to confirm the theoretical results., Comment: there are some mistakes. I need to correct them

This paper investigates projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition. This condition admits some equations with highly nonlinear drift and diffusion coefficients. We appropriately generalized the idea of C-stability and B-consistency given by Beyn et al. [J. Sci. Comput. 67 (2016), no. 3, 955-987] to the case with delay. Moreover, the method is proved to be convergent with order $\frac{1}{2}$ in a succinct way. Finally, some numerical examples are included to illustrate the obtained theoretical results.

We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirichlet (fractional and classical, respectively) and Neumann (fractional) boundary conditions. In particular, we develop a spectral penalty method (SPM) by using the Jacobi poly-fractonomial approximation for the conservative R-L FDEs while using the polynomial approximation for the conservative Caputo FDEs. We establish the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems. This analysis allows us to estimate the proper values of the penalty parameters at boundary points. We present several numerical examples to verify the theory and demonstrate the high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we compare the results against a Petrov-Galerkin spectral tau method (PGS-$\tau$, an extension of [Z. Mao, G.E. Karniadakis, SIAM J. Numer. Anal., 2018]) and demonstrate the superior accuracy of SPM for all cases considered., Comment: 27 pages

This paper proposes and analyzes an efficient difference scheme for the nonlinear complex Ginzburg-Landau equation involving fractional Laplacian. The scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted Gr\"unwald difference operator for the spatial fractional Laplacian. By virtue of a careful analysis of the difference operator, some useful inequalities with respect to suitable fractional Sobolev norms are established. Then the numerical solution is shown to be bounded, and convergent in the $l^2_h$ norm with the optimal order $O(\tau^2+h^2)$ with time step $\tau$ and mesh size $h$. The a priori bound as well as the convergence order hold unconditionally, in the sense that no restriction on the time step $\tau$ in terms of the mesh size $h$ needs to be assumed. Numerical tests are performed to validate the theoretical results and effectiveness of the scheme., Comment: 25 pages, 6 figures, 2 tables