In this paper, a pseudo-spectral method with the Lagrange polynomial basis is proposed to solve the TIME-FRACTIONAL advection-DIFFUSION EQUATION. A semi-discrete approximation scheme is used for conversion of this EQUATION to a system of ordinary FRACTIONAL differential EQUATIONs. Also, to protect the high accuracy of the spectral approximation, the Mittag-Leffler function is used for the integration along the TIME variable. Some examples are performed to illustrate the accuracy and efficiency of the proposed method.

In this paper, we present a well-organized strategy to estimate the FRACTIONAL advection-DIFFUSION EQUATIONs, which is an important class of EQUATIONs that arises in many application fields. Thus,  Lagrange square interpolation is applied in the discretization of the FRACTIONAL temporal derivative, and the weighted and shifted Legendre polynomials as operators are exploited to discretize the spatial FRACTIONAL derivatives of the space-FRACTIONAL term in multi-termTIME FRACTIONAL advection-DIFFUSION model. The privilege of the numerical method is the orthogonality of Legendre polynomials and its operational matrices which reduces TIME computation and increases speed. A second-order implicit technique is given, and its stability and convergence are investigated. Finally, we propose three numerical examples to check the validity and numerical results    to illustrate the precision and efficiency of the new approach.

In this study, one explicit and one implicit finite difference scheme is introduced for the numerical solution of TIME-FRACTIONAL Riesz space DIFFUSION EQUATION. The TIME derivative is approximated by the standard Gr¨, unwald Letnikov formula of order one, while the Riesz space derivative is discretized by Fourier transform-based algorithm of order four. The stability and convergence of the proposed methods are studied. It is proved that the implicit scheme is unconditionally stable, while the explicit scheme is stable conditionally. Some examples are solved to illustrate the efficiency and accuracy of the proposed methods. Numerical results confirm that the accuracy of present schemes is of order one.

FRACTIONAL order DIFFUSION EQUATIONs are generalizations of classical DIFFUSION EQUATIONs which are used to model in physics, finance, engineering, etc. In this paper we present an implicit difference approximation by using the alternating directions implicit (ADI) approach to solve the two-dimensional space-TIME FRACTIONAL DIFFUSION EQUATION (2DSTFDE) on a finite domain. Consistency, unconditional stability, and therefore first-order convergence of the method are proven. Some numerical examples with known exact solution are tested, and the behavior of the errors are analyzed to demonstrate the order of convergence of the method.

The ultimate goal of this performance study is to provide a  proposed scheme for solving the TIME-FRACTIONAL stochastic advection-DIFFUSION EQUATION (TFSADE) of order $\alpha (0\le \alpha <1)$. In this proposed scheme, we utilize an approach based on cubic trigonometric B-spline collocation methods (CTBSCM).  In this study, we replace the existing FRACTIONAL derivative with the FRACTIONAL Caputo derivative for TIME discretization and then replace the first and second derivatives of the EQUATION using cubic trigonometric B-spline functions for spatial discretization. Applying this proposed scheme to TFSADE causes the EQUATION to reduce to the linear system. In the end, the examples show that the order of convergence of the proposed method is $O(\tau ^{2-\alpha}+h^2)$ where $h$ and $\tau$  are the spatial and TIME step lengths, respectively.

In this paper, a modification of the finite integration method (FIM) is combined with the radial basis function (RBF) method to solve a TIME-FRACTIONAL convection DIFFUSION EQUATION with variable coefficients. The FIM transforms partial differential EQUATIONs into integral EQUATIONs and this creates some constants of integration. Unlike the usual FIM, the proposed method computes constants of integration by using initial conditions. This leads to fewer computations rather than the standard FIM. Also, a product Simpson method is used to overcome the singularity included in the definition of FRACTIONAL derivatives, and an integration matrix is obtained. Some numerical examples are provided to show the efficiency of the method. In addition, a comparison is made between the proposed method and the previous ones.

In this paper, we apply a local discontinuous Galerkin (LDG) method to solve some FRACTIONAL inverse problems. In fact, we determine a TIMEdependent source term in an inverse problem of the TIME-FRACTIONAL DIFFUSION EQUATION. The method is based on a finite difference scheme in TIME and a LDG method in space. A numerical stability theorem as well as an error estimate is provided. Finally, some numerical examples are tested to confirm theoretical results and to illustrate effectiveness of the method. It must be pointed out that proposed method generates stable and accurate numerical approximations without using any regularization methods which are necessary for other numerical methods for solving such ill-posed inverse problems.

In this paper, a new and effective optimization algorithm is proposed for solving the nonlinear TIME FRACTIONAL convection-DIFFUSION EQUATION with the concept of variable-order FRACTIONAL derivative in the Caputo sense. For finding the solution, we first introduce the generalized polynomials (GPs) and construct the variable-order operational matrices. In the proposed optimization technique, the solution of the problem under consideration is expanded in terms of GPs with unknown free coefficients and control parameters. The main advantage of the presented method is to convert the variable-order FRACTIONAL partial differential EQUATION to a system of nonlinear algebraic EQUATIONs. Also, we obtain the free coefficients and control parameters optimally by minimizing the error of the approximate solution. Finally, the numerical examples confirm the high accuracy and efficiency of the proposed method in solving the problem under study.