The Riesz FRACTIONAL advection-DIFFUSION is a result of the mechanics of chaotic dynamics. It’, s of preponderant importance to solve this EQUATION numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical EQUATIONs shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [−, 1, 1] × [−, 1, 1]. Then, we use the operational matrix to discretize FRACTIONAL operators. Applying the resulting discretization, we obtain a linear system of EQUATIONs, which leads to the numerical solution. Examples show the effectiveness of the method.

In this work, we apply the radial basis functions for solving the time FRACTIONAL DIFFUSION-wave EQUATION defined by Caputo sense for (1<a£2). The problem is discretized in the time direction based on finite difference scheme and is continuously approximated by using the radial basis functions in the space direction which achieves the semi-discrete solution. Numerical results show the accuracy and efficiency of the presented method.

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.

This paper is concerned with numerical approach for solving space FRACTIONAL DIFFUSION EQUATION using shifted Gegenbauer polynomials, where the FRACTIONAL derivatives are expressed in Caputo sense. The properties of Gegenbauer polynomials are exploited to reduce space FRACTIONAL DIFFUSION EQUATION to a system of ordinary differential EQUATIONs, that are then solved using finite difference method. Some selected numerical simulations of space FRACTIONAL DIFFUSION EQUATIONs are presented and the results are compared with the exact solution, also with the results obtained via other methods in the literature. The comparison reveals that the proposed method is reliable, effective and accurate. All the computations were carried out using Matlab package.

Introduction: FRACTIONAL differential EQUATIONs (FDEs) have attracted much attention and have been widely used in the fields of finance, physics, image processing, and biology, etc. It is not always possible to find an analytical solution for such EQUATIONs. The approximate solution or numerical scheme may be a good approach, particularly, the schemes in numerical linear algebra for solving a system, , emerging by discretizing the partial derivatives, with large and sparse dimensions. In the procedure of solving a specified FDE, if the dimension of the corresponding system of linear EQUATIONs is small, one can use the direct methods or the classical iterative methods for the analysis of these systems. However, if the dimension is large, then the proposed methods are not effective. In this case, we use variants of the Krylov subspace methods that are more robust with respect to the computer memory and time. The GMRES (Generalized Minimal Residual) is a well-known method based on Krylov subspace that is used to solve a system of sparse linear EQUATIONs with an non-symmetric matrix. A main drawback of iterative methods is the slowness of convergence rate which depends on the condition number of the corresponding coefficient matrix. If the condition number of the coefficient matrix is small, then the rate of convergence will be faster. So, we try to convert the original system to another equivalent system, in which the condition number of its coefficient matrix becomes small. A preconditioner matrix is a matrix that performs this transformation. In this paper, we propose the iterative GMRES method, preconditioned GMRES method and examine capability of these methods by solving the space FRACTIONAL advection-DIFFUSION EQUATION. Material and methods: We first introduce a space FRACTIONAL advection-DIFFUSION EQUATION in the sense of the shifted Grü nwald-Letnikov FRACTIONAL derivative. To improve the introduced numerical scheme, we discretize the partial derivatives of EQUATION using the FRACTIONAL Crank-Nicholson finite difference method. Then we use a preconditioner matrix and present preconditioned GMRES method for solving the derived linear system of algebraic EQUATIONs. Results and discussion: In this paper, we use the GMRES and preconditioned GMRES to solve a linear system of EQUATIONs emerging by discretizing partial derivatives appearing in a Advection-DIFFUSION EQUATION and then asset the accuracy of these methods. Numerical results indicate that we derive a smaller condition number of the equivalent coefficient matrix for different values of M and N, as dimensions of the corresponding linear EQUATIONs. Hence the convergence rate increases and consequently the number of iterations and the calculation time decreases. Conclusion: The following conclusions were drawn from this research. The GMRES method is a Kyrlov subspace methods to solve large-dimensions non-symmetric system of linear EQUATIONs, which will be more effective when is applied with preconditioning techniques. One of the common ways to increase the rate of convergence of iterative methods based on the Kyrlov subspace is the applying the preconditioned techniques. An appropriate preconditioner matrix increases the rate of convergence of the iterative 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, the homotopy perturbation transform method (HPTM) is performed to give analytical solutions of the time FRACTIONAL DIFFUSION EQUATION. The HPTM is a combined form of the Laplace transform and homotopy perturbation methods. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is very effective and simple in performing a solution to the FRACTIONAL partial differential EQUATION. A solution has been plotted for different values of a, and some numerical illustrations are given.