Mathematical Researches

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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 Grünwald-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../files/site1/files/51/%DA%86%D8%B1%D8%A7%D8%BA%DB%8C.pdf

Keywords: Fractional advection-diffusion equation, Fractional Crank-Nicholson method, Shifted Grünwald-Letnikov fractional derivative, Preconditioned GMRES method

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