A Pseudo-spectral Method for Time Distributed Order Two-sided Space 
		Fractional Differential Equations

AbstractFractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on FGLFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

Download Full-text

Stability Analysis of Distributed Order Fractional Differential Equations

Abstract and Applied Analysis ◽

10.1155/2011/175323 ◽

2011 ◽

Vol 2011 ◽

pp. 1-12 ◽

Cited By ~ 12

Author(s):

H. Saberi Najafi ◽

A. Refahi Sheikhani ◽

A. Ansari

Keyword(s):

Stability Analysis ◽

Characteristic Function ◽

Differential Equations ◽

Density Function ◽

Robust Stability ◽

Fractional Differential Equations ◽

Stability Condition ◽

Fractional Differential ◽

The Stability ◽

Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

Download Full-text

A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations

Journal of Algorithms & Computational Technology ◽

10.1177/17483026211008409 ◽

2021 ◽

Vol 15 ◽

pp. 174830262110084

Author(s):

Xianjuan Li ◽

Yanhui Su

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Differential Equations ◽

Spectral Method ◽

Fractional Differential Equations ◽

Difference Method ◽

Spectral Collocation ◽

Parallel Method ◽

Fractional Differential ◽

Time Fractional Differential Equations

In this article, we consider the numerical solution for the time fractional differential equations (TFDEs). We propose a parallel in time method, combined with a spectral collocation scheme and the finite difference scheme for the TFDEs. The parallel in time method follows the same sprit as the domain decomposition that consists in breaking the domain of computation into subdomains and solving iteratively the sub-problems over each subdomain in a parallel way. Concretely, the iterative scheme falls in the category of the predictor-corrector scheme, where the predictor is solved by finite difference method in a sequential way, while the corrector is solved by computing the difference between spectral collocation and finite difference method in a parallel way. The solution of the iterative method converges to the solution of the spectral method with high accuracy. Some numerical tests are performed to confirm the efficiency of the method in three areas: (i) convergence behaviors with respect to the discretization parameters are tested; (ii) the overall CPU time in parallel machine is compared with that for solving the original problem by spectral method in a single processor; (iii) for the fixed precision, while the parallel elements grow larger, the iteration number of the parallel method always keep constant, which plays the key role in the efficiency of the time parallel method.

Download Full-text