A spectral method for stochastic fractional differential equations

2019 ◽  
Vol 139 ◽  
pp. 115-119 ◽  
Author(s):  
Angelamaria Cardone ◽  
Raffaele D'Ambrosio ◽  
Beatrice Paternoster
Keyword(s):  
Differential Equations ◽  
Spectral Method ◽  
Fractional Differential Equations ◽  
Fractional Differential ◽  
Stochastic Fractional Differential Equations

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

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.

An averaging principle for neutral stochastic fractional order differential equations with variable delays driven by Lévy noise

2021 ◽  
Author(s):  
Guangjun Shen ◽  
Jiang-Lun Wu ◽  
Ruidong Xiao ◽  
Xiuwei Yin
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lévy Noise ◽  
Averaging Principle ◽  
Fractional Order Differential Equations ◽  
Variable Delays ◽  
Fractional Differential ◽  
Stochastic Fractional Differential Equations ◽  
Levy Noise ◽  
Convergence In Mean

In this paper, we establish an averaging principle for neutral stochastic fractional differential equations with non-Lipschitz coefficients and with variable delays, driven by Lévy noise. Our result shows that the solutions of the equations concerned can be approximated by the solutions of averaged neutral stochastic fractional differential equations in the sense of convergence in mean square. As an application, we present an example with numerical simulations to explore the established averaging principle.

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