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.

scholarly journals On the Distributed Order Fractional Multi-Strain Tuberculosis Model: a Numerical Study

2020 ◽  
Vol 8 (1) ◽  
pp. 175-186
Author(s):  
Nasser Sweilam ◽  
S. M. AL-Mekhlafi ◽  
A. O. Albalawi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Difference Method ◽  
Numerical Study ◽  
Proposed Model ◽  
Fractional Differential ◽  
Distributed Order ◽  
Nonstandard Finite Difference

In this paper, a novel mathematical distributed order fractional model of multistrain Tuberculosis is presented. The proposed model is governed by a system of distributed order fractional differential equations, where the distributed order fractional derivative is defined in the sense of the Grünwald-Letinkov definition. A nonstandard finite difference method is proposed to study the resulting system. The stability analysis of the proposed model is discussed. Numerical simulations show that the nonstandard finite difference method can be applied to solve such distributed order fractional differential equations simply and eectively.

scholarly journals The Existence of Fourier Coefficients and Periodic Multiplicity Based on Initial Values and One-Dimensional Wave Limits Requirements

2020 ◽  
Vol 10 (2) ◽  
pp. 146
Author(s):  
Adi Jufriansah ◽  
Azmi Khusnani ◽  
Arief Hermanto ◽  
Mohammad Toifur ◽  
Erwin Prasetyo
Keyword(s):  
Finite Difference ◽  
Analytical Solution ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Fourier Analysis ◽  
Spectral Method ◽  
Difference Method ◽  
Hyperbolic Model ◽  
Solution Approach ◽  
The Finite Difference Method

Physical systems in partial differential equations can be interpreted in a visual form using a wave simulation. In particular, the interpretation of the differential equations used is in the nonlinear hyperbolic model, but in its completion, there are some limitations to the stability requirements found. The aim of this study is to investigate the analytical and numerical analysis of a wave equation with a similar unit and fractal intervals using the Fourier coefficient. The method in this research is to use the analytical solution approach, the spectral method, and the finite difference method. The hyperbolic wave equation's analytical solution approach, illustrated in the Fourier analysis, uses a pulse triangle. The spectral method minimizes errors when there is the addition of the same sample grid points or the periodic domain's expansion with a trigonometric basis. Meanwhile, different ways offer a more efficient solution. Based on the research results, the information obtained is that the Fourier analysis illustrates the pulse triangle use to solve the solution. These results are also suitable for adding sample points to the same spectra. Fourier analysis requires a relatively long time to solve one pulse triangle graph to need another solution, namely the finite difference method. However, its use is still limited in terms of stability when faced with more complex problems.

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