Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods

The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods for fractional differential equations. The construction of the methods together with the convergence and stability analysis are reported and some numerical experiments are carried out to show the efficiency of the proposed methods.

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SPECTRAL COLLOCATION METHOD FOR MULTI-ORDER FRACTIONAL DIFFERENTIAL EQUATIONS

International Journal of Computational Methods ◽

10.1142/s0219876213500722 ◽

2014 ◽

Vol 11 (05) ◽

pp. 1350072 ◽

Cited By ~ 8

Author(s):

F. GHOREISHI ◽

P. MOKHTARY

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Collocation Method ◽

Fractional Differential Equations ◽

Jacobi Polynomials ◽

Transformation Method ◽

Spectral Collocation Method ◽

Spectral Collocation ◽

Fractional Differential ◽

New Equation

In this paper, the spectral collocation method is investigated for the numerical solution of multi-order Fractional Differential Equations (FDEs). We choose the orthogonal Jacobi polynomials and set of Jacobi Gauss–Lobatto quadrature points as basis functions and grid points respectively. This solution strategy is an application of the matrix-vector-product approach in spectral approximation of FDEs. The fractional derivatives are described in the Caputo type. Numerical solvability and an efficient convergence analysis of the method have also been discussed. Due to the fact that the solutions of fractional differential equations usually have a weak singularity at origin, we use a variable transformation method to change some classes of the original equation into a new equation with a unique smooth solution such that, the spectral collocation method can be applied conveniently. We prove that after this regularization technique, numerical solution of the new equation has exponential rate of convergence. Some standard examples are provided to confirm the reliability of the proposed method.

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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.

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