Volume 12, Issue 1
A Jacobi Collocation Method for the Fractional Ginzburg-Landau Differential Equation

Adv. Appl. Math. Mech., 12 (2020), pp. 57-86.

Published online: 2019-12

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• Abstract

In this paper, we design a collocation method to solve the fractional Ginzburg-Landau equation. A Jacobi collocation method is developed and implemented in two steps. First, we space-discretize the equation by the Jacobi-Gauss-Lobatto collocation (JGLC) method in one- and two-dimensional space. The equation is then converted to a system of ordinary differential equations (ODEs) with the time variable based on JGLC. The second step applies the Jacobi-Gauss-Radau collocation (JGRC) method for the time discretization. Finally, we give a theoretical proof of convergence of this Jacobi collocation method and some numerical results showing the proposed scheme is an effective and high-precision algorithm.

35R35, 65M12, 65M70

yangyinxtu@xtu.edu.cn (Yin Yang)

szhang@udel.edu (Shangyou Zhang)

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@Article{AAMM-12-57, author = {Yang , YinTao , JianyongZhang , Shangyou and V. Sivtsev , Petr}, title = {A Jacobi Collocation Method for the Fractional Ginzburg-Landau Differential Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {12}, number = {1}, pages = {57--86}, abstract = {

In this paper, we design a collocation method to solve the fractional Ginzburg-Landau equation. A Jacobi collocation method is developed and implemented in two steps. First, we space-discretize the equation by the Jacobi-Gauss-Lobatto collocation (JGLC) method in one- and two-dimensional space. The equation is then converted to a system of ordinary differential equations (ODEs) with the time variable based on JGLC. The second step applies the Jacobi-Gauss-Radau collocation (JGRC) method for the time discretization. Finally, we give a theoretical proof of convergence of this Jacobi collocation method and some numerical results showing the proposed scheme is an effective and high-precision algorithm.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0070}, url = {http://global-sci.org/intro/article_detail/aamm/13419.html} }
TY - JOUR T1 - A Jacobi Collocation Method for the Fractional Ginzburg-Landau Differential Equation AU - Yang , Yin AU - Tao , Jianyong AU - Zhang , Shangyou AU - V. Sivtsev , Petr JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 57 EP - 86 PY - 2019 DA - 2019/12 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0070 UR - https://global-sci.org/intro/article_detail/aamm/13419.html KW - The fractional Ginzburg-Landau equation, Jacobi collocation method, convergence. AB -

In this paper, we design a collocation method to solve the fractional Ginzburg-Landau equation. A Jacobi collocation method is developed and implemented in two steps. First, we space-discretize the equation by the Jacobi-Gauss-Lobatto collocation (JGLC) method in one- and two-dimensional space. The equation is then converted to a system of ordinary differential equations (ODEs) with the time variable based on JGLC. The second step applies the Jacobi-Gauss-Radau collocation (JGRC) method for the time discretization. Finally, we give a theoretical proof of convergence of this Jacobi collocation method and some numerical results showing the proposed scheme is an effective and high-precision algorithm.

Yin Yang, Jianyong Tao, Shangyou Zhang & Petr V. Sivtsev. (2019). A Jacobi Collocation Method for the Fractional Ginzburg-Landau Differential Equation. Advances in Applied Mathematics and Mechanics. 12 (1). 57-86. doi:10.4208/aamm.OA-2019-0070
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