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Technical Brief

The Time-Splitting Spectral Method for the Gerdjikov–Ivanov Equation With the Riesz Fractional Derivative in the Quantum Field Theory

[+] Author and Article Information
S. Saha Ray

Department of Mathematics,
National Institute of Technology,
Rourkela 769008, India
e-mail: santanusaharay@yahoo.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 5, 2018; final manuscript received October 24, 2018; published online November 28, 2018. Assoc. Editor: Anindya Chatterjee.

J. Comput. Nonlinear Dynam 14(1), 014502 (Nov 28, 2018) (6 pages) Paper No: CND-18-1202; doi: 10.1115/1.4041891 History: Received May 05, 2018; Revised October 24, 2018

The present work deals with the solutions of the Gerdjikov–Ivanov(G–I) equation with the Riesz fractional derivative by means of the time-splitting spectral approach. In this approach, the G–I equation is split into two equations and the proposed technique viz. time-splitting spectral method is employed for discretizing the equation in space and then subsequently integrating in time exactly. Furthermore, an implicit finite difference method (IMFD) is utilized here to compare the results with the above-mentioned seminumerical method viz. time-splitting spectral technique. Moreover, it has been established that the proposed method is unconditionally stable. In addition to these, the error norms have been also presented here.

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References

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Figures

Grahic Jump Location
Fig. 1

One soliton wave 3D solution of |q(x,t)| of example 1 for the Riesz fractional G–I equation obtained by TSSM for α=1.9

Grahic Jump Location
Fig. 2

Comparison of graphs for the solutions of |q(x,t)| obtained from TSSM and implicit finite difference scheme for the Riesz fractional Gerdjikov–Ivanov Eq. (3.1) with fractional order α=1.9 for example 1 at t=1.0.

Grahic Jump Location
Fig. 3

One soliton wave 3D solution of |q(x,t)| of example 2 for the Riesz fractional G–I equation obtained by TSSM for α=1.9

Grahic Jump Location
Fig. 4

Comparison of graphs for the solutions of |q(x,t)| obtained from TSSM and implicit finite difference scheme for the Riesz fractional Gerdjikov–Ivanov Eq. (3.1) with fractional order α=1.9 for example 2 at t=1.0.

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