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Bell-shaped soliton solutions and travelling wave solutions of the fifth-order nonlinear modified Kawahara equation

  • Ali Başhan ORCID logo EMAIL logo


The main aim of this work is to investigate numerical solutions of the two different types of the fifth-order modified Kawahara equation namely bell-shaped soliton solutions and travelling wave solutions that occur thereby the different type of the Korteweg–de Vries equation. For this approach, we have used an effective and simple type of finite difference method namely Crank-Nicolson scheme for time integration and third-order modified cubic B-spline-based differential quadrature method for space integration. We preferred the third-order modified cubic B-splines to solve the fifth-order partial differential equation because of by using low energy, less algebraic process and produce better results than earlier works. To display the efficiency and accuracy of the present fresh approach famous test problems namely bell-shaped single soliton that has negative amplitude and travelling wave solutions that have the both of the positive and negative amplitudes are solved and the error norms L 2 and L are calculated and compared with earlier works. Comparison of the error norms show that present fresh approach obtained superior results than earlier works by using same parameters. At the same time, two lowest invariants of the test problems during the simulations are calculated and reported. Besides those, relative changes of invariants are computed and reported.

AMS classification: 65N06; 65D32; 35C08; 65D07

Corresponding author: Ali Başhan, Department of Maths, Zonguldak Bulent Ecevit University, Zonguldak, 67100, Turkey, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.


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Received: 2019-02-26
Accepted: 2020-10-13
Published Online: 2020-11-06
Published in Print: 2021-10-26

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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