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Study of coupled nonlinear partial differential equations for finding exact analytical solutions.

Kamruzzaman Khan1, M Ali Akbar2, H Koppelaar3

  • 1Department of Mathematics , Pabna University of Science and Technology , Pabna 6600, Bangladesh.

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|November 21, 2015
PubMed
Summary

The enhanced (G/G)-expansion method efficiently finds exact solutions for nonlinear partial differential equations (NPDEs). This approach is demonstrated on the Drinfel

Keywords:
Drinfel'd–Sokolov–Wilson equationPainlevé integrable Burgers equationenhanced (G′/G)-expansion methodnonlinear partial differential equationssolitary wave

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Computational Mathematics

Background:

  • Nonlinear partial differential equations (NPDEs) are fundamental in describing complex phenomena across science and engineering.
  • Finding exact solutions for NPDEs is crucial for understanding their behavior and validating numerical methods.
  • The Drinfel'd-Sokolov-Wilson (DSW) and (2+1)-dimensional Painlevé integrable Burgers (PIB) equations are significant models in mathematical physics.

Purpose of the Study:

  • To introduce and demonstrate the efficacy of the enhanced (G'/G)-expansion method for obtaining exact solutions of NPDEs.
  • To apply this method to solve the DSW and PIB equations.
  • To highlight the method's applicability to a broader range of NPDEs.

Main Methods:

  • The enhanced (G'/G)-expansion method is employed, which involves a transformation to a system of ordinary differential equations.
  • The solutions are systematically derived by considering different cases for the auxiliary equation.
  • The method's efficiency is validated by successfully applying it to specific NPDEs.

Main Results:

  • Exact solutions for the Drinfel'd-Sokolov-Wilson (DSW) equation were successfully obtained.
  • Exact solutions for the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation were successfully derived.
  • The study demonstrates the power and versatility of the enhanced (G'/G)-expansion method.

Conclusions:

  • The enhanced (G'/G)-expansion method is a powerful and effective technique for finding exact solutions of NPDEs.
  • The method provides a reliable framework for analyzing complex nonlinear systems.
  • This approach holds significant potential for applications in various fields of mathematical physics.