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Homotopic approximate solutions for the perturbed CKdV equation with variable coefficients.

Dianchen Lu1, Tingting Chen1, Baojian Hong1

  • 1Center of Nonlinear Science Research, Jiangsu University, Zhenjiang, Jiangsu 212013, China.

Thescientificworldjournal
|April 17, 2014
PubMed
Summary

This study presents a novel method for finding approximate solutions to the perturbed combined KdV equation with variable coefficients. The homotopic mapping method yields double periodic solutions, which can simplify to hyperbolic or trigonometric forms.

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Area of Science:

  • Nonlinear Partial Differential Equations
  • Mathematical Physics
  • Computational Mathematics

Background:

  • The combined KdV equation with variable coefficients presents challenges in finding exact solutions.
  • Perturbations and variable coefficients complicate the analysis of nonlinear wave phenomena.
  • Approximate solutions are crucial for understanding complex dynamics.

Purpose of the Study:

  • To develop a method for finding double periodic approximate solutions for the perturbed combined KdV equation with variable coefficients.
  • To investigate the behavior of these solutions in limiting cases.
  • To derive first and second-order approximate solutions under perturbation.

Main Methods:

  • Application of the homotopic mapping method.
  • Analysis of approximate solutions in limiting cases (hyperbolic and trigonometric functions).
  • Derivation of perturbation-based approximate solutions.

Main Results:

  • The homotopic mapping method successfully generates double periodic approximate solutions.
  • Solutions demonstrate degeneracy into hyperbolic and trigonometric forms.
  • First and second-order approximate solutions are obtained for the perturbed equation.

Conclusions:

  • The homotopic mapping method is effective for solving the variable coefficients combined KdV equation.
  • The derived solutions offer insights into nonlinear wave behavior under perturbation.
  • The method provides a framework for analyzing similar complex nonlinear systems.