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A Guide to Concentration Alternating Frequency Response Analysis of Fuel Cells
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An efficient technique for higher order fractional differential equation.

Ayyaz Ali1, Muhammad Asad Iqbal1, Qazi Mahmood Ul-Hassan2

  • 1Department of Mathematics, Faculty of Sciences, HITEC University, Taxila, Pakistan.

Springerplus
|April 6, 2016
PubMed
Summary

Researchers found exact solutions for the fractional Kawahara equation using the [Formula: see text]-expansion method. This approach effectively generates solitary wave solutions for nonlinear evolution equations in mathematical physics.

Keywords:
Fractional calculusKawahara equationModified Riemann–Liouville derivativeThe [Formula: see text]-expansion methodTraveling wave solutions

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Fractional Calculus

Background:

  • Nonlinear evolution equations (NLEEs) are crucial in modeling complex phenomena.
  • Finding exact solutions for these equations, especially fractional ones, remains a significant challenge.
  • The Kawahara equation is a notable example within this class of equations.

Purpose of the Study:

  • To establish exact solutions for the fractional Kawahara equation.
  • To demonstrate the efficacy of the [Formula: see text]-expansion method for fractional partial differential equations (PDEs).
  • To explore the applicability of this method to other physical problems.

Main Methods:

  • The study employs the [Formula: see text]-expansion method.
  • This technique is utilized to derive analytical solutions for the fractional Kawahara equation.
  • The method involves expressing solutions in terms of various standard functions.

Main Results:

  • Exact solitary wave solutions for the fractional Kawahara equation were successfully established.
  • Solutions were found in forms including hyperbolic, trigonometric, exponential, and rational functions.
  • Graphical representations and numerical data confirmed the accuracy and effectiveness of the method.

Conclusions:

  • The [Formula: see text]-expansion method is highly effective and expedient for solving fractional PDEs.
  • It offers a valuable alternative for obtaining exact solutions to NLEEs.
  • The method's potential for extension to other physical problems is highlighted.