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Numerical solution to generalized Burgers'-Fisher equation using Exp-function method hybridized with heuristic

Suheel Abdullah Malik1, Ijaz Mansoor Qureshi2, Muhammad Amir1

  • 1Department of Electronic Engineering, Faculty of Engineering and Technology, International Islamic University, Islamabad, Pakistan.

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|March 27, 2015
PubMed
Summary
This summary is machine-generated.

A novel heuristic approach combines the Exp-function method with a genetic algorithm (GA) to approximate solutions for the generalized Burgers-Fisher equation. This hybrid method offers an accurate and viable alternative to traditional techniques for nonlinear partial differential equations (NPDEs).

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Area of Science:

  • Applied Mathematics
  • Numerical Analysis
  • Computational Physics

Background:

  • The generalized Burgers'-Fisher equation is a significant nonlinear partial differential equation (NPDE) modeling various phenomena.
  • Accurate and efficient numerical methods are crucial for solving such complex equations.
  • Existing methods like Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), and Optimal Homotopy Asymptotic Method (OHAM) have limitations.

Purpose of the Study:

  • To propose a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation.
  • To hybridize the Exp-function method with a nature-inspired algorithm for enhanced accuracy.
  • To validate the proposed scheme against exact solutions and established numerical methods.

Main Methods:

  • Transformation of the NPDE into a nonlinear ordinary differential equation (NODE).
  • Approximation of the traveling wave solution using the Exp-function method with unknown parameters.
  • Optimization of unknown parameters via a global error minimization problem solved by the Genetic Algorithm (GA).

Main Results:

  • Successful implementation of the hybrid Exp-function and GA scheme for the generalized Burgers'-Fisher equation.
  • The proposed method demonstrates high accuracy when compared to exact solutions.
  • Numerical results show the scheme's superiority or comparability to ADM, HPM, and OHAM.

Conclusions:

  • The hybridized Exp-function method and Genetic Algorithm provide a robust and accurate approach for solving the generalized Burgers'-Fisher equation.
  • This novel scheme is a viable and efficient tool for tackling complex nonlinear partial differential equations (NPDEs).
  • The method's accuracy and applicability are confirmed through comparative analysis with existing techniques.