In this paper, the Variational Iteration Method (VIM) and the Homotopy Perturbation Method (HPM) are applied to solve the non-linear differential equations. The Newell-Whitehead-Segel equation, the Burgers-Huxley, the Burgers-Fisher equation, the Fitzhugh–Nagumo Equation, the Fisher Type Equation are studied in different chapters and exact solutions are also obtained. A comparison is made between obtained results in finding the exact solution of the equation in order to present precision of the methods. The results prove capability and great potential of the methods as effective algorithms in order to obtain the exact solution of non-linear differential equations.
Table of Contents
Chapter 1
The ideas of variational iteration method and homotopy perturbation method
Chapter 2
The Newell-Whitehead-Segel equation
Chapter 3
The Burgers-Huxley equation
Chapter 4
The Burgers-Fisher equation
Chapter 5
The Fitzhugh–Nagumo equation
Chapter 6
The Fisher Type equation
Objectives and Topics
The primary objective of this work is to demonstrate the effectiveness of the Variational Iteration Method (VIM) and the Homotopy Perturbation Method (HPM) in deriving exact solutions for various non-linear differential equations. The study investigates several specific equations, comparing the precision and convergence rates of these two semi-analytical methods through numerical analysis.
- Application of Variational Iteration Method (VIM) to non-linear differential equations.
- Application of Homotopy Perturbation Method (HPM) for solving non-linear systems.
- Comparative performance analysis of VIM and HPM algorithms.
- Determination of exact closed-form solutions for equations such as Burgers-Huxley and Fitzhugh-Nagumo.
- Assessment of rapid sequence convergence to exact solutions.
Excerpt from the book
1.2. The idea of variational iteration method
The idea of the variational iteration method is based on constructing a correction functional by a general Lagrange multiplier. The multiplier is chosen in such a way that its correction solution is improved with respect to the initial approximation or to the trial function. To illustrate the basic idea of the variational iteration method, consider the following nonlinear equation:
Lu(t) + Nu(t) = g(t), (1.1)
Where L is a linear operator, N is a nonlinear operator, and g (t) is a known analytic function. According to the variational iteration method, we can construct the following correction functional:
un+1(x,t) = un(x,t) + ∫₀ᵗ λ(ξ)(Lun(ξ) + Nũn(ξ) - g(t)) dξ, (1.2)
Where λ is a general Lagrange multiplier which can be identified optimally via variational theory and ũn is considered as a restricted variation which means δũn = 0.
u₀(t) is an initial approximation with possible unknowns. We first determine the Lagrange multiplier λ that will be identified optimally via integration by parts. With λ determined, then several approximations un(t), n ≥ 0 follow immediately. Consequently, the exact solution may be obtained as:
u(t) = limn→∞ un(t), (1.3)
The correction functional of the Eq. (1.1) gives several approximations. Therefore, the exact solution can be obtained as the limit of resulting successive approximations.
Chapter Summaries
Chapter 1, The ideas of variational iteration method and homotopy perturbation method: This chapter introduces the theoretical foundations of the VIM and HPM algorithms used throughout the study to solve linear and non-linear differential equations.
Chapter 2, The Newell-Whitehead-Segel equation: This section applies VIM and HPM to solve multiple case study problems of the Newell-Whitehead-Segel equation, demonstrating their capacity to produce exact solutions.
Chapter 3, The Burgers-Huxley equation: The authors focus on the generalized Burgers-Huxley equation, applying both methods to obtain closed-form solutions and assessing their accuracy through comparative numerical analysis.
Chapter 4, The Burgers-Fisher equation: This chapter extends the application of VIM and HPM to the Burgers-Fisher equation, presenting the convergence trends and comparing results with known exact solutions.
Chapter 5, The Fitzhugh–Nagumo equation: Here, the VIM and HPM are utilized to solve non-linear Fitzhugh–Nagumo equations, highlighting the rapid convergence of sequences toward the exact solution.
Chapter 6, The Fisher Type equation: The final chapter covers the Fisher Type equation, concluding the study by verifying the effectiveness and reliability of VIM and HPM in solving this specific reaction-diffusion model.
Keywords
Nonlinear differential equations, Variational Iteration Method, Homotopy Perturbation Method, Newell-Whitehead-Segel equation, Burgers-Huxley equation, Burgers-Fisher equation, Fitzhugh–Nagumo equation, Fisher Type equation, Semi-analytical methods, Exact solution, Convergence, Lagrange multiplier, Perturbation technique, Maple package, Reaction-diffusion.
Frequently Asked Questions
What is the core focus of this research?
This work focuses on obtaining exact solutions for various non-linear differential equations using semi-analytical techniques, specifically the Variational Iteration Method and the Homotopy Perturbation Method.
Which specific equations are analyzed in the book?
The study examines the Newell-Whitehead-Segel equation, the Burgers-Huxley equation, the Burgers-Fisher equation, the Fitzhugh–Nagumo equation, and the Fisher Type equation.
What is the primary goal of applying these methods?
The goal is to demonstrate that both VIM and HPM are powerful, efficient, and accurate algorithms capable of constructing exact closed-form solutions for complex non-linear problems.
What scientific methodology is utilized?
The authors use semi-analytical approaches involving correction functionals with Lagrange multipliers (VIM) and combinations of classical perturbation and homotopy techniques (HPM).
What content is covered in the main chapters?
Each main chapter introduces the specific non-linear equation, describes its physical or scientific applications, details the mathematical implementation of VIM and HPM, and validates the results through numerical convergence analysis.
Which keywords best characterize this work?
The work is characterized by terms such as Nonlinear differential equations, Variational Iteration Method, Homotopy Perturbation Method, Exact solution, and Reaction-diffusion.
How is the accuracy of the methods evaluated?
The authors evaluate accuracy by calculating the percentage of relative error (%RE) for various successive approximations compared to the exact solutions, presenting these findings in comparative tables.
What role does the Maple package play in this study?
The Maple package is used to solve the recursive sequences generated by the VIM and HPM algorithms to arrive at the final closed-form expressions.
- Citation du texte
- Mohsen Soori (Auteur), S. Salman Nourazar (Auteur), 2019, On The Exact Solution of Nonlinear Differential Equations Using Variational Iteration Method and Homotopy Perturbation Method, Munich, GRIN Verlag, https://www.grin.com/document/455364
