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.
Inhaltsverzeichnis (Table of Contents)
- Chapter 1, The ideas of variational iteration method and homotopy perturbation method
- Introduction
- The idea of variational iteration method
- The idea of 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
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This paper aims to demonstrate the effectiveness of two semi-analytical methods, the Variational Iteration Method (VIM) and the Homotopy Perturbation Method (HPM), in obtaining exact solutions for non-linear differential equations. The study focuses on a variety of nonlinear equations, including the Newell-Whitehead-Segel equation, the Burgers-Huxley, Burgers-Fisher, Fitzhugh-Nagumo, and Fisher Type equations.
- Application of VIM and HPM to solve non-linear differential equations
- Finding exact solutions for various non-linear equations
- Comparison of the precision and effectiveness of VIM and HPM
- Exploring the potential of these methods for solving non-linear problems
- Analyzing the convergence of the methods and their applicability to different equations
Zusammenfassung der Kapitel (Chapter Summaries)
- Chapter 1 introduces the concepts and methodologies of the Variational Iteration Method (VIM) and the Homotopy Perturbation Method (HPM), outlining their application in obtaining exact solutions for nonlinear differential equations. This chapter provides a theoretical foundation for the subsequent chapters.
- Chapter 2 delves into the Newell-Whitehead-Segel equation, applying the VIM and HPM techniques to derive its exact solution. This chapter highlights the methods' ability to address a specific non-linear equation.
- Chapter 3 focuses on the Burgers-Huxley equation, again employing the VIM and HPM to determine its exact solution. This chapter further illustrates the methods' versatility in solving different types of non-linear differential equations.
- Chapter 4 extends the analysis to the Burgers-Fisher equation, presenting the exact solution obtained using VIM and HPM. This chapter continues to demonstrate the effectiveness of the methods across a range of non-linear problems.
- Chapter 5 explores the Fitzhugh-Nagumo equation, applying VIM and HPM to find its exact solution. This chapter showcases the methods' capability in dealing with complex non-linear equations.
Schlüsselwörter (Keywords)
The paper primarily revolves around the application of variational iteration method and homotopy perturbation method for the exact solution of non-linear differential equations. Key themes and concepts include the study of specific non-linear equations such as the Newell-Whitehead-Segel, Burgers-Huxley, Burgers-Fisher, Fitzhugh-Nagumo, and Fisher Type equations. The paper investigates the precision and effectiveness of the methods, highlighting their potential in solving diverse types of non-linear problems.
- Quote paper
- Mohsen Soori (Author), S. Salman Nourazar (Author), 2019, On The Exact Solution of Nonlinear Differential Equations Using Variational Iteration Method and Homotopy Perturbation Method, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/455364
