The current study deals with distinct kinds of solitary wave solutions for the fractional generalized Duffing model and fractional diffusion reaction model with novel truncated M-fractional derivative. We are also interested in studying some special
cases of the fractional generalized Duffing model.
These are known as fractional Landau-Ginzburg-Higgs equation, classical fractional Klein-Gordon equation, the
Phi-4 equation, the Sine-Gordon equation and the Duffing equation. The obtained results can be used in describing these models in some better way. The novel fractional derivative operator namely M-fractional derivative is used to study the above-mentioned models. Also, the obtained results are verified via symbolic software Mathematica. A modified integration method, the extended Sinh-Gordon equation expansion method (EShGEEM) is employed to secure the aforesaid solitary wave solutions. Furthermore, the obtained results show that the suggested approach have broadened capacity to obtain the different wave solutions of the fractional differential equations effectively. At the end, the results are also explained through their graphical representations.
Table of Contents
1 Introduction and Preliminaries
1.1 Truncated M-fractional Derivative and it’s Properties:
1.2 Description of the Scheme Strategy:
2 Main Results-I
2.1 Mathematical Analysis of the Fractional Generalized Reaction Duffing Model
2.2 Solutions to Eq.(2.2) with the aforesaid approach
2.3 The Landua-Ginburg-Higgs equation as a special case of Eq. (1.1)
2.4 The classical fractional Klein-Gordon equation from Eq. (1.1)
2.5 The solutions for Phi-4 equation
2.6 The sine-Gordon equation as a special case of Eq. (1.1)
2.7 The Duffing equation as a special case of Eq. (1.1)
3 Main Results-II
3.1 Mathematical Analysis of the Fractional Diffusion Reaction Model with its Solutions:
3.2 Solutions to the diffusion reaction equation (3.1)
4 Conclusion
Research Objective and Scope
The primary research objective is to derive exact solitary wave solutions for the fractional generalized Duffing model and the fractional diffusion-reaction model, utilizing the novel truncated M-fractional derivative and the extended Sinh-Gordon equation expansion method (EShGEEM).
- Investigation of the fractional generalized Duffing model and its specific special cases.
- Application of the truncated M-fractional derivative operator in modeling physical phenomena.
- Implementation of the extended Sinh-Gordon equation expansion method to find analytic solutions.
- Verification of obtained mathematical results through symbolic software Mathematica.
- Graphical representation and analysis of the derived solitary wave solutions.
Excerpt from the Thesis
1.1 Truncated M-fractional Derivative and it’s Properties:
In this section, we recall the definition of M-fractional derivative for its use in our remaining sections. The main advantages of the aforesaid definition are the product rule, the chain rule, the quotient rule and other important theorems arising in ordinary calculus [31, 32]:.
Definition:
Suppose g(t) : [0, ∞) → ℜ, then the truncated M-fractional derivative of g of power μ is given:
Dμ,β M,tg(t) = limτ→0 g (t iEβ(τ t1−μ)) − g(t) / τ , 0 < μ < 1, β > 0,
here iEβ(.) is a truncated Mittag-Leffler function of one parameter that is defined as:
iEβ(z) = Σ i j=0 zj / Γ(βj + 1), β > 0 and z ∈ C.
Summary of Chapters
1 Introduction and Preliminaries: This chapter introduces nonlinear partial differential equations and defines the truncated M-fractional derivative alongside the methodology used for the subsequent analysis.
2 Main Results-I: This section presents the mathematical derivation of exact solitary wave solutions for the fractional generalized Duffing model and its special cases, including the Landau-Ginzburg-Higgs, Klein-Gordon, Phi-4, sine-Gordon, and Duffing equations.
3 Main Results-II: This chapter focuses on the fractional diffusion-reaction model, providing a detailed mathematical analysis and solving the equation using the defined transformation methods.
4 Conclusion: The final chapter summarizes the findings, confirming the effectiveness of the extended Sinh-Gordon equation expansion method in obtaining wave solutions for fractional models.
Keywords
Nonlinear fractional differential equations, truncated M-fractional derivative, solitary wave solutions, extended Sinh-Gordon equation expansion method, EShGEEM, generalized Duffing model, fractional diffusion-reaction model, Landau-Ginzburg-Higgs equation, Klein-Gordon equation, Phi-4 equation, sine-Gordon equation, mathematical biology, Mathematica, analytical solutions, soliton.
Frequently Asked Questions
What is the core focus of this research?
The research focuses on finding exact solitary wave solutions for various nonlinear fractional differential equations using the truncated M-fractional derivative.
Which models are specifically analyzed in the work?
The study examines the fractional generalized Duffing model, the fractional diffusion-reaction model, and several special cases like the Landau-Ginzburg-Higgs and sine-Gordon equations.
What is the primary objective?
The objective is to provide a broader capacity to obtain diverse wave solutions for fractional differential equations in an effective and verifiable manner.
What scientific method is utilized?
The research employs the extended Sinh-Gordon equation expansion method (EShGEEM) combined with the homogeneous balance method.
What content is covered in the main body?
The main body consists of two chapters detailing the derivation of solution sets for the Duffing model and the diffusion-reaction model, supplemented by graphical 2D and 3D simulations.
Which keywords characterize this thesis?
Key terms include fractional differential equations, solitary wave solutions, M-fractional derivative, and EShGEEM.
How is the accuracy of the results ensured?
All obtained solutions are verified using the symbolic computation software Mathematica.
What is the significance of the truncated M-fractional derivative?
The truncated M-fractional derivative is significant because it allows the application of standard calculus theorems like the product and chain rule to fractional equations, facilitating easier analytical solutions.
- Citation du texte
- Anoosha Qaisar (Auteur), 2021, Nonlinear Fractional Differential Equations. Some Exact Solitary Wave Solutions, Munich, GRIN Verlag, https://www.grin.com/document/1061186
