A HOMOTOPY PERTURBATION METHOD (HPM) WITH THE CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE (CFD) TO ANALYSE AND PREDICT THE BEHAVIOR OF AN ARBITRARY ORDER FRACTIONAL JAULENT-MIODEK SYSTEM

Abstract

The present invention relates to a Homotopy Perturbation Method (HPM) with the Caputo-Fabrizio fractional derivative (CFD) to analyse and predict the behavior of an arbitrary order fractional Jaulent-Miodek system. The Laplace transform with a Homotopy perturbation scheme, and a fractional derivative defined by the Caputo-Fabrizio derivative (CFD) joint by HPTM in an elegant way to study the proposed model in terms of fractional order. FJMS is a nonlinear partial differential equation that describes the behavior of some physical phenomena, including wave propagation in fluids and heat transfer in solids. The FJMS is a significant tool in the perusal of nonlinear waves also other nonlinear dynamic model, and its applications have a wide category of practical applications in different fields of Science and Engineering. The FJMS is of particular interest in the study of nonlinear waves, as it provides a simple and effective mathematical model for the analysis of wave propagation in various medium. The system is also used in the study of various other nonlinear dynamic systems, including nonlinear optical systems and nonlinear control systems.

Specification

Description:Field of the Invention
The present invention relates to a mathematical model, more particular to a Homotopy Perturbation Method (HPM) coupled with the Caputo-Fabrizio fractional derivative (CFD) to analyse and predict the behavior of an arbitrary order fractional Jaulent-Miodek system.
Background of the Invention
The Homotopy perturbation transform method (HPTM) have been used to receive a solution for the fractional Jaulent-Miodek system(FJMS). The fractional Jaulent-Miodek system (FJMS) is of particular interest in the study of nonlinear waves, as it provides a simple and effective mathematical model for the analysis of wave propagation in various medium. The system is also used in the study of various other nonlinear dynamic systems, including nonlinear optical systems and nonlinear control systems.
There are various numerical and approximate methods that can be used to solve nonlinear fractional differential equations (NFDE), including finite difference methods, spectral methods, and variational iteration methods, among others. The choice of method will depend on the specific equation and the desired level of accuracy. Despite the challenges in solving NFDEs, they have been found to provide more accurate and realistic models for many physical and biological systems, making them a valuable tool for understanding and predicting these systems. Mathematics' fractional calculus (FC) is a field that focuses on non-integer order derivatives and integrals. It offers a generalization of the integrals and derivatives of integer order found in classical calculus. In FC, derivatives and integrals of any order between zero and infinity can be defined and studied. Main applications of fractional calculus can be found in various fields, such as physics, Medicine, Geophysics, Engineering, Finance, and Electromagnetics. This fractional derivative buries all appropriate themes and helps us perceive natural phenomena in an orderly and dominant manner. Fractional Jaulent-Miodek System is considered as

Drawings
Figure 1 shows that, for different values of a = 0.6, a = 0.75, a = 0.9, and a = 1, the approximate solutions of FHPTM w(x, t).
Figure 2 shows that, in the case of (a) a = 0.6, (b) a = 0.75, (c) a = 0.9 , and (d) a = 1 for an equation (3), a 3D figure is a graphical representation for w(x, t).
Detailed Description of the Invention
The following description includes the preferred best mode of one embodiment of the present invention. It will be clear from this description of the invention that the invention is not limited to these illustrated embodiments but that the invention also includes a variety of modifications and embodiments thereto. Therefore, the present description should be seen as illustrative and not limiting. While the invention is susceptible to various modifications and alternative constructions, it should be understood, that there is no intention to limit the invention to the specific form disclosed, but, on the contrary, the invention is to cover all modifications, alternative constructions, and equivalents falling within the spirit and scope of the invention as defined in the claims.
In any embodiment described herein, the open-ended terms "comprising," "comprises," and the like (which are synonymous with "including," "having" and "characterized by") may be replaced by the respective partially closed phrases "consisting essentially of," consists essentially of," and the like or the respective closed phrases "consisting of," "consists of, the like. As used herein, the singular forms "a", "an", and "the" designate both the singular and the plural, unless expressly stated to designate the singular only.
In the present invention, the Homotopy perturbation transform method (HPTM) has been used to receive a solution for the fractional Jaulent-Miodek system (FJMS). The Laplace transform with a Homotopy perturbation scheme, and a fractional derivative defined by the Caputo-Fabrizio derivative (CFD) joint by HPTM in an elegant way to study the proposed model in terms of fractional order.
With the help of HPTM we used a numerical technique to find approximate solutions of nonlinear partial differential equations (NPDEs) and other mathematical problems. It is an iterative method that starts with an initial approximation and then perturbs it until a solution is found. Finally, we get some approximations. With the help of this approximations for different values of parameter, we find the approximate solutions of FHPTM, by using the equation
(3)

In one aspect, the HPTM is used swimmingly to find the solution for an arbitrary order fractional Jaulent-Miodek system. Sytem acquired an infinite series solution for FJMS, based on the CFD.
The appropriate model plays a live role that analyzing physiological events and thus for future work we investigated using effects and precise methods recently proposed and nurtured. With the help of the result acquired, we can catch more enthralling results. Finally, the result has been assured that the plan is very effective and more concise and highly organized. Furthermore, the suggested approach solves the linked nonlinear issue without the use of discretization, perturbation, or transformations. The current invention aids researchers in studying the behavior of nonlinear situations, with extremely intriguing and valuable outcomes.
Fractional differential equations that are nonlinear (NFDEs) include differential equations that involve derivatives of fractional order and nonlinear terms. These are used to model a wide category of Physical, Biological, and engineering phenomena, including anomalous diffusion, viscoelasticity, control systems, and dynamic systems. NFDEs is a challenging task, as the fractional derivatives introduce nonlocality and memory effects, making it difficult to use classical methods from integer-order calculus. Additionally, the nonlinear terms introduce further complexity, making it difficult to obtain analytical solutions in many cases.
The FJMS is of particular interest in the study of nonlinear waves, as it provides a simple and effective mathematical model for the analysis of wave propagation in various medium. The system is also used in the study of various other nonlinear dynamic systems, including nonlinear optical systems and nonlinear control systems.
The fractional Jaulent-Miodek system(FJMS) has several applications in different belt of Science and Engineering, containing:
? Nonlinear Optics: The FJMS can be used to model the behavior of nonlinear optical waves, such as solitons, in optical fibers and other nonlinear optical media.
? Fluid Mechanics: The FJMS can be used to model wave propagation in fluids, such as water waves and waves in the ocean.
? Control Systems: The FJMS can be used to model nonlinear control systems, such as those used in aircraft control and automobile suspension systems.
? Heat Transfer: The FJMS can be used to model heat transfer in solids, such as heat conduction in metals and other materials.
? Nonlinear Dynamics: The FJMS is also used in the study of nonlinear dynamics, including the behavior of chaotic systems and the stability of nonlinear systems.

, Claims:
1. A Homotopy Perturbation Method (HPM) with the Caputo-Fabrizio fractional derivative (CFD) to analyse and predict the behavior of an arbitrary order fractional Jaulent-Miodek system, comprising of:
• Homotopy perturbation transform method (HPTM) to receive a solution for the fractional Jaulent-Miodek system (FJMS); and
• Laplace transform with a Homotopy perturbation scheme, and a fractional derivative defined by the Caputo-Fabrizio derivative (CFD) joint by HPTM to study the model in terms of fractional order; and
• Using an iterative numerical technique that starts with an initial approximation and perturbs it in successive steps to approximate the solution for nonlinear partial differential equations (NPDEs) and other complex mathematical problems.
2. The Homotopy Perturbation Method (HPM) with the Caputo-Fabrizio fractional derivative (CFD) to analyse and predict the behavior of an arbitrary order fractional Jaulent-Miodek system as claimed in the claim 1, wherein system is an effective tool for studying and predicting the behavior of nonlinear dynamic systems, including nonlinear waves, nonlinear optical systems, and nonlinear control systems, particularly those modeled by nonlinear fractional differential equations (NFDEs).
3. The Homotopy Perturbation Method (HPM) with the Caputo-Fabrizio fractional derivative (CFD) to analyse and predict the behavior of an arbitrary order fractional Jaulent-Miodek system as claimed in the claim 1, wherein Using the results of the method to analyze physiological events and dynamic systems in various physical, biological, and engineering contexts, yielding valuable insights into phenomena such as anomalous diffusion, viscoelasticity, and control system dynamics.

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