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Keywords = fractional Caudrey–Dodd–Gibbon equation

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27 pages, 1500 KiB  
Article
An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator
by Mashael M. AlBaidani, Abdul Hamid Ganie, Adnan Khan and Fahad Aljuaydi
Fractal Fract. 2025, 9(4), 199; https://doi.org/10.3390/fractalfract9040199 - 25 Mar 2025
Cited by 2 | Viewed by 568
Abstract
The development of numerical or analytical solutions for fractional mathematical models describing specific phenomena is an important subject in physics, mathematics, and engineering. This paper’s main objective is to investigate the approximation of the fractional order Caudrey–Dodd–Gibbon (CDG) nonlinear [...] Read more.
The development of numerical or analytical solutions for fractional mathematical models describing specific phenomena is an important subject in physics, mathematics, and engineering. This paper’s main objective is to investigate the approximation of the fractional order Caudrey–Dodd–Gibbon (CDG) nonlinear equation, which appears in the fields of laser optics and plasma physics. The physical issue is modeled using the Caputo derivative. Adomian and homotopy polynomials facilitate the handling of the nonlinear term. The main innovation in this paper is how the recurrence relation, which generates the series solutions after just a few iterations, is handled. We examined the assumed model in fractional form in order to demonstrate and verify the efficacy of the new methods. Moreover, the numerical simulation is used to show how the physical behavior of the suggested method’s solution has been represented in plots and tables for various fractional orders. We provide three problems of each equation to check the validity of the offered schemes. It is discovered that the outcomes derived are close to the accurate result of the problems illustrated. Additionally, we compare our results with the Laplace residual power series method (LRPSM), the natural transform decomposition method (NTDM), and the homotopy analysis shehu transform method (HASTM). From the comparison, our methods have been demonstrated to be more accurate than alternative approaches. The results demonstrate the significant benefit of the established methodologies in achieving both approximate and accurate solutions to the problems. The results show that the technique is extremely methodical, accurate, and very effective for examining the nature of nonlinear differential equations of arbitrary order that have arisen in related scientific fields. Full article
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26 pages, 15223 KiB  
Article
Construction of Soliton Solutions of Time-Fractional Caudrey–Dodd–Gibbon–Sawada–Kotera Equation with Painlevé Analysis in Plasma Physics
by Khadija Shakeel, Alina Alb Lupas, Muhammad Abbas, Pshtiwan Othman Mohammed, Farah Aini Abdullah and Mohamed Abdelwahed
Symmetry 2024, 16(7), 824; https://doi.org/10.3390/sym16070824 - 1 Jul 2024
Cited by 1 | Viewed by 1779
Abstract
Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE [...] Read more.
Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots. Full article
(This article belongs to the Special Issue Application of Symmetry in Equations)
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12 pages, 1258 KiB  
Article
Exact Solutions to Some Nonlinear Time-Fractional Evolution Equations Using the Generalized Kudryashov Method in Mathematical Physics
by Mustafa Ekici
Symmetry 2023, 15(10), 1961; https://doi.org/10.3390/sym15101961 - 23 Oct 2023
Cited by 11 | Viewed by 1691
Abstract
In this study, we utilize the potent generalized Kudryashov method to address the intricate obstacles presented by fractional differential equations in the field of mathematical physics. Specifically, our focus centers on obtaining novel exact solutions for three pivotal equations: the time-fractional seventh-order Sawada-Kotera-Ito [...] Read more.
In this study, we utilize the potent generalized Kudryashov method to address the intricate obstacles presented by fractional differential equations in the field of mathematical physics. Specifically, our focus centers on obtaining novel exact solutions for three pivotal equations: the time-fractional seventh-order Sawada-Kotera-Ito equation, the time-fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation, and the time-fractional seventh-order Kaup–Kupershmidt equation. The generalized Kudryashov method, celebrated for its versatility and efficacy in addressing intricate nonlinear problems, plays a central role in our research. This method not only simplifies the equations but also unveils their inner dynamics, rendering them amenable to meticulous analysis. It is worth noting that our fractional derivatives are defined in the context of the conformable fractional derivative, providing a solid foundation for our mathematical investigations. One notable aspect of our study is the visual representation of our findings. Graphical representations of the yielded solutions enliven intricate mathematical structures, providing a concrete insight into the dynamics and behaviors of said equations. This paper highlights the proficiency of the generalized Kudryashov method in resolving complex issues presented by fractional differential equations. Our study not only broadens the range of mathematical methods but also enhances our comprehension of the intriguing realm of nonlinear physical phenomena. Full article
(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities, 2nd Edition)
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18 pages, 538 KiB  
Article
An Efficient Analytical Approach to Investigate Fractional Caudrey–Dodd–Gibbon Equations with Non-Singular Kernel Derivatives
by Dowlath Fathima, Reham A. Alahmadi, Adnan Khan, Afroza Akhter and Abdul Hamid Ganie
Symmetry 2023, 15(4), 850; https://doi.org/10.3390/sym15040850 - 2 Apr 2023
Cited by 24 | Viewed by 1879
Abstract
Fractional calculus is at this time an area where many models are still being developed, explored, and used in real-world applications in many branches of science and engineering where non-locality plays a key role. Although many wonderful discoveries have already been reported by [...] Read more.
Fractional calculus is at this time an area where many models are still being developed, explored, and used in real-world applications in many branches of science and engineering where non-locality plays a key role. Although many wonderful discoveries have already been reported by researchers in important monographs and review articles, there is still a great deal of non-local phenomena that have not been studied and are only waiting to be explored. As a result, we can continually learn about new applications and aspects of fractional modelling. In this study, a precise and analytical method with non-singular kernel derivatives is used to solve the Caudrey–Dodd–Gibbon (CDG) model, a modification of the fifth-order KdV equation (fKdV). The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative and the Atangana–Baleanu derivative in the Caputo sense (ABC). This model illustrates the propagation of magneto-acoustic, shallow-water, and gravity–capillary waves in a plasma medium. The dynamic behaviour of the acquired solutions has been represented in a number of two- and three-dimensional figures. A number of simulations are also performed to demonstrate how the resulting solutions physically behave with respect to fractional order. The significance of the current research is that new solutions are obtained by using a strong analytical approach. Utilizing a fractional derivative operator to solve equivalent models is another benefit of this approach. The results of the present work have similar aspects to the symmetry of partial differential equations. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
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