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Search Results (5)

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Keywords = M-truncated derivative (M-TD)

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13 pages, 1745 KiB  
Article
The Analytical Fractional Solutions for Coupled Fokas System in Fiber Optics Using Different Methods
by Wael W. Mohammed, Clemente Cesarano, Elsayed M. Elsayed and Farah M. Al-Askar
Fractal Fract. 2023, 7(7), 556; https://doi.org/10.3390/fractalfract7070556 - 18 Jul 2023
Cited by 14 | Viewed by 1305
Abstract
The Fokas system with M-truncated derivative (FS-MTD) was considered in this study. To get analytical solutions of FS-MTD in the forms of elliptic, rational, hyperbolic, and trigonometric functions, we employed the extend F-expansion approach and the Jacobi elliptic function method. Since nonlinear [...] Read more.
The Fokas system with M-truncated derivative (FS-MTD) was considered in this study. To get analytical solutions of FS-MTD in the forms of elliptic, rational, hyperbolic, and trigonometric functions, we employed the extend F-expansion approach and the Jacobi elliptic function method. Since nonlinear pulse transmission in monomode optical fibers is explained by the Fokas system, the derived solutions may be utilized to analyze a broad range of important physical processes. In order to comprehend the impacts of MTD on the solutions, the dynamic behavior of the various generated solutions are shown using 2D and 3D figures. Full article
(This article belongs to the Special Issue New Insights into Nonlinear Coupled Differential Equations with Its Applications)
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12 pages, 718 KiB  
Article
The Analytical Solutions to the Fractional Kraenkel–Manna–Merle System in Ferromagnetic Materials
by Mohammad Alshammari, Amjad E. Hamza, Clemente Cesarano, Elkhateeb S. Aly and Wael W. Mohammed
Fractal Fract. 2023, 7(7), 523; https://doi.org/10.3390/fractalfract7070523 - 1 Jul 2023
Cited by 19 | Viewed by 1398
Abstract
In this article, we examine the Kraenkel–Manna–Merle system (KMMS) with an M-truncated derivative (MTD). Our goal is to obtain rational, hyperbolic, and trigonometric solutions by using the F-expansion technique with the Riccati equation. To our knowledge, no one has studied the exact [...] Read more.
In this article, we examine the Kraenkel–Manna–Merle system (KMMS) with an M-truncated derivative (MTD). Our goal is to obtain rational, hyperbolic, and trigonometric solutions by using the F-expansion technique with the Riccati equation. To our knowledge, no one has studied the exact solutions to the KMMS in the presence/absence of a damping effect with an M-truncated derivative, using the F-expansion technique. The magnetic field propagation in a zero-conductivity ferromagnet is described by the KMMS; hence, solutions to this equation may provide light on several fascinating scientific phenomena. We use MATLAB to display figures in a variety of 3D and 2D formats to demonstrate the influence of the M-truncated derivative on the exact solutions to the KMMS. Full article
(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)
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28 pages, 18941 KiB  
Article
Dynamical Study of Coupled Riemann Wave Equation Involving Conformable, Beta, and M-Truncated Derivatives via Two Efficient Analytical Methods
by Rimsha Ansar, Muhammad Abbas, Pshtiwan Othman Mohammed, Eman Al-Sarairah, Khaled A. Gepreel and Mohamed S. Soliman
Symmetry 2023, 15(7), 1293; https://doi.org/10.3390/sym15071293 - 21 Jun 2023
Cited by 17 | Viewed by 2244
Abstract
In this study, the Jacobi elliptic function method (JEFM) and modified auxiliary equation method (MAEM) are used to investigate the solitary wave solutions of the nonlinear coupled Riemann wave (RW) equation. Nonlinear coupled partial differential equations (NLPDEs) can be transformed into a collection [...] Read more.
In this study, the Jacobi elliptic function method (JEFM) and modified auxiliary equation method (MAEM) are used to investigate the solitary wave solutions of the nonlinear coupled Riemann wave (RW) equation. Nonlinear coupled partial differential equations (NLPDEs) can be transformed into a collection of algebraic equations by utilising a travelling wave transformation. This study’s objective is to learn more about the non-linear coupled RW equation, which accounts for tidal waves, tsunamis, and static uniform media. The variance in the governing model’s travelling wave behavior is investigated using the conformable, beta, and M-truncated derivatives (M-TD). The aforementioned methods can be used to derive solitary wave solutions for trigonometric, hyperbolic, and jacobi functions. We may produce periodic solutions, bell-form soliton, anti-bell-shape soliton, M-shaped, and W-shaped solitons by altering specific parameter values. The mathematical form of each pair of travelling wave solutions is symmetric. Lastly, in order to emphasise the impact of conformable, beta, and M-TD on the behaviour and symmetric solutions for the presented problem, the 2D and 3D representations of the analytical soliton solutions can be produced using Mathematica 10. Full article
(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)
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23 pages, 17240 KiB  
Article
The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches
by Xiaoming Wang, Rimsha Ansar, Muhammad Abbas, Farah Aini Abdullah and Khadijah M. Abualnaja
Axioms 2023, 12(6), 599; https://doi.org/10.3390/axioms12060599 - 16 Jun 2023
Cited by 5 | Viewed by 1526
Abstract
The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the [...] Read more.
The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions. Full article
(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)
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10 pages, 707 KiB  
Article
Abundant Solitary Wave Solutions for the Boiti–Leon–Manna–Pempinelli Equation with M-Truncated Derivative
by Farah M. Al-Askar, Clemente Cesarano and Wael W. Mohammed
Axioms 2023, 12(5), 466; https://doi.org/10.3390/axioms12050466 - 12 May 2023
Cited by 19 | Viewed by 2049
Abstract
In this work, we consider the Boiti–Leon–Manna–Pempinelli equation with the M-truncated derivative (BLMPE-MTD). Our aim here is to obtain trigonometric, rational and hyperbolic solutions of BLMPE-MTD by employing two diverse methods, namely, He’s semi-inverse method and the extended tanh function method. In addition, [...] Read more.
In this work, we consider the Boiti–Leon–Manna–Pempinelli equation with the M-truncated derivative (BLMPE-MTD). Our aim here is to obtain trigonometric, rational and hyperbolic solutions of BLMPE-MTD by employing two diverse methods, namely, He’s semi-inverse method and the extended tanh function method. In addition, we generalize some previous results. As the Boiti–Leon–Manna–Pempinelli equation is a model for an incompressible fluid, the solutions obtained may be utilized to represent a wide variety of fascinating physical phenomena. We construct a large number of 2D and 3D figures to demonstrate the impact of the M-truncated derivative on the exact solution of the BLMPE-MTD. Full article
(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)
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