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11 pages, 17162 KiB

Open AccessArticle

Analytic Approach Solution to Time-Fractional Phi-4 Equation with Two-Parameter Fractional Derivative

by Youssouf Massoun, Abedel-Karrem Alomari and Clemente Cesarano

Fractal Fract. 2024, 8(10), 576; https://doi.org/10.3390/fractalfract8100576 - 30 Sep 2024

Viewed by 987

This paper is devoted to building a general framework for constructing a solution to fractional Phi-4 differential equations using a Caputo definition with two parameters. We briefly introduce some definitions and properties of fractional calculus in two parameters and the Phi-4 equation. By [...] Read more.

This paper is devoted to building a general framework for constructing a solution to fractional Phi-4 differential equations using a Caputo definition with two parameters. We briefly introduce some definitions and properties of fractional calculus in two parameters and the Phi-4 equation. By investigating the homotopy analysis method, we built the solution algorithm. The two parameters of the fractional derivative gain vary the behavior of the solution, which allows the researchers to fit their data with the proper parameter. To evaluate the effectiveness and accuracy of the proposed algorithm, we compare the results with those obtained using various numerical methods in a comprehensive comparative study. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)

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12 pages, 285 KiB

Open AccessEntry

A Survey on Orthogonal Polynomials from a Monomiality Principle Point of View

by Clemente Cesarano, Yamilet Quintana and William Ramírez

Encyclopedia 2024, 4(3), 1355-1366; https://doi.org/10.3390/encyclopedia4030088 - 20 Sep 2024

Cited by 2 | Viewed by 1220

Definition

This survey highlights the significant role of exponential operators and the monomiality principle in the theory of special polynomials. Using operational calculus formalism, we revisited classical and current results corresponding to a broad class of special polynomials. For instance, we explore the 2D [...] Read more.

This survey highlights the significant role of exponential operators and the monomiality principle in the theory of special polynomials. Using operational calculus formalism, we revisited classical and current results corresponding to a broad class of special polynomials. For instance, we explore the 2D Hermite polynomials and their generalizations. We also present an integral representation of Gegenbauer polynomials in terms of Gould–Hopper polynomials, establishing connections with a simple case of Gegenbauer–Sobolev orthogonality. The monomiality principle is examined, emphasizing its utility in simplifying the algebraic and differential properties of several special polynomial families. This principle provides a powerful tool for deriving properties and applications of such polynomials. Additionally, we review advancements over the past 25 years, showcasing the evolution and extensive applicability of this operational formalism in understanding and manipulating special polynomial families. Full article

(This article belongs to the Section Mathematics & Computer Science)

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11 pages, 268 KiB

Open AccessArticle

New Oscillation Criteria for Sturm–Liouville Dynamic Equations with Deviating Arguments

by Taher S. Hassan, Clemente Cesarano, Loredana Florentina Iambor, Amir Abdel Menaem, Naveed Iqbal and Akbar Ali

Mathematics 2024, 12(10), 1532; https://doi.org/10.3390/math12101532 - 14 May 2024

Viewed by 1220

Abstract

The aim of this study is to refine the known Riccati transformation technique to provide new oscillation criteria for solutions to second-order dynamic equations over time. It is important to note that the convergence or divergence of some improper integrals on time scales [...] Read more.

The aim of this study is to refine the known Riccati transformation technique to provide new oscillation criteria for solutions to second-order dynamic equations over time. It is important to note that the convergence or divergence of some improper integrals on time scales depends not only on the integration function but also on the integration time scale. Therefore, there has been a motivation to find new oscillation criteria that can be applicable regardless of whether

\int_{ζ_{0}}^{\infty} \frac{Δ ξ}{a (ξ)}

is convergent or divergent, in contrast to what has been followed in most previous works in the literature. We have provided an example to illustrate the significance of the obtained results. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena, 2nd Edition)

14 pages, 279 KiB

Open AccessArticle

Second-Order Damped Differential Equations with Superlinear Neutral Term: New Criteria for Oscillation

by Asma Al-Jaser, Clemente Cesarano, Belgees Qaraad and Loredana Florentina Iambor

Axioms 2024, 13(4), 234; https://doi.org/10.3390/axioms13040234 - 1 Apr 2024

Cited by 4 | Viewed by 1224

Abstract

This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. [...] Read more.

This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. To prove these results, we employed various analysis methods, establishing new relationships to address certain problems that have hindered previous research. Consequently, by applying the principles of comparison and the Riccati transformation, we obtained findings that develop and complement those reported in earlier literature. The significance of our results is illustrated with several examples. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics)

16 pages, 1040 KiB

Open AccessArticle

Impact of White Noise on the Exact Solutions of the Stochastic Riemann Wave Equation in Quantum Mechanics

by Wael W. Mohammed, Clemente Cesarano, Doaa Rizk, Elkhateeb S. Aly and Mahmoud El-Morshedy

Symmetry 2023, 15(11), 2070; https://doi.org/10.3390/sym15112070 - 16 Nov 2023

Cited by 4 | Viewed by 1609

Abstract

In this article, the stochastic Riemann wave equation (SRWE) forced by white noise in the Itô sense is considered. The extended tanh function and mapping methods are applied to obtain new elliptic, rational, hyperbolic, and trigonometric stochastic solutions. Furthermore, we generalize some previous [...] Read more.

In this article, the stochastic Riemann wave equation (SRWE) forced by white noise in the Itô sense is considered. The extended tanh function and mapping methods are applied to obtain new elliptic, rational, hyperbolic, and trigonometric stochastic solutions. Furthermore, we generalize some previous studies. The obtained solutions are important in explaining some exciting physical phenomena, since the SRWE is required for describing wave propagation. We plot numerous 3D and 2D graphical representations to explain how the multiplicative white noise influences the exact solutions of the SRWE. We can infer that the introduction of multiplicative white noise disrupts the symmetry of the solutions and serves to stabilize the solutions of the SRWE. Full article

(This article belongs to the Section Mathematics)

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18 pages, 333 KiB

Open AccessArticle

Kneser-Type Oscillation Criteria for Half-Linear Delay Differential Equations of Third Order

by Fahd Masood, Clemente Cesarano, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani and Hamdy El-Metwally

Symmetry 2023, 15(11), 1994; https://doi.org/10.3390/sym15111994 - 29 Oct 2023

Cited by 10 | Viewed by 1313

Abstract

This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, [...] Read more.

This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

15 pages, 280 KiB

Open AccessArticle

On Some Quasi-Curves in Galilean Three-Space

by Ayman Elsharkawy, Yusra Tashkandy, Walid Emam, Clemente Cesarano and Noha Elsharkawy

Axioms 2023, 12(9), 823; https://doi.org/10.3390/axioms12090823 - 27 Aug 2023

Cited by 8 | Viewed by 1496

Abstract

In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute [...] Read more.

In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute is studied. Moreover, we prove that there is no quasi-evolute curve in Galilean three-space. Also, we introduce quasi-Smarandache curves in Galilean three-space. Finally, we demonstrate an illustrated example to present our findings. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)

17 pages, 1633 KiB

Open AccessArticle

On the Dynamical Behavior of Solitary Waves for Coupled Stochastic Korteweg–De Vries Equations

by Wael W. Mohammed, Farah M. Al-Askar and Clemente Cesarano

Mathematics 2023, 11(16), 3506; https://doi.org/10.3390/math11163506 - 14 Aug 2023

Cited by 14 | Viewed by 1415

Abstract

In this paper, we take into account the coupled stochastic Korteweg–De Vries (CSKdV) equations in the Itô sense. Using the mapping method, new trigonometric, rational, hyperbolic, and elliptic stochastic solutions are obtained. These obtained solutions can be applied to the analysis of a [...] Read more.

In this paper, we take into account the coupled stochastic Korteweg–De Vries (CSKdV) equations in the Itô sense. Using the mapping method, new trigonometric, rational, hyperbolic, and elliptic stochastic solutions are obtained. These obtained solutions can be applied to the analysis of a wide variety of crucial physical phenomena because the coupled KdV equations have important applications in various fields of physics and engineering. Also, it is used in the design of optical fiber communication systems, which transmit information using soliton-like waves. The dynamic performance of the various obtained solutions are depicted using 3D and 2D curves in order to interpret the effects of multiplicative noise. We conclude that multiplicative noise influences the behavior of the solutions of CSKdV equations and stabilizes them. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)

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11 pages, 2677 KiB

Open AccessArticle

Effects of the Wiener Process and Beta Derivative on the Exact Solutions of the Kadomtsev–Petviashvili Equation

by Farah M. Al-Askar, Clemente Cesarano and Wael W. Mohammed

Axioms 2023, 12(8), 748; https://doi.org/10.3390/axioms12080748 - 29 Jul 2023

Cited by 11 | Viewed by 1073

Abstract

We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, the Riccati equation and Jacobi elliptic function methods are employed. Because the KP equation is required for [...] Read more.

We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, the Riccati equation and Jacobi elliptic function methods are employed. Because the KP equation is required for explaining the development of quasi-one-dimensional shallow-water waves, the solutions obtained can be used to interpret various attractive physical phenomena. To display how the multiplicative white noise and beta-derivative impact the exact solutions of the SKPE-BD, we plot a few graphs in MATLAB and display different 3D and 2D figures. We deduce how multiplicative noise stabilizes the solutions of SKPE-BD at zero. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

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13 pages, 1745 KiB

Open AccessArticle

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 1311

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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15 pages, 1818 KiB

Open AccessArticle

Solitary Solutions for the Stochastic Fokas System Found in Monomode Optical Fibers

by Wael W. Mohammed, Farah M. Al-Askar and Clemente Cesarano

Symmetry 2023, 15(7), 1433; https://doi.org/10.3390/sym15071433 - 17 Jul 2023

Cited by 14 | Viewed by 1301

Abstract

The stochastic Fokas system (SFS), driven by multiplicative noise in the Itô sense, was investigated in this study. Novel trigonometric, rational, hyperbolic, and elliptic stochastic solutions are found using a modified mapping method. Because the Fokas system is used to explain nonlinear pulse [...] Read more.

The stochastic Fokas system (SFS), driven by multiplicative noise in the Itô sense, was investigated in this study. Novel trigonometric, rational, hyperbolic, and elliptic stochastic solutions are found using a modified mapping method. Because the Fokas system is used to explain nonlinear pulse propagation in monomode optical fibers, the solutions provided may be utilized to analyze a broad range of critical physical phenomena. In order to explain the impacts of multiplicative noise, the dynamic performances of the different found solutions are illustrated using 3D and 2D curves. We conclude that multiplicative noise eliminates the symmetry of the solutions of the SFS and stabilizes them. Full article

(This article belongs to the Special Issue Recent Advances in Symmetries Methods and Other Approaches to Nonlinear Differential Equations)

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12 pages, 718 KiB

Open AccessArticle

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 1406

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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11 pages, 288 KiB

Open AccessArticle

Oscillation Criteria for Qusilinear Even-Order Differential Equations

by Mnaouer Kachout, Clemente Cesarano, Amir Abdel Menaem, Taher S. Hassan and Belal A. Glalah

Mathematics 2023, 11(12), 2782; https://doi.org/10.3390/math11122782 - 20 Jun 2023

Viewed by 3140

Abstract

In this study, we extended and improved the oscillation criteria previously established for second-order differential equations to even-order differential equations. Some examples are given to demonstrate the significance of the results accomplished. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

10 pages, 1428 KiB

Open AccessArticle

The Solitary Solutions for the Stochastic Jimbo–Miwa Equation Perturbed by White Noise

by Farah M. Al-Askar, Clemente Cesarano and Wael W. Mohammed

Symmetry 2023, 15(6), 1153; https://doi.org/10.3390/sym15061153 - 26 May 2023

Cited by 16 | Viewed by 1771

Abstract

We study the (3+1)-dimensional stochastic Jimbo–Miwa (SJM) equation induced by multiplicative white noise in the Itô sense. We employ the Riccati equation mapping and He’s semi-inverse techniques to provide trigonometric, hyperbolic, and rational function solutions of SJME. Due to the applications of the [...] Read more.

We study the (3+1)-dimensional stochastic Jimbo–Miwa (SJM) equation induced by multiplicative white noise in the Itô sense. We employ the Riccati equation mapping and He’s semi-inverse techniques to provide trigonometric, hyperbolic, and rational function solutions of SJME. Due to the applications of the Jimbo–Miwa equation in ocean studies and other disciplines, the acquired solutions may explain numerous fascinating physical phenomena. Using a variety of 2D and 3D diagrams, we illustrate how white noise influences the analytical solutions of SJM equation. We deduce that the noise destroys the symmetry of the solutions of SJM equation and stabilizes them at zero. Full article

(This article belongs to the Special Issue Partial Differential Equations and Their Applications in Nonlinear Optics)

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9 pages, 862 KiB

Open AccessArticle

On the Dynamics of Solitary Waves to a (3+1)-Dimensional Stochastic Boiti–Leon–Manna–Pempinelli Model in Incompressible Fluid

by Wael W. Mohammed, Farah M. Al-Askar, Clemente Cesarano and M. El-Morshedy

Mathematics 2023, 11(10), 2390; https://doi.org/10.3390/math11102390 - 22 May 2023

Cited by 4 | Viewed by 1772

Abstract

We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He’s semi-inverse method and the Riccati equation mapping method, we can acquire the solutions of the SBLMPE. Since the Boiti–Leon–Manna–Pempinelli equation is utilized to [...] Read more.

We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He’s semi-inverse method and the Riccati equation mapping method, we can acquire the solutions of the SBLMPE. Since the Boiti–Leon–Manna–Pempinelli equation is utilized to explain incompressible liquid in fluid mechanics, the acquired solutions may be applied to explain a lot of fascinating physical phenomena. To address how Brownian motion effects the exact solutions of the SBLMPE, we present some 2D and 3D diagrams. Full article

(This article belongs to the Special Issue Spectral Theory and Its Applications in Problems of Mathematical Physics)

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