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10 pages, 674 KiB

Open AccessArticle

Abundant Exact Traveling-Wave Solutions for Stochastic Graphene Sheets Model

by Wael W. Mohammed, Taher S. Hassan, Rabeb Sidaoui, Hijyah Alshammary and Mohamed S. Algolam

Axioms 2025, 14(6), 477; https://doi.org/10.3390/axioms14060477 - 19 Jun 2025

Viewed by 223

Here, we consider the stochastic graphene sheets model (SGSM) forced by multiplicative noise in the Itô sense. We show that the exact solution of the SGSM may be obtained by solving some deterministic counterparts of the graphene sheets model and combining the result [...] Read more.

Here, we consider the stochastic graphene sheets model (SGSM) forced by multiplicative noise in the Itô sense. We show that the exact solution of the SGSM may be obtained by solving some deterministic counterparts of the graphene sheets model and combining the result with a solution of stochastic ordinary differential equations. By applying the extended tanh function method, we obtain the soliton solutions for the deterministic counterparts of the graphene sheets model. Because graphene sheets are important in many fields, such as electronics, photonics, and energy storage, the solutions of the stochastic graphene sheets model are beneficial for understanding several fascinating scientific phenomena. Using the MATLAB program, we exhibit several 3D graphs that illustrate the impact of multiplicative noise on the exact solutions of the SGSM. By incorporating stochastic elements into the equations that govern the evolution of graphene sheets, researchers can gain insights into how these fluctuations impact the behavior of the material over time. Full article

(This article belongs to the Special Issue Advances in Partial Differential Equations: Qualitative Analysis and Numerical Methods)

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12 pages, 1132 KiB

Open AccessArticle

Random Traveling Wave Equations for the Heisenberg Ferromagnetic Spin Chain Model and Their Optical Stochastic Solutions in a Ferromagnetic Materials

by Wael W. Mohammed, Fakhr Gassem and Rabeb Sidaoui

Axioms 2024, 13(12), 864; https://doi.org/10.3390/axioms13120864 - 10 Dec 2024

Cited by 1 | Viewed by 764

Abstract

In this paper, we investigate the stochastic Heisenberg ferromagnetic equation (SHFE) derived by a multiplicative Wiener process. We use a suitable transformation to change the SHF equation into another Heisenberg ferromagnetic equation with random variable coefficients (HFE-RVCs). We employ the mapping approach to [...] Read more.

In this paper, we investigate the stochastic Heisenberg ferromagnetic equation (SHFE) derived by a multiplicative Wiener process. We use a suitable transformation to change the SHF equation into another Heisenberg ferromagnetic equation with random variable coefficients (HFE-RVCs). We employ the mapping approach to obtain novel rational, trigonometric, elliptic and hyperbolic function solutions for HFE-RVCs. Following that, we can attain the solutions of the SHFE. For the first time in the Heisenberg ferromagnetic equation, we postulate that the solution to the wave equation is stochastic, whereas all previous investigations supposed that it was deterministic. Moreover, we give various visual representations to demonstrate the impact of the multiplicative Wiener process on the exact solutions to the stochastic Heisenberg ferromagnetic equation. Full article

(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)

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12 pages, 275 KiB

Open AccessArticle

Travelling Solitary Wave Solutions to Non-Gaussian χ-Wick-Type Stochastic Burgers’ Equation with Variable Coefficients

by Mohammed Zakarya, Manal Al-Qarni and Tahani Al-Qahtani

Symmetry 2024, 16(12), 1572; https://doi.org/10.3390/sym16121572 - 24 Nov 2024

Viewed by 665

Abstract

In this work, we obtain non-Gaussian (NG) stochastic solutions to

χ

-Wick-type stochastic (

χ

-Wk-TS) Burgers’ equations with variable coefficients. An Exp-function method, the connection between white noise theory and hypercomplex systems (HCSs), the

χ

-Wick product (

χ

-Wk-product) and an [...] Read more.

In this work, we obtain non-Gaussian (NG) stochastic solutions to

χ

-Wick-type stochastic (

χ

-Wk-TS) Burgers’ equations with variable coefficients. An Exp-function method, the connection between white noise theory and hypercomplex systems (HCSs), the

χ

-Wick product (

χ

-Wk-product) and an

χ

-Hermite transform (

χ

-Hr-transform) are proposed. We provide a new set of non-Gaussian solitary wave solutions (NG-SWSs) to Burgers’ equations with variable coefficients. NG white noise functional solutions (NG-WNFSs) to

χ

-Wk-TS Burgers’ equations with variable coefficients are shown. The symmetry coefficients of partial differential equations and the symmetrical properties of SPDEs are critical in determining the best solution. Full article

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

13 pages, 251 KiB

Open AccessArticle

Asymptotic Behavior of Stochastic Reaction–Diffusion Equations

by Hao Wen, Yuanjing Wang, Guangyuan Liu and Dawei Liu

Mathematics 2024, 12(9), 1284; https://doi.org/10.3390/math12091284 - 24 Apr 2024

Viewed by 998

Abstract

In this paper, we concentrate on the propagation dynamics of stochastic reaction–diffusion equations, including the existence of travelling wave solution and asymptotic wave speed. Based on the stochastic Feynman–Kac formula and comparison principle, the boundedness of the solution of stochastic reaction–diffusion equations can [...] Read more.

In this paper, we concentrate on the propagation dynamics of stochastic reaction–diffusion equations, including the existence of travelling wave solution and asymptotic wave speed. Based on the stochastic Feynman–Kac formula and comparison principle, the boundedness of the solution of stochastic reaction–diffusion equations can be obtained so that we can construct a sup-solution and a sub-solution to estimate the upper bound and the lower bound of wave speed. Full article

(This article belongs to the Special Issue Dynamics of Predator-Prey and Infectious Disease Models)

13 pages, 565 KiB

Open AccessArticle

Characteristics of Solitary Stochastic Structures for Heisenberg Ferromagnetic Spin Chain Equation

by Munerah Almulhem, Samia Z. Hassan, Alanwood Al-buainain, Mohammed A. Sohaly and Mahmoud A. E. Abdelrahman

Symmetry 2023, 15(4), 927; https://doi.org/10.3390/sym15040927 - 17 Apr 2023

Cited by 1 | Viewed by 1437

Abstract

The impact of Stratonovich integrals on the solutions of the Heisenberg ferromagnetic spin chain equation using the unified solver approach is examined in this study. In particular, using arbitrary parameters, the traveling wave arrangements of rational, trigonometric, and hyperbolic functions are developed. The [...] Read more.

The impact of Stratonovich integrals on the solutions of the Heisenberg ferromagnetic spin chain equation using the unified solver approach is examined in this study. In particular, using arbitrary parameters, the traveling wave arrangements of rational, trigonometric, and hyperbolic functions are developed. The detailed arrangements are exceptionally critical for clarifying diverse complex wonders in plasma material science, optical fiber, quantum mechanics, super liquids and so on. Here, the Itô stochastic calculus and the Stratonovich stochastic calculus are considered. To describe the dynamic behaviour of random solutions, some graphical representations for these solutions are described with appropriate parameters. Full article

(This article belongs to the Section Mathematics)

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13 pages, 598 KiB

Open AccessArticle

Hypercomplex Systems and Non-Gaussian Stochastic Solutions with Some Numerical Simulation of χ-Wick-Type (2 + 1)-D C-KdV Equations

by Mohammed Zakarya, Mahmoud A. Abd-Rabo and Ghada AlNemer

Axioms 2022, 11(11), 658; https://doi.org/10.3390/axioms11110658 - 21 Nov 2022

Cited by 1 | Viewed by 1770

Abstract

In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the

χ

-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, [...] Read more.

In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the

χ

-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefficients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively. Full article

(This article belongs to the Special Issue Recent Advances in Stochastic Differential Equations)

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19 pages, 1351 KiB

Open AccessArticle

Fractional Biswas–Milovic Equation in Random Case Study

by Abdulwahab Almutairi

Fractal Fract. 2022, 6(11), 687; https://doi.org/10.3390/fractalfract6110687 - 19 Nov 2022

Viewed by 1279

Abstract

We apply two mathematical techniques, specifically, the unified solver approach and the exp

(- φ (ξ))

-expansion method, for constructing many new solitary waves, such as bright, dark, and singular soliton solutions via the fractional Biswas–Milovic (FBM) model in [...] Read more.

We apply two mathematical techniques, specifically, the unified solver approach and the exp

(- φ (ξ))

-expansion method, for constructing many new solitary waves, such as bright, dark, and singular soliton solutions via the fractional Biswas–Milovic (FBM) model in the sense of conformable fractional derivative. These solutions are so important for the explanation of some practical physical problems. Additionally, we study the stochastic modeling for the fractional Biswas–Milovic, where the parameter and the fraction parameters are random variables. We consider these parameters via beta distribution, so the mathematical methods that were used in this paper may be called random methods, and the exact solutions derived using these methods may be called stochastic process solutions. We also determined some statistical properties of the stochastic solutions such as the first and second moments. The proposed techniques are robust and sturdy for solving wide classes of nonlinear fractional order equations. Finally, some selected solutions are illustrated for some special values of parameters. Full article

(This article belongs to the Section Mathematical Physics)

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15 pages, 316 KiB

Open AccessArticle

Exact Solutions for a Class of Wick-Type Stochastic (3+1)-Dimensional Modified Benjamin–Bona–Mahony Equations

by Praveen Agarwal, Abd-Allah Hyder, M. Zakarya, Ghada AlNemer, Clemente Cesarano and Dario Assante

Axioms 2019, 8(4), 134; https://doi.org/10.3390/axioms8040134 - 3 Dec 2019

Cited by 22 | Viewed by 3133

Abstract

In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling [...] Read more.

In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)

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