Analytical Solutions for Generalized Stochastic HSC-KdV Equations with Variable Coefficients Using Hermite Transform and F-Expansion Method

Zakarya, Mohammed; Al-Shehri, Nadiah Zafer; Ali, Hegagi M.; Abd-Rabo, Mahmoud A.; Rezk, Haytham M.

doi:10.3390/axioms14080624

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Analytical Solutions for Generalized Stochastic HSC-KdV Equations with Variable Coefficients Using Hermite Transform and F-Expansion Method

by

Mohammed Zakarya

¹

,

Nadiah Zafer Al-Shehri

¹,

Hegagi M. Ali

²

,

Mahmoud A. Abd-Rabo

³

and

Haytham M. Rezk

^4,*

¹

Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia

²

Department of Mathematics, College of Science, University of Bisha, Bisha 61922, Saudi Arabia

³

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt

⁴

Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt

^*

Author to whom correspondence should be addressed.

Axioms 2025, 14(8), 624; https://doi.org/10.3390/axioms14080624

Submission received: 26 June 2025 / Revised: 5 August 2025 / Accepted: 6 August 2025 / Published: 10 August 2025

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Abstract

This study focuses on analyzing the generalized HSC-KdV equations characterized by variable coefficients and Wick-type stochastic (Wt.S) elements. To derive white noise functional (WNF) solutions, we employ the Hermite transform, the homogeneous balance principle, and the

F_{e}

(F-expansion) technique. Leveraging the inherent connection between hypercomplex system (HCS) theory and white noise (WN) analysis, we establish a comprehensive framework for exploring stochastic partial differential equations (PDEs) involving non-Gaussian parameters (N-GP). As a result, exact solutions expressed through Jacobi elliptic functions (JEFs) and trigonometric and hyperbolic forms are obtained for both the variable coefficients and stochastic forms of the generalized HSC-KdV equations. An illustrative example is included to validate the theoretical findings.

Keywords: generalized HSC-KdV equations; Hermite transform; F-expansion technique; Wick-type Stochastic analysis; non-Gaussian parameters; white noise functional approach

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MDPI and ACS Style

Zakarya, M.; Al-Shehri, N.Z.; Ali, H.M.; Abd-Rabo, M.A.; Rezk, H.M. Analytical Solutions for Generalized Stochastic HSC-KdV Equations with Variable Coefficients Using Hermite Transform and F-Expansion Method. Axioms 2025, 14, 624. https://doi.org/10.3390/axioms14080624

AMA Style

Zakarya M, Al-Shehri NZ, Ali HM, Abd-Rabo MA, Rezk HM. Analytical Solutions for Generalized Stochastic HSC-KdV Equations with Variable Coefficients Using Hermite Transform and F-Expansion Method. Axioms. 2025; 14(8):624. https://doi.org/10.3390/axioms14080624

Chicago/Turabian Style

Zakarya, Mohammed, Nadiah Zafer Al-Shehri, Hegagi M. Ali, Mahmoud A. Abd-Rabo, and Haytham M. Rezk. 2025. "Analytical Solutions for Generalized Stochastic HSC-KdV Equations with Variable Coefficients Using Hermite Transform and F-Expansion Method" Axioms 14, no. 8: 624. https://doi.org/10.3390/axioms14080624

APA Style

Zakarya, M., Al-Shehri, N. Z., Ali, H. M., Abd-Rabo, M. A., & Rezk, H. M. (2025). Analytical Solutions for Generalized Stochastic HSC-KdV Equations with Variable Coefficients Using Hermite Transform and F-Expansion Method. Axioms, 14(8), 624. https://doi.org/10.3390/axioms14080624

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Analytical Solutions for Generalized Stochastic HSC-KdV Equations with Variable Coefficients Using Hermite Transform and F-Expansion Method

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