A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

Mohammed Djaouti, Abdelhamid; Khan, Zareen A.; Imran Liaqat, Muhammad; Al-Quran, Ashraf

doi:10.3390/math12111654

Mathematics

2024

DOI: 10.3390/math12111654

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A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

Abdelhamid Mohammed Djaouti,

Zareen A. Khan,

Muhammad Imran Liaqat

et al.

Abstract: Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p=2 to results in the Lp space; second, by incorporating the Caputo–… Show more

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Cited by 4 publications

References 64 publications

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A novel technique using integral transforms and residual functions for nonlinear partial fractional differential equations involving Caputo derivatives

Khan,

Riaz,

Liaqat

et al. 2024

PLoS ONE

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Fractional nonlinear partial differential equations are used in many scientific fields to model various processes, although most of these equations lack closed-form solutions. For this reason, methods for approximating solutions that occasionally yield closed-form solutions are crucial for solving these equations. This study introduces a novel technique that combines the residual function and a modified fractional power series with the Elzaki transform to solve various nonlinear problems within the Caputo derivative framework. The accuracy and effectiveness of our approach are validated through analyses of absolute, relative, and residual errors. We utilize the limit principle at zero to identify the coefficients of the series solution terms, while other methods, including variational iteration, homotopy perturbation, and Adomian, depend on integration. In contrast, the residual power series method uses differentiation, and both approaches encounter difficulties in fractional contexts. Furthermore, the effectiveness of our approach in addressing nonlinear problems without relying on Adomian and He polynomials enhances its superiority over various approximate series solution techniques.

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A novel technique using integral transforms and residual functions for nonlinear partial fractional differential equations involving Caputo derivatives

Khan,

Riaz,

Liaqat

et al. 2024

PLoS ONE

View full text Add to dashboard Cite

show abstract

Analytical Analysis for Space Fractional Helmholtz Equations by Using The Hybrid Efficient Approach

Khan,

Liaqat,

Mushtaq

2024

Acta Mechanica Et Automatica

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The Helmholtz equation is an important differential equation. It has a wide range of uses in physics, including acoustics, electro-statics, optics, and quantum mechanics. In this article, a hybrid approach called the Shehu transform decomposition method (STDM) is implemented to solve space-fractional-order Helmholtz equations with initial boundary conditions. The fractional-order derivative is regarded in the Caputo sense. The solutions are provided as series, and then we use the Mittag-Leffler function to identify the exact solutions to the Helmholtz equations. The accuracy of the considered problem is examined graphically and numerically by the absolute, relative, and recurrence errors of the three problems. For different values of fractional-order derivatives, graphs are also developed. The results show that our approach can be a suitable alternative to the approximate methods that exist in the literature to solve fractional differential equations.

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Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations

Mohammed Djaouti,

Imran Liaqat

2024

Axioms

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Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in Lp spaces with p≥2 under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end.

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A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

Cited by 4 publications

References 64 publications

A novel technique using integral transforms and residual functions for nonlinear partial fractional differential equations involving Caputo derivatives

A novel technique using integral transforms and residual functions for nonlinear partial fractional differential equations involving Caputo derivatives

Analytical Analysis for Space Fractional Helmholtz Equations by Using The Hybrid Efficient Approach

Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations

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