A new class of fractional Orlicz-Sobolev space and singular elliptic problems

Boujemaa, Hamza; Oulgiht, Badr; Ragusa, Maria Alessandra

doi:10.1016/j.jmaa.2023.127342

Cited by 9 publications

(3 citation statements)

References 36 publications

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“…Stability theory in mathematics studies the behavior and robustness of solutions in differential equations, integral equations, fractional integral equations, and dynamical systems under small perturbations of initial conditions (see [33][34][35]). It explores whether solutions remain bounded and predictable or exhibit unpredictable and divergent behavior.…”

Section: Stability Analysis Of the Solutionmentioning

confidence: 99%

A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

Mamatova,

Eshkuvatov,

Ismail

2023

Mathematics

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In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted Lp space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method’s validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others.

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Section: Stability Analysis Of the Solutionmentioning

confidence: 99%

A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

Mamatova,

Eshkuvatov,

Ismail

2023

Mathematics

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show abstract

“…Fractional calculus, with its three-century history, was long confined to the realm of pure mathematical analysis and largely ignored by engineers [1][2][3][4][5]. It was not until Mandelbrot introduced fractal theory, linking Riemann-Liouville fractional calculus to Brownian motion in fractal media [6][7][8], that the field captured the attention of engineering technologists.…”

Section: Introductionmentioning

confidence: 99%

Research on Application of Fractional Calculus Operator in Image Underlying Processing

Huang,

Qin,

Chen

et al. 2024

Fractal Fract

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Fractional calculus extends traditional, integer-based calculus to include non-integer orders, offering a powerful tool for a range of engineering applications, including image processing. This work delves into the utility of fractional calculus in two crucial aspects of image processing: image enhancement and denoising. We explore the foundational theories of fractional calculus together with its amplitude–frequency characteristics. Our focus is on the effectiveness of fractional differential operators in enhancing image features and reducing noise. Experimental results reveal that fractional calculus offers unique benefits for image enhancement and denoising. Specifically, fractional-order differential operators outperform their integer-order counterparts in accentuating details such as weak edges and strong textures in images. Moreover, fractional integral operators excel in denoising images, not only improving the signal-to-noise ratio but also better preserving essential features such as edges and textures when compared to traditional denoising techniques. Our empirical results affirm the effectiveness of the fractional-order calculus-based image-processing approach in yielding optimal results for low-level image processing.

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“…Fractional calculus, initially developed to construct non-integer derivatives and integrals, constitutes a powerful mathematical framework for elucidating diverse phenomena across various scientific domains [1][2][3][4]. In the realm of scientific modeling, it has become increasingly valuable, particularly for simulating numerous physical processes characterized by power law behaviors, multi-scale media, or non-Gaussian statistics, as evidenced by various studies [5,6].…”

Section: Introductionmentioning

confidence: 99%

The Fractional Soliton Wave Propagation of Non-Linear Volatility and Option Pricing Systems with a Sensitive Demonstration

Riaz,

Ansari,

Jhangeer

et al. 2023

Fractal Fract

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In this study, we explore a fractional non-linear coupled option pricing and volatility system. The model under consideration can be viewed as a fractional non-linear coupled wave alternative to the Black–Scholes option pricing governing system, introducing a leveraging effect where stock volatility corresponds to stock returns. Employing the inverse scattering transformation, we find that the Cauchy problem for this model is insolvable. Consequently, we utilize the Φ6-expansion algorithm to generate generalized novel solitonic analytical wave structures within the system. We present graphical representations in contour, 3D, and 2D formats to illustrate how the system’s behavior responds to the propagation of pulses, enabling us to predict suitable parameter values that align with the data. Finally, a conclusion is given.

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A new class of fractional Orlicz-Sobolev space and singular elliptic problems

Cited by 9 publications

References 36 publications

A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

Research on Application of Fractional Calculus Operator in Image Underlying Processing

The Fractional Soliton Wave Propagation of Non-Linear Volatility and Option Pricing Systems with a Sensitive Demonstration

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