Browsing by Author "Aydin, Mustafa"

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A Couple of Novel Image Enhancement Methods Depending on the Prabhakar Fractional Approaches
(Springer London Ltd, 2024) Topal, Ahmet; Aydin, Mustafa
Integrating fractional calculus into image processing techniques offers a useful and robust approach. In this study, we proposed contrast enhancement filters using Prabhakar fractional integral operator based on Grunwald-Letnikov and forward Euler. We evaluated the performance of the proposed enhancement methods on both high and low contrast images and compared them with fractional and non-fractional contrast enhancement methods. To demonstrate the superiority of our methods, we employed five different image quality metrics: PSNR, MSE, SSIM, FSIM, and entropy. For low contrast images, our methods not only achieved acceptable results for each metric-PSNR values above 25, SSIM values above 0.9, MSE values below 200, FSIM values above 0.97, and entropy values above 7-but also demonstrated better performance compared to other methods. In high contrast images, despite an overall decline in metric scores, the Grunwald-Letnikov based method remains the leading approach among both fractional and non-fractional methods. Additionally, empirical results provide evidence that the proposed methods are more effective in enhancing low contrast images compared to high contrast images.
Langevin Delayed Equations With Prabhakar Derivatives Involving Two Generalized Fractional Distinct Orders
(Tubitak Scientific & Technological Research Council Turkey, 2024) Aydin, Mustafa
This paper is devoted to defining the delayed analogue of the Mittag-Leffler type function with three parameters and investigating a representation of a solution to Langevin delayed equations with Prabhakar derivatives involving two generalized fractional distinct orders, which are first introduced and investigated, by means of the Laplace integral transform. It is verified by showing the solution satisfies the introduced system. Special cases which are also novel are presented as examples. The findings are illustrated with the help of the RLC circuits.
Relative Controllability of Fractional Dynamical Systems With a Delay in State and Multiple Delays in Control
(Wiley, 2025) Aydin, Mustafa; Mahmudov, Nazim I.
This work is devoted to the study of the relative controllability of fractional dynamic systems in finite-dimensional spaces with a state delay and multiple delays in control. For linear systems to be relatively controllable, necessary and sufficient conditions are determined by defining and using the Gramian matrix. The controllability conditions for semilinear systems are determined on the basis of Schauder's fixed point theorem.
Representation of Solutions To Tempered Delayed Ψ-Fractional Systems With Noncommutative Coefficients
(Pergamon–Elsevier Science Ltd, 2025) Aydin, Mustafa; Mahmudov, Nazim I.
This paper focuses on deriving explicit solutions for tempered delayed fractional differential systems that utilize Caputo fractional derivatives in relation to another function. To achieve this, we define tempered yr-delayed perturbations of Mittag-Leffler type functions and explore their yr-Laplace transforms. Additionally, we discuss theorems related to shifting and time-delay in the context of yr-Laplace transforms. Utilizing the tempered yr-delayed perturbational function, we establish a representation of explicit solutions for the system through the Laplace transform method. This representation is validated by demonstrating that it satisfies the system, alongside employing the method of variation of constants. Several novel special cases are introduced, and a numerical example is provided to demonstrate the practical application of the results obtained.
Some Applications of the Generalized Laplace Transform and the Representation of a Solution To Sobolev-Type Evolution Equations With the Generalized Caputo Derivative
(Polska Akad Nauk, Polish Acad Sci, Div Iv Technical Sciences Pas, 2024) Aydin, Mustafa; Mahmudov, Nazim I.
We introduce the Sobolev-type multi -term mu-fractional evolution with generalized fractional orders with respect to another function. We make some applications of the generalized Laplace transform. In the sequel, we propose a novel type of Mittag-Leffler function generated by noncommutative linear bounded operators with respect to the given function and give a few of its properties. We look for the mild solution formula of the Sobolev-type evolution equation by building on the aforementioned Mittag-Leffler-type function with the aid of two different approaches. We share new special cases of the obtained findings.
A Study on Linear Prabhakar Fractional Systems With Variable Coefficients
(Springer Basel Ag, 2024) Aydin, Mustafa; Mahmudov, N. I.
The focus of this paper is on addressing the initial value problem related to linear systems of fractional differential equations characterized by variable coefficients, incorporating Prabhakar fractional derivatives of Riemann-Liouville and Caputo types. Utilizing the generalized Peano-Baker series technique, the state-transition matrix is acquired. The paper presents closed form solutions for both homogeneous and inhomogeneous cases, substantiated by illustrative examples.
The Μ-Neutral Fractional Multi-Delayed Differential Equations
(Univ Miskolc inst Math, 2024) Aydin, Mustafa; Mahmudov, Nazim i.
The mu -neutral linear fractional multi -delayed differential nonhomogeneous system with noncommutative coefficient matrices is introduced. The novel mu -neutral multi -delayed perturbation of Mittag-Leffler type matrix function is proposed. Based on this, an explicit solution to the system is investigated step by step. The existence uniqueness of solutions to mu -neutral nonlinear fractional multi -delayed differential system is obtained with regard to the supremum norm. The notion of stability analysis in the sense of solutions to the described system is discussed on the grounds of the fixed point approach.
ψ-Caputo Type Time-Delay Langevin Equations With Two General Fractional Orders
(Wiley, 2023) Aydin, Mustafa; Mahmudov, Nazim I.
In the present paper, first, a Psi-delayed Mittag-Leffler type function is introduced, which generalizes the existing delayed Mittag-Leffler type function. Second, by means of Psi-delayed Mittag-Leffler type function, an exact analytical solution formula to non-homogeneous linear delayed Langevin equations involving two distinct Psi-Caputo type fractional derivatives of general orders is obtained. Moreover, existence and uniqueness, stability of solution to nonlinear delayed Langevin fractional differential equations is obtained in the weighted space. Numerical and simulated examples are shared to exemplify the theoretical findings.