Search Results (19)

This paper introduces a novel numerical approach for solving fractional stochastic differential equations (FSDEs) using bilinear time-series models, driven by the Caputo–Katugampola (C-K) fractional derivative. The C-K operator generalizes classical fractional derivatives by incorporating an additional parameter, enabling the enhanced modeling of memory effects and hereditary properties in stochastic systems. The primary contribution of this work is the development of an efficient numerical framework that combines bilinear time-series discretization with the C-K derivative to approximate solutions for FSDEs, which are otherwise analytically intractable due to their nonlinear and memory-dependent nature. We rigorously analyze the impact of fractional-order dynamics on system behavior. The bilinear time-series framework provides a computationally efficient alternative to traditional methods, leveraging multiplicative interactions between past observations and stochastic innovations to model complex dependencies. A key advantage of our approach is its flexibility in handling both stochasticity and fractional-order effects, making it suitable for applications in a famous nuclear physics model. To validate the method, we conduct a comparative analysis between exact solutions and numerical approximations, evaluating convergence properties under varying fractional orders and discretization steps. Our results demonstrate robust convergence, with simulations highlighting the superior accuracy of the C-K operator over classical fractional derivatives in preserving system dynamics. Additionally, we provide theoretical insights into the stability and error bounds of the discretization scheme. Using the changes in the number of simulations and the operator parameters of Caputo–Katugampola, we can extract some properties of the stochastic fractional differential model, and also note the influence of Brownian motion and its formulation on the model, the main idea posed in our contribution based on constructing the fractional solution of a proposed fractional model using known bilinear time series illustrated by application in nuclear physics models. Full article

In this work, we derive novel theoretical results concerning well-posedness and Ulam–Hyers stability. Specifically, we investigate the well-posedness of Caputo–Katugampola fractional stochastic delay integro-differential equations. Additionally, we develop a generalized Grönwall inequality and apply it to prove Ulam–Hyers stability in ${\mathcal{L}}^{\mathrm{p}}$ space. Our findings generalize existing results for fractional derivatives and space, as we formulate them in the Caputo–Katugampola fractional derivative and ${\mathcal{L}}^{\mathrm{p}}$ space. To support our theoretical results, we present an illustrative example. Full article

The importance of this research comes from the several applications of the Mathieu equation and its generalizations in many scientific fields. Two models of fractional Mathieu equations are provided using Katugampola fractional derivatives in the sense of Riemann-Liouville and Caputo. Each model contains two fractional derivatives with unique fractional orders, periodic forcing of the cosine stiffness coefficient, and many extensions and generalizations. The Banach contraction principle is used to prove that each model under consideration has a unique solution. Our results are applied to four real-life problems: the nonlinear Mathieu equation for parametric damping and the Duffing oscillator, the quadratically damped Mathieu equation, the fractional Mathieu equation’s transition curves, and the tempered fractional model of the linearly damped ion motion with an octopole. Full article

Stochastic pantograph fractional differential equations (SPFDEs) combine three intricate components: stochastic processes, fractional calculus, and pantograph terms. These equations are important because they allow us to model and analyze systems with complex behaviors that traditional differential equations cannot capture. In this study, we achieve significant results for these equations within the context of Caputo–Katugampola derivatives. First, we establish the existence and uniqueness of solutions by employing the contraction mapping principle with a suitably weighted norm and demonstrate that the solutions continuously depend on both the initial values and the fractional exponent. The second part examines the regularity concerning time. Third, we illustrate the results of the averaging principle using techniques involving inequalities and interval translations. We generalize these results in two ways: first, by establishing them in the sense of the Caputo–Katugampola derivative. Applying condition $\beta =1$, we derive the results within the framework of the Caputo derivative, while condition $\beta \to 0^{+}$ yields them in the context of the Caputo–Hadamard derivative. Second, we establish them in $L^p$ space, thereby generalizing the case for $p=2$. Full article

The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing $\epsilon$ in front of the drift term and $\sqrt{\epsilon }$ in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in ${\mathrm{L}}^{\mathrm{p}}$ space, generalizing the case of $\mathrm{p}=2$. Second, we establish the result using the Caputo–Katugampola operator, which generalizes the results of the Caputo and Caputo–Hadamard derivatives. Full article

Two novel crossover models for breast cancer that incorporate $\Psi$-Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion and the crossover model for breast cancer that incorporates Atangana–Baleanu Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion are presented here, where we used a simple nonstandard kernel function $\Psi \left(t\right)$ in the first model and a non-singular kernel in the second model. Moreover, we evaluated our models using actual statistics from Saudi Arabia. To ensure consistency with the physical model problem, the symmetry parameter $\zeta$ is introduced. We can obtain the fractal variable-order fractional Caputo and Caputo–Katugampola derivatives as special cases from the proposed $\Psi$-Caputo derivative. The crossover dynamics models define three alternative models: fractal variable-order fractional model, fractal fractional-order model, and variable-order fractional stochastic model over three-time intervals. The stability of the proposed model is analyzed. The $\Psi$-nonstandard finite-difference method is designed to solve fractal variable-order fractional and fractal fractional models, and the Toufik–Atangana method is used to solve the second crossover model with the non-singular kernel. Also, the nonstandard modified Euler–Maruyama method is used to study the variable-order fractional stochastic model. Numerous numerical tests and comparisons with real data were conducted to validate the methods’ efficacy and support the theoretical conclusions. Full article

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 $\mathrm{p}=2$ to results in the ${\mathbb{L}}^{\mathrm{p}}$ space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. Full article

The purpose of this paper is to study nonlinear implicit differential equations with the Caputo–Katugampola fractional derivative. By using Gronwall inequality and Banach fixed-point theorem, the existence of the solution of the implicit equation is proved, and the relevant conclusions about the stability of Ulam–Hyers are obtained. Finally, the correctness of the conclusions is verified by an example. Full article

Fractional calculus, which deals with the concept of fractional derivatives and integrals, has become an important area of research, due to its ability to capture memory effects and non-local behavior in the modeling of real-world phenomena. In this work, we study a new class of fractional Volterra–Fredholm integro-differential equations, involving the Caputo–Katugampola fractional derivative. By applying the Krasnoselskii and Banach fixed-point theorems, we prove the existence and uniqueness of solutions to this problem. The modified Adomian decomposition method is used, to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to the given problem; therefore, we investigate the convergence of approximate solutions, using the modified Adomian decomposition method. Finally, we provide an example, to demonstrate our results. Our findings contribute to the current understanding of fractional integro-differential equations and their solutions, and have the potential to inform future research in this area. Full article

In this work, we examine the existence of weak solution for a class of boundary value problems involving fractional Langevin inclusion with the Katugampola–Caputo fractional derivative under specified conditions contain the Pettis integrability assumption. The Mönch fixed point theorem is used with the weak noncompactness measure approach to investigate the existence results. In order to illustrate our results, we present an example. Full article

We investigate the existence and uniqueness results for coupled Langevin differential equations of fractional order with Katugampola integral boundary conditions involving generalized Liouville–Caputo fractional derivatives. Furthermore, we discuss Ulam–Hyers stability in the context of the problem at hand. The results are shown with examples. Results are asymmetric when a generalised Liouville–Caputo fractional derivative $\left(\rho \right)$ parameter is changed. With its novel results, this paper makes a significant contribution to the relevant literature. Full article

The fractional model of diffusion equations is very important in the study of oil pollution in the water. The key objective of this article is to analyze a fractional modification of diffusion equations occurring in oil pollution associated with the Katugampola derivative in the Caputo sense. An effective and reliable computational method q-homotopy analysis generalized transform method is suggested to obtain the solutions of fractional order diffusion equations. The results of this research are demonstrated in graphical and tabular descriptions. This study shows that the applied computational technique is very effective, accurate, and beneficial for managing such kind of fractional order nonlinear models occurring in oil pollution. Full article

We consider the following stochastic fractional differential equation ${}^C{\mathcal{D}}_{0^{+}}^{\alpha ,\rho }\phi \left(t\right)=\kappa \vartheta (t,\phi \left(t\right))\dot{w}\left(t\right)$, $0<t\le T,$ where $\phi \left(0\right)={\phi }_0$ is the initial function, ${}^C{\mathcal{D}}_{0^{+}}^{\alpha ,\rho }$ is the Caputo–Katugampola fractional differential operator of orders $0<\alpha \le 1,\rho >0$, the function $\vartheta :[0,T]\times \mathbb{R}\to \mathbb{R}$ is Lipschitz continuous on the second variable, $\dot{w}\left(t\right)$ denotes the generalized derivative of the Wiener process $w\left(t\right)$ and $\kappa >0$ represents the noise level. The main result of the paper focuses on the energy growth bound and the asymptotic behaviour of the random solution. Furthermore, we employ Banach fixed point theorem to establish the existence and uniqueness result of the mild solution. Full article

In the last few years, a new class of fractional-order (FO) systems, known as Katugampola FO systems, has been introduced. This class is noteworthy to investigate, as it presents a generalization of the well-known Caputo fractional-order systems. In this paper, a novel lemma for the analysis of a function with a bounded Katugampola fractional integral is presented and proven. The Caputo–Katugampola fractional derivative concept, which involves two parameters 0 < α < 1 and ρ > 0, was used. Then, using the demonstrated barbalat-like lemma, two identification problems, namely, the “Fractional Error Model 1” and the “Fractional Error Model 1 with parameter constraints”, were studied and solved. Numerical simulations were carried out to validate our theoretical results. Full article

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville, $\psi$-Caputo, $\psi$-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators. Full article