Fractional Differential Equations Research Articles

Analysis of the non-linear higher dimensional fractional differential equations arising in dusty plasma using the Atangana–Baleanu fractional derivative

Analysis of the non-linear higher dimensional fractional differential equations arising in dusty plasma using the Atangana–Baleanu fractional derivative

On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability

On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability

Advancing Solutions for Fractional Differential Equations: Integrating the Sawi Transform with Iterative Methods

Advancing Solutions for Fractional Differential Equations: Integrating the Sawi Transform with Iterative Methods

A Model Combining Sensitive Vegetation Indices and Fractional-Order Differential Characteristic Bands for SPAD Value Estimation in Cd-Contaminated Rice Leaves

A Model Combining Sensitive Vegetation Indices and Fractional-Order Differential Characteristic Bands for SPAD Value Estimation in Cd-Contaminated Rice Leaves

A fractional hybrid function composed of block-pulses and fractional Fibonacci polynomials for solving multiterm fractional variable-order differential equations

This paper presents the development of a fractional hybrid function composed of block-pulse functions and Fibonacci polynomials (FHBPF) for the numerical solution of multiterm variable-order fractional differential equations. By replacing x→xα in FHBPF and utilizing incomplete beta functions, we construct the method with a focus on fractional derivatives in the Caputo sense and fractional integrals in the Riemann–Liouville sense. A key advantage of the proposed method is its exact computation of the Riemann–Liouville fractional integral operator. Using the Newton–Cotes collocation method, we transform the differential equations into systems of algebraic equations, which are then solved with traditional techniques such as Newton’s iterative method. An error analysis method is also introduced. Several numerical examples are provided to demonstrate the effectiveness and simplicity of the approach, highlighting its efficiency even for relatively small base sizes.

The time-fractional (2+1)-dimensional Heisenberg ferromagnetic spin chain equation: its Lie symmetries, exact solutions and conservation laws

Abstract In this paper, the Lie symmetry analysis method is applied to the (2+1)-dimensional time-fractional Heisenberg ferromagnetic spin chain equation. We obtain all the Lie symmetries admitted by the governing equation and reduce the corresponding (2+1)-dimensional fractional partial differential equations with the Riemann–Liouville fractional derivative to (1+1)-dimensional counterparts with the Erdélyi–Kober fractional derivative. Then, we obtain the power series solutions of the reduced equations, prove their convergence and analyze their dynamic behavior graphically. In addition, the conservation laws for all the obtained Lie symmetries are constructed using the new conservation theorem and the generalization of Noether operators.

Strict Stability of Fractional Differential Equations with a Caputo Fractional Derivative with Respect to Another Function

In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider a system of two scalar fractional equations with CFDF and define a strict stability in the couple. We illustrate both definitions with several examples and, in these examples, we show that the applied function in the fractional derivative has a huge influence on the stability properties of the solutions. In addition, we use Lyapunov functions and their CFDF to obtain several sufficient conditions for strict stability.

On boundary value problems with implicit random Caputo tempered fractional differential equations

On boundary value problems with implicit random Caputo tempered fractional differential equations

Discussion on the Optimal Controls Results for Fractional Systems With Atangana–Baleanu Derivative

ABSTRACTIn this paper, we present the optimal controls for a system governed by Atangana–Baleanu fractional differential systems in Banach spaces. We verify the existence of the mild solutions through the concepts of fractional calculus, semigroup theory, and Leray–Schauder's fixed point theorem. Likewise, we discuss the existence of optimal controls for the corresponding fractional control system. Furthermore, an example is offered to define the theoretical result.

Evaluating the stability and efficacy of fractal-fractional models in reproductive cancer apoptosis with ABT-737

Cancer refers to a group of diseases characterized by the uncontrolled growth of abnormal cells. Conventional cancer therapies, such as chemotherapy and radiation therapy, often encounter issues like toxicity and resistance, primarily due to their inability to effectively differentiate between cancerous and normal cells. As a result, these treatments frequently harm healthy cells, underscoring the urgent need for innovative and more effective cancer treatment strategies. This article aims to investigate the changes in the Bcl-2/Bax ratio over time and its stability, using the Atangana–Baleanu fractional derivative operators as a key factor in reproductive cancer. The article also discusses the potential application of this model to investigate the effects of the ABT-737 inhibitor on mitochondrial apoptosis under initial conditions. The methodology entails using a fixed-point approach to investigate existence and uniqueness, as well as exploring Hyers–Ulam stability results for the given model. The numerical schemes utilize the basic concepts of fractional calculus and Lagrange polynomial interpolation. We generate simulated graphical representations of these models for a variety of fractional-order values. Finally, we present the simulation results to confirm the effectiveness and practicality of the theoretical findings.

Projective synchronization of fractional-order quaternion-valued neural networks with Mittag-Leffler kernel

We consider fractional-order quaternion-valued neural networks (FO-QVNNs) with Mittag-Leffler kernel and study projective synchronization. We establish Mittag-Leffler projective synchronization (MLPS) and asymptotic projective synchronization for the FO-QVNNs. Initially, a novel fractional differential inequality for the fractional derivative is established to study the synchronization of fractional-order systems. Subsequently, quaternion-valued feedback controllers are designed. Conditions for MLPS are then formulated using quaternion techniques, the proposed inequality and the properties of fractional calculus. The effectiveness and practicality of the proposed methodologies are displayed through numerical simulations in a concrete case.

Dynamical investigation of the new (3+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation with conformable fractional derivative

This paper examines the conformable version of a recently proposed novel [Formula: see text]-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation. First, some basic definitions and properties of the conformable derivative are provided. To find exact solutions to this problem, the modified extended tanh-function, exp([Formula: see text]-expansion, and Kudryashov R function methods are applied. The results are illustrated by utilizing some of the gathered data’s physical 3D, contour, and 2D surfaces, which sheds light on how geometric patterns are physically comprehended. These solutions contribute to understanding the potential practical applications of the examined model and other nonlinear representations in the physical sciences. These techniques can provide significant solutions for various fractional differential equations.

Exploring Fixed-Point Results Using Random Sehgal Contraction in Symmetric Random Cone Metric Spaces with Applications

This paper introduces a new concept of random Sehgal contraction in the setting of random cone metric spaces. We explore the modern advancements of traditional fixed-point theorems in a random setting, elaborating on the Sehgal–Guseman fixed-point theorem within the realm of random cone metric spaces. A significant aspect of our research is the interplay between symmetry and randomness; while symmetry provides a framework for understanding structural properties, randomness introduces complexity, which can lead to unexpected behaviors. Our research provides a deeper understanding of the classical results and incorporates a detailed example to illustrate our findings. In addition, major random fixed-point results are also established, which could be applied to nonlinear random fractional differential equations (FDEs) and integral equations as well as to random boundary value problems (BVPs) related to homogeneous random transverse bars.

Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations

Abstract In this work, we prove some fixed point results in generalized Banach spaces ( G B S {\mathfrak{G}}{\mathfrak{B}}{\mathfrak{S}} s) in the sense of Perov using the vector degree of nondensifiability tools. The given result generalizes Darbo’s and Krasnoselskii’s theorems, which are connected with the vector measure of noncompactness. An existence result for coupled Caputo fractional delay differential equations in the G B S C ( [ − τ , T ] , R ) × C ( [ − τ , T ] , R ) {\mathfrak{G}}{\mathfrak{B}}{\mathfrak{S}}\hspace{0.33em}{\mathcal{C}}\left(\left[-\tau ,T],{\mathbb{R}})\times {\mathcal{C}}\left(\left[-\tau ,T],{\mathbb{R}}) is given to show the significance and the applicability of our theoretical results.

On Multi Order Nonlinear Langevin Type of FDE Subject to Multi-Point Boundary Conditions

This research paper is about studying complicated equations called multi-order nonlinear fractional Langevin differential equations. These equations are analyzed when they have specific conditions at multiple points. The paper uses two mathematical tools, the Banach contraction principle and Schaefer’s fixed point theorem, to show that there is only one solution for the given equation. Moreover, we thoroughly study and analyze the stability of the solutions. To help understand the ideas we talked about, we will give related examples.

Terminal value problem for the system of fractional differential equations with additional restrictions

This paper deals with the study of terminal value problem for the system of fractional differential equations with Caputo derivative. Additional conditions are imposed on the solutions of this problem in the form of a linear vector functional. Using the theory of pseudo-inverse matrices, we obtain the necessary and sufficient conditions for the solvability and the general form of the solution of this boundary-value problem. In the one-dimensional case, the obtained results are generalized to the case of a multi-point boundary-value problem. The issue of obtaining similar results for the terminal value problem for the system of fractional differential equations with tempered and Ψ–tempered fractional derivatives of Caputo type is considered.

Discrete Gronwall’s inequality for Ulam stability of delay fractional difference equations

This paper investigates Ulam stability of delay fractional difference equations. First, a useful equality of double fractional sums is employed and discrete Gronwall’s inequality of delay type is provided. A delay discrete-time Mittag-Leffler function is used and its non-negativity condition is given. With the solutions’ existences, Ulam stability condition is presented to discuss the error estimation of exact and approximate solutions.

Analytic solutions of a generalized complex multi-dimensional system with fractional order

Abstract The Duhamel principle is a mathematical principle that allows us to solve linear partial differential equations. This system is generalized by the concept of the k k -symbol fractional calculus. We demonstrate that in specific functional domains, the suggested system produces a global solution. Convergence toward the stable point is investigated. Some special cases are illustrated including the rotated Koebe function, which is used for creating the optimal solutions, where these functions are convex univalent in the open unit disk.

On a Preloaded Compliance System of Fractional Order: Numerical Integration

In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, the discontinuous right-hand side into a set-valued function. Next, via Cellina’s Theorem, the obtained set-valued differential inclusion of fractional order can be restarted as a single-valued continuous differential equation of fractional order, to which the existing numerical schemes for fractional differential equations can be applied. In this way, the delicate problem of integrating discontinuous problems of fractional order, as well as integer order, is solved by transforming the discontinuous problem into a continuous one. Also, it is noted that even the numerical methods for fractional-order differential equations can be applied abruptly to the discontinuous problem, without considering the underlying discontinuity, so the results could be incorrect. The technical example of a single-degree-of-freedom preloaded compliance system of fractional order is presented.

SENSITIVITY ANALYSIS AND NUMERICAL MODELING OF INFLUENZA PROPAGATION AND INTERVENTION STRATEGIES UNDER THE FRACTAL-FRACTIONAL OPERATOR

This paper introduces a novel nonlinear fractal fractional model to comprehensively analyze influenza epidemics. By integrating fractional calculus and considering nonlinear interactions among individuals, the model utilizes the Fractal-Fractional (FF) operator. This operator, combining fractal and fractional calculus, establishes a unique framework for investigating influenza virus propagation dynamics and potential vaccination strategies. Amid growing concerns over influenza outbreaks, fractional derivatives are employed to address intricate challenges. The proposed model sheds light on virus spread dynamics and countermeasures. The integration of the FF operator enriches analysis, while the application of the fixed point theory of Schauder and Banach demonstrates solution existence and uniqueness. Model validation employs numerical simulations with MATLAB12̂ and the Adams–Bashforth method, confirming its ability to capture influenza propagation dynamics accurately. Ulam–Hyers stability techniques ensure the model’s reliability. Beyond its scientific contributions, the model underscores the significance of studying influenza epidemics via mathematical modeling in understanding disease dynamics and guiding effective intervention strategies. Through a synergy of mathematical innovation and epidemiological insights, this study establishes a robust foundation for more effective strategies against influenza epidemics.