Fractional Differential Systems Research Articles

Exploring the dynamics of leprosy transmission with treatment through a fractal–fractional differential model

Leprosy continues to be a significant public health challenge in many parts of the world, necessitating novel approaches to understanding and controlling its transmission. The dynamics of leprosy are examined in this study by means of a caputo fabrizio fractal–fractional differential system model. Leprosy transmission and treatment are complex and non-linear processes that can be captured by fractal–fractional derivatives. The temporal progression of the disease is the primary focus of our mathematical analysis, which evaluates a variety of parameter values, including initial population densities and compartmental transitions. Through simulations, we examine the impact of critical parameters on the severity and spread of diseases, as well as the rates of recovery and treatment. Numerical results of the model offer valuable insights into the impact of these parameters on leprosy dynamics, which is beneficial for public health interventions. The stability analysis of the model identifies supplementary conditions that are necessary for the success of disease control. By incorporating these novel mathematical techniques, we hope to improve our understanding of leprosy transmission and ultimately contribute to more effective control strategies. Based on our findings, additional studies investigating the relation between population density, treatment accessibility, and recovery rates are warranted. We hope that by graphically representing the relationships between these factors, we can draw attention to the possibility of targeted interventions that can reduce the transmission of leprosy. This study provides a strong basis for future studies on infectious disease modeling and aids leprosy-affected communities in developing strategies to mitigate the disease’s impact.

Solutions for a New Fractional Differential Dynamical System and Yosida Quasi-Inverse Variational Inequality in Hilbert Space

In this article, first we introduce and study a Yosida Quasi-inverse variational inequality problem (in short, YQIVI) in Hilbert space and then developed a new fractional differential dynamical system for the YQIVI. We prove the existence and uniqueness of solution for the suggested dynamical system. Further, using the Lyapunov function we also prove the asymptotic stability of the new dynamical system at the equilibrium point. Furthermore, using Rothe's time discretization method we investigate existence and uniqueness of solution of the proposed dynamical system. Finally, we provide a numerical example to demonstrate the credibility and efficacy of the dynamical system in solving the YQIVI.

Existence and uniqueness for a mixed fractional differential system with slit-strips conditions

This paper studies the existence and uniqueness of solutions for a new kind of mixed fractional differential systems with slit-strips conditions, containing Caputo-type fractional derivatives. The first result on the existence and uniqueness is based on Banach’s fixed point theorem, and the second result on the existence and uniqueness of the solution is proved by using a Schaefer-type fixed point theorem. The applicability of our primary results is finally illustrated by some examples.

Improved order in Hilfer fractional differential systems: Solvability and optimal control problem for hemivariational inequalities

The main objective of this study is to analyze the optimal control results and solvability concerning Hilfer fractional differential hemivariational inequalities within the range of (1<ϱ<2). Initially, we explore the solvability of a mild solutions for the Hilfer fractional hemivariational inequality. Subsequently, we investigate the optimal control strategies for the given problems by employing cost functionals, cosine operators, mild solutions, the fixed point method for multivalued functions, and a generalized Clarke subdifferential approach. To illustrate our findings, we provide a practical example and a filter diagram.

On stochastic fractional differential variational inequalities general system with Lévy jumps

This paper is focused on investigating a stochastic fractional differential variational inequalities general system possessing Lévy jumps (SFDVIGS possessing Lévy jumps), which consists of two systems, i.e., a stochastic variational inequalities general system (SVIGS) and a stochastic fractional differential equations general system (SFDEGS) possessing Lévy jumps. Applying Picard successively iterative technique and the projection method, we derive the unique existence of solutions to SFDVIGS possessing Lévy jumps under certain mild assumptions.

The Mathematical Models To Analyze The Impact Of An Adoptive T - Cells Therapy On The Coexistence Of Chronic Lymphocytic And Acute Myelogenous Leukemia

Leukemia is a cancerous and progressive disease that affects the myeloid and lymphoid cells of the bone marrow. It causes a lot of problems for families, patients as well as the healthcare systems around the world. This study investigated how adoptive T - cell therapy affects the growth of coexistence of Acute Myelogenous Leukemia (AML) and Chronic Lymphocytic Leukemia (CLL) in a single patient. The methodology involves the development of both classical and fractional differential systems to analyze the coexistence of CLL and AML in a single patient. The numerical simulations were also performed to confirm the results of the analysis. The studies showed that the concentration levels of both types of leukemia cells were very high before the introduction of immunotherapy of the adoptive T cells. The disease free and endemic equilibrium points were proved to be globally asymptotically stable

Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability

Abstract Developing a model of fractional differential systems and studying the existence and stability of a solution is considebly one of the most important topics in the field of analysis. Therefore, this manuscript was dedicated to deriving a new type of fractional system that arises from the combination of three sequential fractional derivatives with fractional pantograph equations. Also, the fixed-point technique was used to evaluate the existence and uniqueness of solutions to the supposed hybrid model. Furthermore, stability results for the intended system in the sense of the Mittag-Leffler-Ulam have been investigated. Ultimately, an illustrative example has been highlighted in order to reinforce the theoretical results and suggest applications for this article.

On a class of generalized capillarity system involving fractional ψ$$ \psi $$‐Hilfer derivative with p(·)$$ p\left(\cdotp \right) $$‐Laplacian operator

This research delves into a comprehensive investigation of a class of ‐Hilfer generalized fractional nonlinear differential system originated from a capillarity phenomena with Dirichlet boundary conditions, focusing on issues of existence and multiplicity of nonnegative solutions. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti–Rabinowitz type condition. We use minimization arguments of Nehari manifold together with variational approach to show the existence and multiplicity of positive solutions of our problem with respect to the parameter in appropriate fractional ‐Hilfer spaces. Our main result is novel, and its investigation will enhance the scope of the literature on coupled systems of fractional ‐Hilfer generalized capillary phenomena.

LQ optimal control of uncertain fractional differential systems

LQ optimal control of uncertain fractional differential systems

Study of a Coupled Ψ–Liouville–Riemann Fractional Differential System Characterized by Mixed Boundary Conditions

Study of a Coupled Ψ–Liouville–Riemann Fractional Differential System Characterized by Mixed Boundary Conditions

Some Existence Results of Coupled Hilfer Fractional Differential System and Differential Inclusion on the Circular Graph

Some Existence Results of Coupled Hilfer Fractional Differential System and Differential Inclusion on the Circular Graph

A COMPUTATIONAL AND FRACTIONAL MATHEMATICAL MODEL FOR ASSESSING HEPATITIS B WITH OPTIMAL THERAPY

This paper presents a mathematical study applied to Hepatitis B disease utilizing fractional calculus and numerical methods. We begin by discussing mathematical fundamentals, focusing on fractional calculus concepts and relevant numerical aspects for disease modeling. We then formulate the problem, describing a nonlinear fractional differential system that models hepatitis B virus (HBV) infection and its response to drug therapy. A detailed theoretical analysis explores system properties and biological implications. We provide a qualitative analysis of the model, discussing important dynamic behaviors such as stability and parameter sensitivity. Subsequently, numerical analysis is performed to validate the model and estimate unknown parameters from patient data. Our findings include significant insights into biological parameters and their impact on therapeutic response. We conclude that the proposed fractional-order model is effective, offering optimal therapeutic strategies for hepatitis B.

Stability analysis of damped fractional stochastic differential systems with Poisson jumps: an successive approximation approach

This paper aims to investigate a new class of fractional stochastic differential systems under the influence of damping and Poisson jumps. First, the existence and uniqueness of mild solutions for the proposed system are investigated under the non-Lipschitz conditions. The results are formulated and proved by using the ( β , δ ) -regularised family, Grönwall's inequality, and successive approximation technique, which is different from the fixed point approach. Moreover, novel stability criteria for the considered system are obtained by utilising the corollary of the Bihari inequality. Finally, the correctness of the obtained results is verified by example.

Stability Analysis of a Class of Nonlinear Fractional Differential Systems With Riemann-Liouville Derivative

This paper investigates the stability of n-dimensional nonlinear fractional differential systems with Riemann-Liouville derivative. By using the Mittag-Leffler function, Laplace transform and the Gronwall-Bellman lemma, one sufficient condition is attained for the asymptotical stability of a class of nonlinear fractional differential systems whose order lies in (0, 2). According to this theory, if the nonlinear term satisfies some conditions, then the stability condition for nonlinear fractional differential systems is the same as the ones for corresponding linear systems. Several examples are provided to illustrate the applications of our result.

Chaotic Dynamics of the Fractional Order Predator-Prey Model Incorporating Gompertz Growth on Prey with Ivlev Functional Response

This paper examines dynamic behaviours of a two-species discrete fractional order predator-prey system with functional response form of Ivlev along with Gompertz growth of prey population. A discretization scheme is first applied to get Caputo fractional differential system for the prey-predator model. This study identifies certain conditions for the local asymptotic stability at the fixed points of the proposed prey-predator model. The existence and direction of the period-doubling bifurcation, Neimark-Sacker bifurcation, and Control Chaos are examined for the discrete-time domain. As the bifurcation parameter increases, the system displays chaotic behaviour. For various model parameters, bifurcation diagrams, phase portraits, and time graphs are obtained. Theoretical predictions and long-term chaotic behaviour are supported by numerical simulations across a wide variety of parameters. This article aims to offer an OGY and state feedback strategy that can stabilize chaotic orbits at a precarious equilibrium point.

T-controllability of higher-order fractional stochastic delay system via integral contractor

The existence, uniqueness and trajectory (T)-controllability results for mild solutions to the fractional neutral stochastic integro-delay differential system are presented in this article. To demonstrate the results, the concept of bounded integral contractors is combined with the stochastic result and sequencing technique. In contrast to previous publications, we do not need to specify the induced inverse of the controllability operator to prove the controllability results, and the relevant non-linear function does not have to meet the Lipschitz condition. Furthermore, T-controllability results for neutral integro-delay differential systems with non-local conditions have been established. Finally, an application to demonstrate the acquired results is discussed.

Numerical and analytical approach of nonlinear fractional pantograph nonlocal differential systems with non-singular kernel

Numerical and analytical approach of nonlinear fractional pantograph nonlocal differential systems with non-singular kernel

Existence, Uniqueness, and Stability of a Nonlinear Tripled Fractional Order Differential System

This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is used to establish their uniqueness. Stability conditions are derived utilizing classical nonlinear functional analysis techniques. Theoretical findings are illustrated with an example. The proposed system generalizes third-order ordinary differential equations (ODEs) with different boundary conditions (BCs).

On the existence of solutions for nonlocal sequential boundary fractional differential equations via ψ-Riemann–Liouville derivative

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder’s and Perov’s fixed-point theorems. Examples are given to validate the theoretical results.

A constructive approach for investigating the stability of incommensurate fractional differential systems

This paper is devoted to studying the asymptotic behaviour of solutions to generalized incommensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional order pseudospectrum definition proposed by Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of incommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a incommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. It is effective and widely applicable in studying the asymptotic behaviour of solutions to linear incommensurate fractional differential systems with constant coefficient matrices and linearized stability theory for nonlinear incommensurate fractional differential systems. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.