Caputo Fractional Differential Equations Research Articles

Existence and uniqueness of solutions for $$\Psi $$-Caputo fractional neutral sequential differential equations on time scales

Existence and uniqueness of solutions for $$\Psi $$-Caputo fractional neutral sequential differential equations on time scales

Hyers–Ulam–Rassias stability of fractional delay differential equations with Caputo derivative

This paper is devoted to the study of Hyers–Ulam–Rassias (HUR) stability of a nonlinear Caputo fractional delay differential equation (CFrDDE) with multiple variable time delays. We obtain two new theorems with regard to HUR stability of the CFrDDE on bounded and unbounded intervals. The method of the proofs is based on the fixed point approach. The HUR stability results of this paper have indispensable contributions to theory of Ulam stabilities of CFrDDEs and some earlier results in the literature.

Analyzing co-infection dynamics: A mathematical approach using fractional order modeling and Laplace-Adomian decomposition

The co-infection of HIV and COVID-19 is a pressing health concern, carrying substantial potential consequences. This study focuses on the vital task of comprehending the dynamics of HIV-COVID-19 co-infection, a fundamental step in formulating efficacious control strategies and optimizing healthcare approaches. Here, we introduce an innovative mathematical model grounded in Caputo fractional order differential equations, specifically designed to encapsulate the intricate dynamics of co-infection. This model encompasses multiple critical facets: the transmission dynamics of both HIV and COVID-19, the host’s immune responses, and the influence of treatment interventions. Our approach embraces the complexity of these factors to offer an exhaustive portrayal of co-infection dynamics. To tackle the fractional order model, we employ the Laplace-Adomian decomposition method, a potent mathematical tool for approximating solutions in fractional order differential equations. Utilizing this technique, we simulate the intricate interactions between these variables, yielding profound insights into the propagation of co-infection. Notably, we identify pivotal contributors to its advancement. In addition, we conduct a meticulous analysis of the convergence properties inherent in the series solutions acquired through the Laplace-Adomian decomposition method. This examination assures the reliability and accuracy of our mathematical methodology in approximating solutions. Our findings hold significant implications for the formulation of effective control strategies. Policymakers, healthcare professionals, and public health authorities will benefit from this research as they endeavor to curtail the proliferation and impact of HIV-COVID-19 co-infection.

A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations

In this paper, the tempered Ψ–Riemann–Liouville fractional derivative and the tempered Ψ–Caputo fractional derivative of order n−1<α<n∈N are introduced for Cn−1–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered Ψ–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered Ψ–Caputo fractional differential equations are proved. Illustrative examples are given.

Recent advances in employing the Laplace homotopy analysis method to nonlinear fractional models for evolution equations and heat-typed problems

This study stems from the growing need for effective mathematical tools to tackle nonlinear fractional evolution equations, which find wide applications in physics and engineering. Traditional methods often struggle to provide exact closed-form solutions for such equations, leading to the development of novel approaches. In this context, the combination of the homotopy analysis method with the Laplace transform presents a promising avenue for obtaining exact solutions to these complex equations. By exploring this hybrid approach, the study aims to address the existing challenges in solving nonlinear fractional models and provide a reliable method for obtaining closed-form solutions. In this way, this study focuses on rendering the coupling between the homotopy analysis approach and the Laplace transform in the search for exact closed-form solutions for the class of fractional evolution and heat-typed equations. In fact, this class of equations features some interesting nonlinear Caputo fractional partial differential equations that model dissimilar physical processes in physics and engineering. A generalized recurrent scheme is derived, which is applicable to all the members of the governing class, and further analyzed the application of this very scheme on some test fractional initial-value problems of much concern. Eventually, the method has been found to be advantageous, with various advantages, including convergence to available exact solutions. Thus, of the advantage of the current study is innovative combination of mathematical methods, the applicability of these methods to a special class of fractional equations, the development of a generalized scheme, the successful solution of prototype problems, and the practical advantages of the method.

A fractional order model for the transmission dynamics of shigellosis

Shigellosis, a highly contagious bacterial infection causing diarrhea, fever, and abdominal pain, necessitates a deep understanding of its transmission dynamics to devise effective control measures. Our study takes a novel approach, employing a fractional order framework to explore the influence of memory and control measures on Shigellosis transmission dynamics, thereby making a unique contribution to the field. The model is presented as a system of Caputo fractional differential equations capturing time constant controls. The Caputo derivatives are chosen for their inherent benefits. The qualitative features of the model, such as the solutions' existence and uniqueness, positivity, and boundedness, are thoroughly investigated. Moreover, the equilibria of the model are derived and analyzed for their stability using suitable theorems. In particular, local stability was proved through Routh's criteria, while global stability results were established in the Ulam-Hyers sense. The model is then solved numerically with the help of the predict-evaluate-correct-evaluate method of Adams-Bashforth-Moulton. The numerical results underscore the significant impact of memory on disease evolution, highlighting the novelty of integrating memory-related aspects into the meticulous planning of effective disease control strategies. Using fractional-order derivatives is more beneficial for understanding the dynamics of Shigellosis transmission than integral-order models. The advantage of fractional derivatives is their ability to provide numerous degrees of freedom, allowing for a broader range of analysis of the system's dynamic behaviour, including nonlocal solutions. Also, an investigation on the impacts of control measures via parameter variation is done; the findings show that applying treatment and sanitation minimizes disease eruption.

New Results on the Solvability of Abstract Sequential Caputo Fractional Differential Equations with a Resolvent-Operator Approach and Applications

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative.

Finite-time stability of Caputo fractional fuzzy differential equations with delay in granular sense

This manuscript focuses on investigating a class of Caputo fractional fuzzy differential system with time delay. Firstly, we understand the granular form of fuzzy numbers from a novel perspective, which contains more information than the usual membership function. Subsequently, using a successive approximation approach under the granular arithmetic context, we demonstrate the existence of the solution to this system, and the uniqueness is obtained by the completeness of fuzzy space. Furthermore, we establish a criterion for determining the finite-time stability of the system, and which is an evaluation function containing optional parameter ζ∈(0,ν) by employing a Gronwall inequality with delay form, where ν is the order of the Caputo derivative. Finally, we present a numerical example to verify our primary findings and discuss the appropriate selection of parameter ζ to optimize the evaluation function.

On the Cauchy-Kovalevskaya theorem for Caputo fractional differential equations

We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or compartmental. Essentially, the theorem states that if the input function has a Taylor series, then the solution can be locally expressed as a fractional power series. We use, in the real field, the method of majorants and the analytic version of the implicit-function theorem, in a way that circumvents difficulties associated to fractional calculus. Some corollaries on continuity are derived, with computational examples for illustration, and a discussion on fractional partial differential equations is included with a case study and counterexamples. Open problems are raised at the end.

Caputo fractional backward stochastic differential equations driven by fractional Brownian motion with delayed generator

Over the years, the research of backward stochastic differential equations (BSDEs) has come a long way. As a extension of the BSDEs, the BSDEs with time delay have played a major role in the stochastic optimal control, financial risk, insurance management, pricing, and hedging. In this paper, we study a class of BSDEs with time-delay generators driven by Caputo fractional derivatives. In contrast to conventional BSDEs, in this class of equations, the generator is also affected by the past values of solutions. Under the Lipschitz condition and some new assumptions, we present a theorem on the existence and uniqueness of solutions.

An outlook on stability of implicit θ -Caputo fractional differential equation with application involving the Riemann-Stieltjes type

An outlook on stability of implicit θ -Caputo fractional differential equation with application involving the Riemann-Stieltjes type

Modeling and Analysis of Fractional Order Logistic Equation Incorporating Additive Allee Effect

This study investigates the logistic model of a single species incorporating the additive Allee effect using Caputo fractional order differential equations. The Allee effect describes a positive correlation between individual fitness and population density at low densities. Populations subjected to the strong Allee effect can move towards extinction when their population is below a critical level. This study calculates the threshold level of the population suffering from the strong Allee effect. Various published studies are showing that fractional order models are more appropriate for explaining real-world phenomena than ordinary integer-order systems; therefore, this study involves the use of the Caputo fractional order derivative. Single-species models have been extensively used in mathematical biology, such as insect control, optimal biological resource planning, epidemic avoidance and control, and cell growth regulation. This study can help save vulnerable species from extinction and eliminate unwanted species by subjecting them to a strong Allee effect using artificial strategies.

A fixed point result on an extended neutrosophic rectangular metric space with application

In this paper, we propose the notion of extended neutrosophic rectangular metric space and prove some fixed point results under contraction mapping. Finally, as an application of the obtained results, we prove the existence and uniqueness of the Caputo fractional differential equation.

A Novel Numerical Method for Solving Nonlinear Fractional-Order Differential Equations and Its Applications

In this article, a considerably efficient predictor-corrector method (PCM) for solving Atangana–Baleanu Caputo (ABC) fractional differential equations (FDEs) is introduced. First, we propose a conventional PCM whose computational speed scales with quadratic time complexity O(N2) as the number of time steps N grows. A fast algorithm to reduce the computational complexity of the memory term is investigated utilizing a sum-of-exponentials (SOEs) approximation. The conventional PCM is equipped with a fast algorithm, and it only requires linear time complexity O(N). Truncation and global error analyses are provided, achieving a uniform accuracy order O(h2) regardless of the fractional order for both the conventional and fast PCMs. We demonstrate numerical examples for nonlinear initial value problems and linear and nonlinear reaction-diffusion fractional-order partial differential equations (FPDEs) to numerically verify the efficiency and error estimates. Finally, the fast PCM is applied to the fractional-order Rössler dynamical system, and the numerical results prove that the computational cost consumed to obtain the bifurcation diagram is significantly reduced using the proposed fast algorithm.

Theoretical analysis of a class of $ \varphi $-Caputo fractional differential equations in Banach space

<abstract><p>A study of a class of nonlinear differential equations involving the $ \varphi $-Caputo type derivative in a Banach space framework is presented. Weissinger's and Meir-Keeler's fixed-point theorems are used to achieve some quantitative results. Two illustrative examples are provided to justify the theoretical results.</p></abstract>

Value functional and optimal feedback control in linear-quadratic optimal control problem for fractional-order system

In this paper, a finite-horizon optimal control problem involving a dynamical system described by a linear Caputo fractional differential equation and a quadratic cost functional is considered. An explicit formula for the value functional is given, which includes a solution of a certain Fredholm integral equation. A step-by-step feedback control procedure for constructing $ \varepsilon $-optimal controls with any accuracy $ \varepsilon > 0 $ is proposed. The basis for obtaining these results is the study of a solution of the associated Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives.

Initial value problems for fractional p-Laplacian equations with singularity

<abstract><p>We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper.</p></abstract>

A Generalized Hybrid Method for Handling Fractional Caputo Partial Differential Equations via Homotopy Perturbed Analysis

This article describes a novel hybrid technique known as the Sawi transform homotopy perturbation method for solving Caputo fractional partial differential equations. Combining the Sawi transform and the homotopy perturbation method, this innovative technique approximates series solutions for fractional partial differential equations. The Sawi transform is a recently developed integral transform that may successfully manage recurrence relations and integro-differential equations. Using a homotopy parameter, the homotopy perturbation method is a potent semi-analytical tool for constructing approximate solutions to nonlinear problems. The suggested method offers various advantages over existing methods, including high precision, rapid convergence, minimal computing expense, and broad applicability. The new method is used to solve the convection–reaction–diffusion problem using fractional Caputo derivatives.

Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ-tempered Caputo fractional derivative.

Existence and stability results for implicit impulsive convex combined Caputo fractional differential equations

This paper deals with the existence and uniqueness results for a class of impulsive implicit fractional initial value problems of the convex combined Caputo fractional derivative. The arguments are based on Banach's contraction principle, Schauder's and Mönch's fixed point theorems. We will also establish the Ulam stability and give some examples to illustrate our results.