Caputo Fractional Differential Equations Research Articles

Comparison principles for systems of Caputo fractional order ordinary differential equations

The comparison principle for systems of Caputo fractional order ordinary differential equations still remains blank. By applying the Caputo solution continuation and Vainikko’s definition, and reconstructing the framework of system comparison, here we state and prove general comparison principles for systems of Caputo fractional order ordinary differential equations, and illustrate them by examples with numerical simulations.

The asymptotic solutions of two-term linear fractional differential equations via Laplace transform

In this paper, the asymptotic solutions about the origin and infinity are formulated via Laplace transform for a two-term linear Caputo fractional differential equation. The asymptotic expansion about the origin describes the complete singular information of the solution, which is also a good approximation of the solution near the origin. The expansion at infinity exhibits the structure of the solution, as well as the stable or unstable property of the solution, which becomes more accurate as the variable tends to larger. Based on the asymptotic solution about the origin, a singularity-separation Legendre collocation method is designed to validate the methods in this paper. Numerical examples show the easy calculation and high accuracy of the truncated expansions and their Padé approximations when the variable is suitably small or sufficiently large. As an application, the method is used to solve the initial value problem of the Bagley–Torvik equation, and the oscillatory property of the solution is displayed.

Coupled system of three sequential Caputo fractional differential equations: Existence and stability analysis

Recently, many studies on fractional coupled systems involving different sequential fractional derivatives have appeared during the past several years. The paper is dealing with a coupled system of three sequential Caputo fractional differential equations, and the designed system absorbs none of the commutativity and the semigroup properties. The Banach contraction principle is used for proving the existence and uniqueness results. We prove the existence of at least one is obtained by using the Leray–Schauder alternative. The Ulam–Hyers–Rassias stability of the considered system is defined and discussed. An illustrative example is also presented.

Leray–Schauder Alternative for the Existence of Solutions of a Modified Coupled System of Caputo Fractional Differential Equations with Two Point’s Integral Boundary Conditions

In this paper, a coupled system of differential equations involving fractional order with integral boundary conditions is discussed. In the problem at hand, three main aspects that are existence, uniqueness, and stability have been investigated. Firstly, the contraction mapping principle is used to discuss the uniqueness of solutions for the proposed fractional system, and secondly, the existence of solutions for the problem is investigated based on Leray–Schauder’s alternative. Thirdly, the stability of the presented coupled system is discussed based on the Hyers–Ulam stability method. Finally, some examples have been given to confirm and illustrate the conclusion. The comparison between the current symmetrical results and the existing literature is deemed satisfactory. It was found that the presented fractional coupled system with two with integral boundary conditions is existent, unique, and stable.

Existence results for anti-periodic fractional coupled systems with $$p-$$Laplacian operator via measure of noncompactness in Banach spaces

The aim of this paper is to investigate the existence of anti-periodic solutions for a nonlinear coupled system of $$\Psi -$$ Caputo fractional differential equations with $$p-$$ Laplacian operator in a Banach space. The proofs of our main results are based on Mönch’s fixed point theorem and some basic tools of $$\Psi -$$ Caputo fractional calculus paired with measures of noncompactness. As an application, our theoretical results have been illustrated by providing an interesting example.

On The Solutions of Fractional Boundary Value Problems for a Combined Caputo Fractional Differential Equations

In this paper, a new class of fractional boundary value problem with the combined Caputo derivative is proposed and the physical interpretation of this new derivative has been explained. Under some assumptions, the positive solutions of the fractional differential equation with the help of Leray-Schauder and Krasnoselskii's fixed point theorems in a cone have been investigated. Moreover, the solution of the fractional Maxwell models involving the combined Caputo derivative by using the extended Laplace transform is obtained. Finally, some examples are given to support theoretical findings.

Lyapunov inequality for a Caputo fractional differential equation with Riemann–Stieltjes integral boundary conditions

In this a Lyapunov‐type inequality is obtained for the fractional differential equation with Caputo derivative together with the boundary conditions where are functions of bounded variation, , and are nonnegative constants, and is a continuous function. We have divided the Greens function of this problem into two parts, and their upper bounds are obtained separately in order to obtain our results. The classical celebrated result of Lyapunov has become a consequence of our result.

Chaotic dynamics of the fractional order Schnakenberg model and its control

The Schnakenberg model is thought to be the Caputo fractional derivative. In order to create caputo fractional differential equations for the Schnakenberg model, a discretization process is first used. The fixed points in the model are categorized topologically. Then, we show analytically that, under certain parametric conditions, a Neimark-Sacker (NS) bifurcation and a Flip-bifurcation are supported by a fractional order Schnakenberg model. Using central manifold and bifurcation theory, we demonstrate the presence and direction of NS and Flip bifurcations. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order Schnakenberg model. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, period 2, 4, 7, 8, 10, 16, 20 and 40 orbits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. In order to support the system’s chaotic characteristics, we also compute the maximal Lyapunov exponents and fractal dimensions quantitatively. Finally, the chaotic trajectory of the system is stopped using the OGY approach, hybrid control method, and state feedback method.

Solution processes for second-order linear fractional differential equations with random inhomogeneous parts

The novelty of this paper is to derive a mean square solution of a second-order (Caputo) fractional linear differential equation in which the coefficients and initial conditions are random variables, and the forcing term is a second order stochastic process. Using the so-called mean square calculus and assuming mild conditions on the random variables of the equation together with an exponential growth condition on the forcing term, a mean square convergent generalized power series solution is constructed. As a result of this convergence, the sequences of the mean and correlation obtained from the truncated power series solution are convergent as well. The theory is illustrated with several examples in which different kind of distributions on the input parameters are assumed.

Convergence Analysis of a New Implicit Iterative Scheme and Its Application to Delay Caputo Fractional Differential Equations

This article presents a new three-step implicit iterative method. The proposed method is used to approximate the fixed points of a certain class of pseudocontractive-type operators. Additionally, the strong convergence results of the new iterative procedure are derived. Some examples are constructed to authenticate the assumptions in our main result. At the end, we use our new method to solve a fractional delay differential equation in the sense of Caputo. Our main results improve and generalize the results of many prominent authors in the existing literature.

Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study

Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type stability, and it is a special type of data dependence of solutions of differential equations. For boundary value problems, this type of stability requires some additional understanding, and, in connection with this, we discuss the Ulam-Hyers stability for different types of differential equations, such as ordinary differential equations and generalized proportional Caputo fractional differential equations. To propose an appropriate idea of Ulam-type stability, we consider a boundary condition with a parameter, and the value of the parameter depends on the chosen arbitrary solution of the corresponding differential inequality. Several examples are given to illustrate the theoretical considerations.

Analytical solution of the Atangana–Baleanu–Caputo fractional differential equations using Pythagorean fuzzy sets

Analytical solution of the Atangana–Baleanu–Caputo fractional differential equations using Pythagorean fuzzy sets

On a System of Sequential Caputo Fractional Differential Equations with Nonlocal Boundary Conditions

We obtain existence and uniqueness results for the solutions of a system of Caputo fractional differential equations which contain sequential derivatives, integral terms, and two positive parameters, supplemented with general coupled Riemann–Stieltjes integral boundary conditions. The proofs of our results are based on the Banach fixed point theorem and the Leray–Schauder alternative.

Design of a fractional-order atmospheric model via a class of ACT-like chaotic system and its sliding mode chaos control.

Investigation of the dynamical behavior related to environmental phenomena has received much attention across a variety of scientific domains. One such phenomenon is global warming. The main causes of global warming, which has detrimental effects on our ecosystem, are mainly excess greenhouse gases and temperature. Looking at the significance of this climatic event, in this study, we have connected the ACT-like model to three climatic components, namely, permafrost thaw, temperature, and greenhouse gases in the form of a Caputo fractional differential equation, and analyzed their dynamics. The theoretical aspects, such as the existence and uniqueness of the obtained solution, are examined. We have derived two different sliding mode controllers to control chaos in this fractional-order system. The influences of these controllers are analyzed in the presence of uncertainties and external disturbances. In this process, we have obtained a new controlled system of equations without and with uncertainties and external disturbances. Global stability of these new systems is also established. All the aspects are examined for commensurate and non-commensurate fractional-order derivatives. To establish that the system is chaotic, we have taken the assistance of the Lyapunov exponent and the bifurcation diagram with respect to the fractional derivative. To perform numerical simulation, we have identified certain values of the parameters where the system exhibits chaotic behavior. Then, the theoretical claims about the influence of the controller on the system are established with the help of numerical simulations.

Existence of solutions for Caputo fractional delay differential equations with nonlocal and integral boundary conditions

In this paper, the existence and uniqueness of the solutions of Caputo fractional delay differential equations under nonlocal and integral boundary value conditions are studied. By using the Banach contraction principle and the Burton and Kirk fixed-point theorem, some new conclusions about the existence and uniqueness of solutions are obtained. An example is given to illustrate the main results.

Asymptotic Behavior of Delayed Reaction-Diffusion Neural Networks Modeled by Generalized Proportional Caputo Fractional Partial Differential Equations

In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditions for approaching zero are obtained. Lyapunov functions defined as a sum of squares with their generalized proportional Caputo fractional derivatives are applied and a comparison result for a scalar linear generalized proportional Caputo fractional differential equation with several constant delays is presented. Lyapunov functions and the comparison principle are then combined to establish our main results.

On the Optimality Condition for Optimal Control of Caputo Fractional Differential Equations with State Constraints

We consider the fractional optimal control problem with state constraints. The fractional calculus of derivatives and integrals can be viewed as generalizations of their classical ones to any arbitrary real order. In our problem setup, the dynamic constraint is captured by the Caputo fractional differential equation with order α ∈ (0, 1), and the objective functional is formulated by the left Riemann-Liouville fractional integral with order β ≥ 1. In addition, there are terminal and running state constraints; while the former is described by initial and final states within a convex set, the latter is given by an explicit instantaneous inequality state constraint. We obtain the maximum principle, the first-order necessary optimality condition, for the problem of this paper. Due to the inherent complex nature of the fractional control problem, the presence of the terminal and running state constraints, and the generalized standing assumptions, the maximum principle of this paper is new in the optimal control problem context, and its proof requires to develop new variational and duality analysis using fractional calculus and functional analysis, together with the Ekeland variational principle and the spike variation.

A Pontryagin maximum principle for terminal state-constrained optimal control problems of Volterra integral equations with singular kernels

<abstract><p>We consider the terminal state-constrained optimal control problem for Volterra integral equations with singular kernels. A singular kernel introduces abnormal behavior of the state trajectory with respect to the parameter of $ \alpha \in (0, 1) $. Our state equation covers various state dynamics such as any types of classical Volterra integral equations with nonsingular kernels, (Caputo) fractional differential equations, and ordinary differential state equations. We prove the maximum principle for the corresponding state-constrained optimal control problem. In the proof of the maximum principle, due to the presence of the (terminal) state constraint and the control space being only a separable metric space, we have to employ the Ekeland variational principle and the spike variation technique, together with the intrinsic properties of distance function and the generalized Gronwall's inequality, to obtain the desired necessary conditions for optimality. The maximum principle of this paper is new in the optimal control problem context and its proof requires a different technique, compared with that for classical Volterra integral equations studied in the existing literature.</p></abstract>

Controllability results of neutral Caputo fractional functional differential equations

<abstract><p>In this paper, using the properties of the phase space on infinite delay, generalized Gronwall inequality and fixed point theorems, the existence and controllability results of neutral fractional functional differential equations with multi-term Caputo fractional derivatives were obtained under Lipschitz and non-Lipschitz conditions.</p></abstract>

Existence Results for Systems of Nonlinear Caputo Fractional Differential Equations

We aim, in this work, to demonstrate the existence of minimal and maximal coupled quasi-solutions for nonlinear Caputo fractional differential systems with order q ∈ (1,2). Our approach is based on mixed monotone iterative techniques developed under the concept of lower and upper quasi-solutions. Our results extend those obtained for ordinary differential equations and fractional ones.