Caputo-type Derivative Research Articles

Modelling epidemiological dynamics with pseudo-recovery via fractional-order derivative operator and optimal control measures.

In this study, a new deterministic mathematical model based on fractional-order derivative operator that describes the pseudo-recovery dynamics of an epidemiological process is developed. Fractional-order derivative of Caputo type is used to examine the effect of memory in the spread process of infectious diseases with pseudo-recovery. The well-posedness of the model is qualitatively investigated through Banach fixed point theory technique. The spread of the disease in the population is measured by analysing the basic reproduction of the model with respect to its parameters through the sensitivity analysis. Consequently, the analysis is extended to the fractional optimal control model where time-dependent preventive strategy and treatment measure are characterized by Pontryagin's maximum principle. The resulting Caputo fractional-order optimality system is simulated to understand how both preventive and treatment controls affect the pseudo-recovery dynamics of infectious diseases in the presence of memory. Graphical illustrations are shown to corroborate the qualitative results, and to demonstrate the importance of memory effects in infectious disease modelling. It is shown that time-dependent preventive strategy and treatment measure in the presence of memory engenders significant reduction in the spread of the disease when compared with memoryless situation.

Tempered fractional differential equations on hyperbolic space

Abstract In this paper we study linear fractional differential equations involving tempered Caputo-type derivatives in the hyperbolic space. We consider in detail the three-dimensional case for its simple and useful structure. We also discuss the probabilistic meaning of our results in relation to the distribution of an hyperbolic Brownian motion time-changed with the inverse of a tempered stable subordinator. The generalization to an arbitrary dimension n can be easily obtained. We also show that it is possible to construct a particular solution for the non-linear porous-medium type tempered equation by using elementary functions.

On the Numerical Discretization of the Fractional Advection-Diffusion Equation with Generalized Caputo-Type Derivatives on Non-uniform Meshes

On the Numerical Discretization of the Fractional Advection-Diffusion Equation with Generalized Caputo-Type Derivatives on Non-uniform Meshes

Fundamental Matrix, Integral Representation and Stability Analysis of the Solutions of Neutral Fractional Systems with Derivatives in the Riemann—Liouville Sense

The paper studies a class of nonlinear disturbed neutral linear fractional systems with derivatives in the the Riemann–Liouville sense and distributed delays. First, it is proved that the initial problem for these systems with discontinuous initial functions under some natural assumptions possesses a unique solution. The assumptions used for the proof are similar to those used in the case of systems with first-order derivatives. Then, with the obtained result, we derive the existence and uniqueness of a fundamental matrix and a generalized fundamental matrix for the homogeneous system. In the linear case, via these fundamental matrices we obtain integral representations of the solutions of the homogeneous system and the corresponding inhomogeneous system. Furthermore, for the fractional systems with Riemann–Liouville derivatives we introduce a new concept for weighted stabilities in the Lyapunov, Ulam–Hyers, and Ulam–Hyers–Rassias senses, which coincides with the classical stability concepts for the cases of integer-order or Caputo-type derivatives. It is proved that the zero solution of the homogeneous system is weighted stable if and only if all its solutions are weighted bounded. In addition, for the homogeneous system it is established that the weighted stability in the Lyapunov and Ulam–Hyers senses are equivalent if and only if the inequality appearing in the Ulam–Hyers definition possess only bounded solutions. Finally, we derive natural sufficient conditions under which the property of weighted global asymptotic stability of the zero solution of the homogeneous system is preserved under nonlinear disturbances.

Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues.

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>

Analytical Study to Systems of Fractional Differential Equations with Prabhakar Derivative

We study systems of fractional order differential equations involving the Prabhakar derivative of Caputo type. For commensurate systems we obtain their solutions in closed forms using the eigenvevtors and the eigenvalues of the associated square matrix in the system. We discuss the solutions under the cases where the eigenvalues are distinct, repeated or complex. We present several examples to illustrate the efficiency of the obtained results. For incommensurate systems we apply the Laplace transform to obtain their solutions. As the Prabhakar kernels involve many fractional kernels as particular cases, the obtained results will generalize several existing results in the literature.

Dynamic analysis and optimal control of leptospirosis based on Caputo fractional derivative

<p>Caputo fractional derivative solves the fractional initial value problem in Riemann-Liouville (R-L) fractional calculus. The definition of a Caputo-type derivative is in the same form as the definition of an integral differential equation, including the restriction of the value of the integral derivative to the value of the unknown function at the endpoint $ t = a $. Therefore, this paper introduced the Caputo fractional derivative (CFD) to establish the transmission model of leptospirosis. First, to ensure that the model had a particular significance, we proved the dynamic properties of the model, such as nonnegative, boundedness, and stability of the equilibrium point. Second, according to the existence mode and genetic characteristics of pathogenic bacteria of leptospirosis, and from the perspective of score optimal control, we put forward measures such as wearing protective clothing, hospitalization, and cleaning the environment to prevent and control the spread of the disease. According to the proposed control measures, a control model of leptospirosis was established, and a forward-backward scanning algorithm (FB algorithm) was introduced to optimize the control function. Three different disease control strategies were proposed. Finally, the numerical simulation of different fractional orders used the fde12 (based on Adams–Bashforth–Moulton scheme) solver. The three optimized strategies, A, B, and C, were compared and analyzed. The results showed that the optimized control strategy could shorten the transmission time of the disease by about 80 days. Therefore, the above methods contributed to the study of leptospirosis and the World Health Organization.</p>

Continuous Dependence on the Initial Functions and Stability Properties in Hyers–Ulam–Rassias Sense for Neutral Fractional Systems with Distributed Delays

We study several stability properties on a finite or infinite interval of inhomogeneous linear neutral fractional systems with distributed delays and Caputo-type derivatives. First, a continuous dependence of the solutions of the corresponding initial problem on the initial functions is established. Then, with the obtained result, we apply our approach based on the integral representation of the solutions instead on some fixed-point theorems and derive sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of the investigated systems. A number of connections between each of the Hyers–Ulam, Hyers–Ulam–Rassias, and finite-time Lyapunov stability and the continuous dependence of the solutions on the initial functions are established. Some results for stability of the corresponding nonlinear perturbed homogeneous fractional linear neutral systems are obtained, too.

Well-posedness of fractional Moreau’s sweeping processes of Caputo type

In this paper, we study the well-posedness (in the sense of existence and uniqueness of a solution) of a sweeping process involving fractional order derivative of Caputo type in Hilbert spaces. The convex constraint set C(t) is assumed to satisfy a Hölder continuous variation. The notion of a solution of this fractional sweeping process is proposed, the existence and uniqueness of such solution are proved and a stability result is provided, as well. More precisely, we present new arguments for the proof of existence of the fractional sweeping process. It is still based on the ideas of the catching-up algorithm of J.J. Moreau (see Kunze and Monteiro Marques (2000) and Moreau (1977)). We establish a catching-up algorithm for fractional sweeping process. This new algorithm gives us a sequence of approximate solutions. The technical problems are to prove that we can extract a weakly convergent subsequence, and to check that this limit function is a solution of the fractional sweeping process. Here we propose a new approach to overcome these problems. The existence of solutions follows, to some extent, from an important differential inequality on the Caputo fractional derivative of the squared norm CD0α‖u(t)‖2≤2u(t),CD0αu(t), which was studied in Aguila-Camacho et al. (2014), Gomoyunov (2018) and Kamenskii et al. (2021), that applied to two approximate solutions, by the monotonicity of normal cone NC(t)(⋅) and the minus sign on the right hand side, yields that their distance is nonincreasing. This reasoning allows us to prove that the sequence of approximate solutions is of the fractional sweeping process, converging to a solution. Our approach is constructive and the algorithm is implemented numerically. The results of the paper are applied in the study of an application to non-regular electrical circuit containing nonsmooth electronic device like diode and fractional inductor, which represent an additional novelty of our paper. Finally, two numerical simulations to valid the theoretical results are given.

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A, fractional order α and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side f(t). The resulting method possesses exponential convergence for positive sectorial A, any finite t, including t=0 and the whole range α∈(0,2). It is suitable for a practically important case, when no knowledge of f(t) is available outside the considered interval t∈[0,T]. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.

On abstract Cauchy problems in the frame of a generalized Caputo type derivative

In this paper, we consider a class of abstract Cauchy problems in the framework of a generalized Caputo type fractional. We discuss the existence and uniqueness of mild solutions to such a class of fractional differential equations by using properties found in the related fractional calculus, the theory of uniformly continuous semigroups of operators and the fixed point theorem. Moreover, we discuss the continuous dependence on parameters and Ulam stability of the mild solutions. At the end of this paper, we bring forth some examples to endorse the obtained results

Numerical solutions of fractional epidemic models with generalized Caputo-type derivatives

Recently, a general framework of fractional operators, that includes the Caputo model as a particular case, has been introduced and some applications in the area of fractional calculus have been presented. In this paper, novel fractional epidemic models with generalized Caputo-type derivatives were proposed. The universal predictor-corrector method was modified here to deal with the considered epidemic models for the purposes of simulation. The behavior and complex dynamic of these hybrid fractional epidemic models were studied using the modified method. The dynamics of the generalized Caputo-type fractional SIR, HIV and SEIR models were investigated by numerical simulation. Basically, the effect of generalized Caputo-type fractional derivative operator parameters on the dynamic behavior of the proposed epidemic models was discussed.

A fractional optimal control model for a simple cash balance problem

In the paper, a fractional optimal control model is constructed for a simple cash balance problem by introducing two new factors the net profit of firms and the long-term memory of the problem. The long-term memory, described by the fractional derivative of Caputo type, is characterized by a memory kernel that satisfies the law of power decline. Using the minimization sequence methods, together with techniques of nonlinear analysis, the existence of the optimal solution is obtained. Through theoretical analysis and numerical simulation, it is concluded that the strength of memory is described by the size of α, the order of the fractional derivative of cash and securities balances, and further comes to the conclusion that the smaller the value, the stronger the memory, and vice versa. Moreover, it also reveals that the memory of the problem makes it sticky, the inertia to preserve it in the previous state, which leads to a change in the balance of cash and securities being flat rather than sharp, and the total balance of cash and securities decreases with the enhancement of memory. Additionally, it suggests that when α reaches 1, the memory weakens and the viscosity becomes tinier, and the fractional model of order α tends to the integer one under α=1, which is a Markov process without memory, and at this point the model is no longer viscous. Besides that, in another perspective, the parameter α can be defined as decision makers’ reliance on historical data and experience, under which hypothesis the value of α can be used to measure the conservatism of policymakers. It is consistent with the actual, and shows the rationality and validity of the fractional model from the side.

Oscillation results for a nonlinear fractional differential equation

<abstract><p>In this paper, the authors work with a general formulation of the fractional derivative of Caputo type. They study oscillatory solutions of differential equations involving these general fractional derivatives. In particular, they extend the Kamenev-type oscillation criterion given by Baleanu et al. in 2015. In addition, we prove results on the existence and uniqueness of solutions for many of the equations considered. Also, they complete their study with some examples.</p></abstract>

EXTENSION OF HAAR WAVELET TECHNIQUES FOR MITTAG-LEFFLER TYPE FRACTIONAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

Fractional order integro-differential equation (FOIDE) of Fredholm type is considered in this paper. The mentioned equations have many applications in mathematical modeling of real world phenomenon like image and signal processing. Keeping the aforementioned importance, we study the considered problem from two different aspects which include the existence theory and computation of numerical approximate solution. FOIDEs have been investigated very well by using Caputo-type derivative for the existence theory and numerical solutions. But the mentioned problems have very rarely considered under the Mittage-Leffler-type derivative. Also, for FOIDE of Fredholm type under Mittage-Leffler-type derivative has not yet treated by using Haar wavelet (HW) method. The aforementioned derivative is non-singular and nonlocal in nature as compared to classical Caputo derivative of fractional order. In many cases, the nonsingular nature is helpful in numerical computation. Therefore, we develop the existence theory for the considered problem by using fixed point theory. Sufficient conditions are established which demonstrate the existence and uniqueness of solution to the proposed problem. Further on utilizing HW method, a numerical scheme is developed to compute the approximate solution. Various numerical examples are given to demonstrate the applicability of our results. Also, comparison between exact and numerical solution for various fractional orders in the considered examples is given. Numerical results are displayed graphically.

New criteria on global Mittag-Leffler synchronization for Caputo-type delayed Cohen-Grossberg Inertial Neural Networks

<abstract><p>Our focus of this paper is on global Mittag-Leffler synchronization (GMLS) of the Caputo-type Inertial Cohen-Grossberg Neural Networks (ICGNNs) with discrete and distributed delays. This model takes into account the inertial term as well as the two types of delays, which greatly reduces the conservatism with respect to the model. A change of variables transforms the $ 2\beta $ order inertial frame into $ \beta $ order ordinary frame in order to deal with the effect of the inertial term. In the following steps, two novel types of delay controllers are designed for the purpose of reaching the GMLS. In conjunction with the novel controllers, utilizing differential mean-value theorem and inequality techniques, several criteria are derived to determine the GMLS of ICGNNs within the framework of Caputo-type derivative and calculus properties. At length, the feasibility of the results is further demonstrated by two simulation examples.</p></abstract>

ON FRACTIONAL INTEGRALS AND DERIVATIVES OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION

In this paper, we present new definitions of generalized fractional integrals and derivatives with respect to another function and derive some of their properties, such as their inter-relationship and semigroup law. Caputo-type generalized fractional derivative with respect to another function is also defined and its properties are derived. A Cauchy problem involving the new Caputo-type generalized fractional derivative is also studied. We also provide an expansion formula for Caputo-type derivative and apply it to solve a fractional-order problem.

Maximum Principles for Fractional Differential Inequalities with Prabhakar Derivative and Their Applications

This paper is devoted to studying a class of fractional differential equations (FDEs) with the Prabhakar fractional derivative of Caputo type in an analytical manner. At first, an estimate of the Prabhakar fractional derivative of a function at its extreme points is obtained. This estimate is used to formulate and prove comparison principles for related fractional differential inequalities. We then apply these comparison principles to derive pre-norm estimates of solutions and to obtain a uniqueness result for linear FDEs. The solution of linear FDEs with constant coefficients is obtained in closed form via the Laplace transform. For linear FDEs with variable coefficients, we apply the obtained comparison principles to establish an existence result using the method of lower and upper solutions. Two well-defined monotone sequences that converge uniformly to the actual solution of the problem are generated.

Fractional Fourier Transform and Ulam Stability of Fractional Differential Equation with Fractional Caputo-Type Derivative

In this paper, we study the Ulam-Hyers-Mittag-Leffler stability for a linear fractional order differential equation with a fractional Caputo-type derivative using the fractional Fourier transform. Finally, we provide an enumeration of the chemical reactions of the differential equation.