Caputo Fractional Differential Research Articles

Some properties and stability of Helmholtz model involved with nonlinear fractional difference equations and its relevance with quadcopter

This study is devoted to developing mathematical models associated with the Helmholtz equation as a second-order oscillator involved with nonlinear Caputo fractional difference equations. This study also focuses on determining the approximate solution of this model via the Ulam stability conception. Some properties of the mathematical model dealt with in this study are also presented. Numerical simulations are presented to justify the existence of stability results.

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.

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.

К вопросу о решении смешанной краевой задачи для уравнения с производными дробного порядка с различными началами

An approach to solving boundary value problems for an ordinary differential equation of fractional order containing a composition of left- and right-handed Riemann-Liouville and Caputo fractional differentiation operators is proposed. The approach is based on the reduction of the equation under consideration to the study of integral equations of fractional order with involution. As an example, a mixed problem has been solved. In particular, the proposed approach allows us to improve the previously obtained conditions for the solvability of this problem.

Computational Algorithm for Approximating Fractional Derivatives of Functions

This paper presents an algorithmic approach for numerically solving Caputo fractional differentiation. The trapezoidal rule was modified, the new modification was used to derive an algorithm to approximate fractional derivatives of order α > 0, the fractional derivative used was based on Caputo definition for a given function by a weighted sum of function and its ordinary derivatives values at specified points. The trapezoidal rule was used in conjunction with the finite difference scheme which is the forward, backward and central difference to derive the computational algorithm for the numerical approximation of Caputo fractional derivative for evaluating functions of fractional order. The study was conducted through some illustrative examples and analysis of error.

Asymptotic Constancy for the Solutions of Caputo Fractional Differential Equations with Delay

In this paper, we aim to study the neutral-type delayed Caputo fractional differential equations of the form CDαxt−gt,xt=ft,xt,t∈t0,∞,t0≥0 with order 0<α<1, which can be used to describe the growth processes in real-life sciences at which the present growth depends on not only the past state but also the past growth rate. Our ultimate goal in this study is to concentrate on the convergence of the solutions to a predetermined constant by establishing a linkage between the delayed fractional differential equation and an integral equation. In our analysis, the sufficient conditions for the asymptotic results are obtained due to fixed point theory. The utilization of the contraction mapping principle is a convenient approach in obtaining technical conditions that guarantee the asymptotic constancy of the solutions.

Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications

It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is better than the corresponding integer model, we need to compute the solutions of the fractional differential equations and compare them with an integer model relative to the data available. In this work, we will illustrate that the linear nq-order sequential Caputo fractional differential equations, which are sequential of order q where q<1 with fractional initial conditions and/or boundary conditions, can be solved. The reason for choosing sequential fractional dynamic equations is that linear non-sequential Caputo fractional dynamic equations with constant coefficients cannot be solved in general. We used the Laplace transform method to solve sequential Caputo fractional initial value problems. We used fractional boundary conditions to compute Green’s function for sequential boundary value problems. In addition, the solution of the sequential dynamic equations yields the solution of the corresponding integer-order differential equations as a special case as q→1.

Ulam-Hyers Stability of Four-Point Boundary Value Problem for Caputo Fractional Differential Equations with a Parameter

Fractional calculus is considered to be a powerful tool in describing complex systems with a~wide range of applicability in many fields of science and engineering. The behavior of many systems can be described by using fractional differential equations with boundary conditions. In this sense, the study on the stability of fractional boundary value problems is of high importance.\\ The main goal of this paper is to investigate the Ulam--Hyers stability and Ulam--Hyers--Rassias stability of a class of fractional four-point boundary value problem involving Caputo derivative and with a given parameter. By using contraction principles, sufficient conditions are obtained to guarantee the uniqueness of solution. Therefore, we obtain sufficient conditions for the stability of that class of nonlinear fractional boundary value problems on the space of continuous functions. The presented results improve and extend some previous research. Finally, we construct some examples to illustrate the theoretical results.

Magnetic Field, Variable Thermal Conductivity, Thermal Radiation, and Viscous Dissipation Effect on Heat and Momentum of Fractional Oldroyd-B Bio Nano-Fluid within a Channel

This study deals with the analysis of the heat and velocity profile of the fractional-order Oldroyd-B bio-nanofluid within a bounded channel. The study has a wide range of scope in modern fields of basic science such as medicine, the food industry, electrical appliances, nuclear as well as industrial cooling systems, reducing pollutants, fluids used in the brake systems of vehicles, etc. Oldroyd-B fluid is taken as a bio-nanofluid composed of base fluid (blood) and copper as nanoparticles. Using the fractional-order Oldroyd-B parameter, the governing equation is generalized from an integer to a non-integer form. A strong approach, i.e., a finite difference scheme, is applied to discretize the model, because the fractional approach can well address the physical phenomena and memory effect of the flow regime. Therefore, a Caputo fractional differentiation operator is used for the purpose. The transformations for the channel flow are utilized to transfigure the fractional-order partial differential equations (PDEs) into non-dimension PDEs. The graphical outcomes for non-integer ordered Oldroyd-B bio-nanofluid dynamics and temperature profiles are navigated using the numerical technique. These results are obtained under some very important physical conditions applied as a magnetic field effect, variable thermal conductivity, permeable medium, and heat source/sink. The results show that the addition of (copper) nanoparticles to (blood) base fluids enhances the thermal conductivity. For a comparative study, the obtained results are compared with the built-in results using the mathematical software MAPLE 2016.

Boundary Value Problem for Multi-Term Nonlinear Delay Generalized Proportional Caputo Fractional Differential Equations

A nonlocal boundary value problem for a couple of two scalar nonlinear differential equations with several generalized proportional Caputo fractional derivatives and a delay is studied. The exact solution of the scalar nonlinear differential equation with several generalized proportional Caputo fractional derivatives with different orders is obtained. A mild solution of the boundary value problem for the multi-term nonlinear couple of the given fractional equations is defined. The connection between the mild solution and the solution of the studied problem is discussed. As a partial case, several results for the nonlocal boundary value problem for the linear and non-linear multi-term Caputo fractional differential equations are provided. The results generalize several known results in the literature.

An Optimal Adaptive Grid Method Based on L1 Scheme for a Nonlinear Caputo Fractional Differential Equation

A nonlinear fractional differential equation with a Caputo derivative of order α is studied. This problem is discretized by using the L1 scheme on an arbitrary nonuniform mesh. By utilizing the Taylor expansion with integral remainder term, an optimal local truncation error estimation of L1 scheme is proved. Based on this truncation error estimation and the mesh equidistribution principle, a new monitor function is constructed to construct an adaptive grid generation algorithm. Numerical experiments are performed to confirm the accuracy of our new adaptive grid algorithm.

Proportional Caputo Fractional Differential Inclusions in Banach Spaces

In this work, we introduce the notion of a (weak) proportional Caputo fractional derivative of order α∈(0,1) for a continuous (locally integrable) function u:[0,∞)→E, where E is a complex Banach space. In our definition, we do not require that the function u(·) is continuously differentiable, which enables us to consider the wellposedness of the corresponding fractional relaxation problems in a much better theoretical way. More precisely, we systematically investigate several new classes of (degenerate) fractional solution operator families connected with the use of this type of fractional derivatives, obeying the multivalued linear approach to the abstract Volterra integro-differential inclusions. The quasi-periodic properties of the proportional fractional integrals as well as the existence and uniqueness of almost periodic-type solutions for various classes of proportional Caputo fractional differential inclusions in Banach spaces are also considered.

Exponential Convergence of hp-discontinuous Galerkin Method for Nonlinear Caputo Fractional Differential Equations

Exponential Convergence of hp-discontinuous Galerkin Method for Nonlinear Caputo Fractional Differential Equations

The effect of the Caputo fractional difference operator on a new discrete COVID-19 model

This study aims to generalize the discrete integer-order SEIR model to obtain the novel discrete fractional-order SEIR model of COVID-19 and study its dynamic characteristics. Here, we determine the equilibrium points of the model and discuss the stability analysis of these points in detail. Then, the non-linear dynamic behaviors of the suggested discrete fractional model for commensurate and incommensurate fractional orders are investigated through several numerical techniques, including maximum Lyapunov exponents, phase attractors, bifurcation diagrams and C0 algorithm. Finally, we fitted the model with actual data to verify the accuracy of our mathematical study of the stability of the fractional discrete COVID-19 model.

Generalized Proportional Caputo Fractional Differential Equations with Noninstantaneous Impulses: Concepts, Integral Representations, and Ulam-Type Stability

The generalized proportional Caputo fractional derivative is a comparatively new type of derivative that is a generalization of the classical Caputo fractional derivative, and it gives more opportunities to adequately model complex phenomena in physics, chemistry, biology, etc. In this paper, the presence of noninstantaneous impulses in differential equations with generalized proportional Caputo fractional derivatives is discussed. Generalized proportional Caputo fractional derivatives with fixed lower limits at the initial time as well as generalized proportional Caputo fractional derivatives with changeable lower limits at each impulsive time are considered. The statements of the problems in both cases are set up and the integral representation of the solution of the defined problem in each case is presented. Ulam-type stability is also investigated and some examples are given illustrating these concepts.

Proportional Caputo Fractional Differential Equations

Proportional Caputo Fractional Differential Equations

Discrete Memristor and Discrete Memristive Systems.

In this paper, we investigate the mathematical models of discrete memristors based on Caputo fractional difference and G–L fractional difference. Specifically, the integer-order discrete memristor is a special model of those two cases. The “∞”-type hysteresis loop curves are observed when input is the bipolar periodic signal. Meanwhile, numerical analysis results show that the area of hysteresis decreases with the increase of frequency of input signal and the decrease of derivative order. Moreover, the memory effect, characteristics and physical realization of the discrete memristors are discussed, and a discrete memristor with short memory effects is designed. Furthermore, discrete memristive systems are designed by introducing the fractional-order discrete memristor and integer-order discrete memristor to the Sine map. Chaos is found in the systems, and complexity of the systems is controlled by the parameter of the memristor. Finally, FPGA digital circuit implementation is carried out for the integer-order and fractional-order discrete memristor and discrete memristive systems, which shows the potential application value of the discrete memristor in the engineering application field.

Generalized Proportional Caputo Fractional Differential Equations with Delay and Practical Stability by the Razumikhin Method

Practical stability properties of generalized proportional Caputo fractional differential equations with bounded delay are studied in this paper. Two types of stability, practical stability and exponential practical stability, are defined and considered, and also some sufficient conditions to guarantee stability are presented. The study is based on the application of Lyapunov like functions and their generalized proportional Caputo fractional derivatives among solutions of the studied system where appropriate Razumikhin like conditions are applied (appropriately modified in connection with the fractional derivative considered). The theory is illustrated with several nonlinear examples.

A Caputo discrete fractional-order thermostat model with one and two sensors fractional boundary conditions depending on positive parameters by using the Lipschitz-type inequality

A thermostat model described by a second-order fractional difference equation is proposed in this paper with one sensor and two sensors fractional boundary conditions depending on positive parameters by using the Lipschitz-type inequality. By means of well-known contraction mapping and the Brouwer fixed-point theorem, we provide new results on the existence and uniqueness of solutions. In this work by use of the Caputo fractional difference operator and Hyer–Ulam stability definitions we check the sufficient conditions and solution of the equations to be stable, while most researchers have examined the necessary conditions in different ways. Further, we also establish some results regarding Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability for our discrete fractional-order thermostat models. To support the theoretical results, we present suitable examples describing the thermostat models that are illustrated by graphical representation.

Stability of Gene Regulatory Networks Modeled by Generalized Proportional Caputo Fractional Differential Equations

A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functions are pointed out. The equilibrium of the generalized proportional Caputo fractional model and its generalized exponential stability are defined, and sufficient conditions for the generalized exponential stability and asymptotic stability of the equilibrium are obtained. As a special case, the stability of the equilibrium of the Caputo fractional model is discussed. Several examples are provided to illustrate our theoretical results and the influence of the type of fractional derivative on the stability behavior of the equilibrium.