Caputo-type Fractional Derivatives Research Articles

Analysis of some generalized [formula omitted] – Fractional logistic models

In this article, some logistic models in the settings of Caputo fractional operators with multi-parametered Mittag-Leffler kernels (ABC) are studied. This study mainly focuses on modified quadratic and cubic logistic models in the presence of a Caputo type fractional derivative. Existence and uniqueness theorems are proved and stability analysis is discussed by perturbing the equilibrium points. Numerical illustrative examples are discussed for the studied models.

A reliable treatment of residual power series method for time-fractional Black–Scholes European option pricing equations

Approximate analytical solution of a fractional Black–Scholes pricing model is much relevant due to its practical importance to financial markets. In this article, a powerful approximate iterative mathematical scheme based on residual power series (RPS) algorithm is presented to achieve the numerical results of the nonlinear time fractional Black–Scholes (BS) equations based on European options. The Caputo-type fractional derivatives are considered in the present article. The residual power series method (RPSM) developed by Arqub (2013) is the novel technique for finding the analytical Taylor series solutions of systems of linear and nonlinear ODEs and PDEs. The residual power series technique supplies the approximate analytical solutions of the problem in truncated series form using residual error concept along with given initial conditions. The numerical procedures reveal that only a few iterations are sufficient for better approximations of the solutions, which clearly exhibits the reliability and effectiveness of this iterative scheme. This article shows that the adopted scheme is quite systematic as well as computationally attractive regarding solution procedure. In addition, effects of fractional order of time derivatives on the solutions for various particular cases are also depicted through graphs.

Approximate analytical solution of fractional order biochemical reaction model and its stability analysis

Approximate analytical solution of the system of coupled nonlinear Ordinary Differential Equations (ODEs) of a biochemical reaction model is much relevant due to its practical significance to biochemists. In this paper, an effective and powerful mathematical technique, viz. fractional homotopy analysis transform method (FHATM), is employed to get the numerical solutions of biochemical reaction model with time fractional derivatives. The adopted scheme is the beautiful copulation of homotopy analysis technique and Laplace transform algorithm. This paper shows that the adopted scheme is quite easy as well as computationally attractive in the context of a solution procedure. The Caputo-type fractional derivatives are considered in the present paper. Approximate results of the probability density functions of the time fractional biochemical reaction model are computed for miscellaneous fractional Brownian motions as well as for classical motion and are presented graphically. The time fractional biochemical reaction model with respect to stability analysis for various values of fractional order [Formula: see text] is also analyzed. In the context of stability discussion, we have used the fractional Routh–Hurwitz stability criterion to establish the local stability of the biochemical reaction model of fractional order.

A finite difference scheme on graded meshes for time-fractional nonlinear Korteweg-de Vries equation

In this paper, a finite difference scheme for time-fractional nonlinear Korteweg-de Vries (KdV) problems with the Caputo-type fractional derivative is presented. To deal with the weak singularity caused by the fractional derivative that the solution has in the initial layer, the well-known L1 scheme on graded meshes has been used for time discretization. Meanwhile, a nonlinear finite difference approximation on uniform meshes is proposed for spatial discretization. The existence, stability and convergence of the numerical solution are studied by the energy method. It is proved that the scheme is of min{2−α,rα} order convergence in time and of first-order convergence in space, where α is the order of fractional derivative and r is a parameter of graded meshes. Furthermore, in order to increase computational efficiency, the corresponding fast algorithm of the presented scheme has been considered. In numerical simulation, a valid approach based on linear interpolation is designed to test convergence rate for the scheme on temporal graded meshes. Numerical results demonstrate sharpness of the theoretical convergence estimate and effectiveness of the fast algorithm.

Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation

In this paper, the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order 0<α<1 is considered. The uniqueness of solution is established by using the contraction mapping principle and the existence of solution is obtained by Schauder fixed point theorem.

On the theoretical basis of memory-free approaches for fractional differential equations

PurposeA nonclassical method, usually called memory-free approach, has shown promising potential to release arithmetic complexity and meets high memory-storage requirements in solving fractional differential equations. Though many successful applications indicate the validity and effectiveness of memory-free methods, it has been much less understood in the rigorous theoretical basis. This study aims to focus on the theoretical basis of the memory-free Yuan–Agrawal (YA) method [Journal of Vibration and Acoustics 124 (2002), pp. 321-324].Design/methodology/approachMathematically, the YA method is based on the validity of two fundamental procedures. The first is to reverse the integration order of an improper quadrature deduced from the Caputo-type fractional derivative. And, the second concerns the passage to the limit under the integral sign of the improper quadrature.FindingsThough it suffices to verify the integration order reversibility, the uniform convergence of the improper integral is proved to be false. Alternatively, this paper proves that the integration order can still be reversed, as the target solution can be expanded as Taylor series on [0, ∞). Once the integration order is reversed, the paper presents a sufficient condition for the passage to the limit under the integral sign such that the target solution is continuous on [0, ∞). Both positive and counter examples are presented to illustrate and validate the theoretical analysis results.Originality/valueThis study presents some useful results for the real performance for the YA and some similar memory-free approaches. In addition, it opens a theoretical question on sufficient and necessary conditions, if any, for the validity of memory-free approaches.

A Novel Method for Solving the Bagley-Torvik Equation as Ordinary Differential Equation

We present a novel method to solve the Bagley-Torvik equation by transforming it into ordinary differential equations (ODEs). This method is based on the equivalence between the Caputo-type fractional derivative (FD) of order 3/2 and the solution of a diffusion equation subjected to certain initial and boundary conditions. The key procedure is to approximate the infinite boundary condition by a finite one, so that the diffusion equation can be solved by separation of variables. By this procedure, the Bagley-Torvik and the diffusion equations together are transformed to be a set of ODEs, which can be integrated numerically by the Runge-Kutta scheme. The presented method is tested by various numerical cases including linear, nonlinear, nonsmooth, or multidimensional equations, respectively. Importantly, high computational efficiency is achieved as this method is at the expense of linearly increasing computational cost with the solution domain being enlarged.

Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel

Most disordered dielectrics especially solid dielectrics show non-Debye laws, and many empirical approximation models are proposed to describe these anomalous relaxation processes. Fractional calculus approach is a popular approach to analyze the anomalous relaxation processes and has been intensively studied in various dielectric materials. However, Capelas de Oliveira et al. proved that the memory kernels of Caputo type fractional derivatives must satisfy the initial value condition (IVC). But both kernels of the Caputo–Fabrizio derivative and the Atangana–Baleanu derivative do not satisfy this condition. In this paper, we prove that the Caputo type derivative with a Prabhakar-like kernel satisfies the IVC and this derivative is in the framework of general Caputo fractional derivative (GC derivative). Corresponding anomalous relaxation model and its solution are discussed. Analysis result shows our model, as a direct extension of the Cole–Cole model, contains the Debye model and the fractional relaxation model as particular cases.

Analytical study of fractional Bratu-type equation arising in electro-spun organic nanofibers elaboration

In this article, we present a hybrid analytical scheme, namely, homotopy perturbation transform method to achieve the solutions of the non-linear Bratu-type problem with fractional order derivatives. The Caputo type fractional derivatives are considered in the present article. He’s polynomial is used to tackle the nonlinearity which arise in our considered problem. Using the initial conditions, the numerical solutions of the problem are computed. The numerical procedure reveals that only a few iterations are needed for better approximation of the solutions, which illustrates the effectiveness and reliability of the method. Effects of fractional order derivatives on the solutions for various particular cases are depicted through graphs.

Lyapunov-type inequalities for nonlinear fractional differential equations and systems involving Caputo-type fractional derivatives

A Lyapunov-type inequality is derived for a nonlinear fractional boundary value problem involving Caputo-type fractional derivative. The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem. Next, we extend our study to the case of systems.

A comparative study of analytical solutions of space-time fractional hyperbolic-like equations with two reliable methods

This paper deals with a comparative study of analytical solutions of -dimensional space-time fractional hyperbolic-like equations (with Caputo type fractional derivatives) using two reliable semi-analytical methods: “new integral projected differential transform method (NIPDTM)” and “fractional reduced differential transform method (FRDTM)”. Three test problems are carried out in order to illustrate the efficiency of these methods. The computed results are also compared with the results fromo various schemes, in the literature. The computed series solutions from either method converge to the exact solutions. FRDTM solutions are easy to compute without using any transformation as compared to NIPDTM, but FRDTM needs more iterations to obtain the solutions with the same accuracy.

A study of fractional differential equations and inclusions involving generalized Caputo-type derivative equipped with generalized fractional integral boundary conditions

In this paper, we introduce a new kind of generalized fractional integral boundary conditions and develop the existence theory for a fractional di erential equation involving generalized Caputotype fractional derivative equipped with these conditions. We also study the inclusion case of the given problem. Examples are constructed to demonstrate the application of the obtained results.

Scott-Blair models with time-varying viscosity

In a recent paper, Zhou et al. (2017) studied the time-dependent properties of Glass Fiber Reinforced Polymers composites by employing a new rheological model with a time-dependent viscosity coefficient. This rheological model is essentially based on a generalized Scott-Blair body with a time-dependent viscosity coefficient. Motivated by this study, in this note we suggest a different generalization of the Scott-Blair model based on the application of Caputo-type fractional derivatives of a function with respect to another function. This new mathematical approach can be useful in viscoelasticity and diffusion processes in order to model systems with time-dependent features. In this paper we also provide the general solution of the creep experiment for our improved Scott-Blair model together with some explicit examples and illuminating plots.

WEll-Posedness of General Caputo-Type Fractional Differential Equations

Abstract In the present paper, initial value problems of fractional differential equations are considered. The fractional derivatives are defined as the general Caputo-type fractional derivatives proposed by Anatoly Kochubei. By using the Schauder fixed point theorem, the local existence, the global existence and the uniqueness of solutions are obtained under appropriate conditions. In addition, it is proved that the solution depends on the parameters of the equation in a continuous way.

Taylor’s formula involving generalized fractional derivatives

In this paper, we establish a generalized Taylor expansion of a given function f in the formf(x)=∑j=0mcjα,ρ(xρ−aρ)jα+em(x)with m∈N0,cjα,ρ∈R,x > a > 0 and 0 < α ≤ 1. In case ρ=α=1, this expression coincides with the classical Taylor formula. The coefficients cjα,ρ,j=0,…,m as well as the residual term em(x) are given in terms of the generalized Caputo-type fractional derivatives. Several examples and applications of these results for the approximation of functions and for solving some fractional differential equations in series form are given in illustration.

An insight into Newton's cooling law using fractional calculus

For small temperature differences between a heated body and its environment, Newton's law of cooling predicts that the instantaneous rate of change of temperature of any heated body with respect to time is proportional to the difference in temperature of the body with the ambient, time being measured in integer units. Our experiments on the cooling of different liquids (water, mustard oil, and mercury) did not fit the theoretical predictions of Newton's law of cooling in this form. The solution was done using both Caputo and Riemann-Liouville type fractional derivatives to check if natural phenomena showed any preference in mathematics. In both cases, we find that cooling of liquids has an identical value of the fractional derivative of time that increases with the viscosity of the liquid. On the other hand, the cooling studies on metal alloys could be fitted exactly by integer order time derivative equations. The proportionality constant between heat flux and temperature difference was examined with respect to variations in the depth of liquid and exposed surface area. A critical combination of these two parameters signals a change in the mode of heat transfer within liquids. The equivalence between the proportionality constants for the Caputo and Riemann-Liouville type derivatives is established.

On a Caputo-type fractional derivative

Abstract In this paper, we present a new differential operator of arbitrary order defined by means of a Caputo-type modification of the generalized fractional derivative recently proposed by Katugampola. The generalized fractional derivative, when convenient limits are considered, recovers the Riemann–Liouville and the Hadamard derivatives of arbitrary order. Our differential operator recovers as limiting cases the arbitrary order derivatives proposed by Caputo and by Caputo–Hadamard. Some properties are presented as well as the relation between this differential operator of arbitrary order and the Katugampola generalized fractional operator. As an application we prove the fundamental theorem of fractional calculus associated with our operator.

Steady-state response of systems with fractional dampers

In this paper, steady state vibrations of systems with built-up viscoelastic dampers are considered. The dampers are modeled using the fractional-derivative rheological models. The Caputo type fractional derivative definition is used. In particular, the steady state vibrations of systems are analysed. The solution to the steady state vibrations is written using real quantities. The effects induced by changes of environmental temperature are also considered and, in this context, the time-temperature superposition principle is adopted. The results of several parametric studies are also described and discussed in detail.

Existence of solutions for a mixed fractional boundary value problem

In this paper, we prove the existence of solutions for a boundary value problem involving both left Riemann-Liouville and right Caputo-type fractional derivatives. For this, we convert the posed problem to a sum of two integral operators, then we apply Krasnoselskii’s fixed point theorem to conclude the existence of nontrivial solutions.

Modified Adomian decomposition method for solving fractional optimal control problems

In this study, we use the modified Adomian decomposition method to solve a class of fractional optimal control problems. The performance index of a fractional optimal control problem is considered as a function of both the state and the control variables, and the dynamical system is expressed in terms of a Caputo type fractional derivative. Some properties of fractional derivatives and integrals are used to obtain Euler–Lagrange equations for a linear tracking fractional control problem and then, the modified Adomian decomposition method is used to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to a linear tracking fractional optimal control problem. We compare the proposed technique with some numerical methods to demonstrate the accuracy and efficiency of the modified Adomian decomposition method by examining several illustrative test problems.