A sufficient condition of viability for fractional differential equations with the Caputo derivative

In this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give the sufficient condition that guarantees fractional viability of a locally closed set with respect to nonlinear function. As an example we discuss positivity of solutions, particularly in linear case.     

On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process

Consider an inverse problem for the time-fractional diffusion equation in one dimensional spatial space. The aim is to determine the initial status and heat flux on the boundary simultaneously from heat measurement data given on the other boundary. Using the Laplace transform and the unique extension technique, the uniqueness for this inverse problem is proven. Then we construct a regularizing scheme for the reconstruction of boundary flux for known initial status. The convergence rate of the regularizing solution is established under some a priori information about the exact solution. Moreover, the initial distribution can also be recovered approximately from our regularizing scheme. Finally we present some numerical examples, which show the validity of the proposed reconstruction scheme.     

Analysis of the time fractional nonlinear diffusion equation from diffusion process

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.     

A nonlinear viscoelastic fractional derivative model of infant hydrocephalus

Infant communicating hydrocephalus is a clinical condition where the cerebral ventricles become enlarged causing the developing brain parenchyma of the newborn to be displaced outwards into the soft, unfused skull. In this paper, a hyperelastic, fractional derivative viscoelastic model is derived to describe infant brain tissue under conditions consistent with the development of hydrocephalus. An incremental numerical technique is developed to determine the relationship between tissue deformation and applied pressure gradients. Using parameter values appropriate for infant parenchyma, it is shown that pressure gradients of the order of 1 mm Hg are sufficient to cause hydrocephalus. Predicting brain tissue deformations resulting from pressure gradients is of interest and relevance to the treatment and management of hydrocephalus, and to the best of our knowledge, this is the first time that results of this nature have been established.     

An approach to differential geometry of fractional order via modified Riemann-Liouville derivative

In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouville definition of fractional derivative, one (Jumarie) has proposed recently an alternative referred to as (local) modified Riemann-Liouville definition, which directly, provides a Taylor’s series of fractional order for non differentiable functions. We examine here in which way this calculus can be used as a framework for a differential geometry of fractional order. One will examine successively implicit function, manifold, length of curves, radius of curvature, Christoffel coefficients, velocity, acceleration. One outlines the application of this framework to Lagrange optimization in mechanics, and one concludes with some considerations on a possible fractional extension of the pseudo-geodesic of thespecial relativity and of the Lorentz transformation.     

Positive solutions of fractional differential equations with the Riesz space derivative

A new class of fractional differential equations with the Riesz–Caputo derivative is proposed and the physical meaning is introduced in this paper. The boundary value problem is investigated under some conditions. Leray–Schauder and Krasnoselskii’s fixed point theorems in a cone are adopted. Existence of positive solutions is provided. Finally, two examples with numerical solutions are given to support theoretical results.     

An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator

The heat equation is parabolic partial differential equation and occurs in the characterization of diffusion progress. In the present work, a new fractional operator based on the Rabotnov fractional-exponential kernel is considered. Next, we conferred some fascinating and original properties of nominated new fractional derivative with some integral transform operators where all results are significant. The fundamental target of the proposed work is to solve the multidimensional heat equations of arbitrary order by using analytical approach homotopy perturbation transform method and residual power series method, where new fractional operator has been taken in new Yang-Abdel-Aty-Cattani (YAC) sense. The obtained results indicate that solution converges to the original solution in language of generalized Mittag-Leffler function. Three numerical examples are discussed to draw an effective attention to reveal the proficiency and adaptability of the recommended methods on new YAC operator.     

Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method

We use Adomian decomposition method for solving the fractional nonlinear two-point boundary value problem

where D is Caputo fractional derivative, c is a constant, μ > 0, and F:[0,1]×[0,∞)→[0,∞) a continuous function. The fractional Bratu problem is solved as an illustrative example.     

Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation

The principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order α (0 < α < 1) subject to the narrow-band random parametric excitation are investigated. The method of multiple scales is developed to derive two first order stochastic differential equation of amplitude and phase, and then to examine the influences of fractional order and intensity of random excitation on the first-order and second-order moment. As an example, the stochastic Duffing oscillator with fractional derivative damping is considered. The effects of detuning frequency parameter, the intensity of random excitation and the fractional order derivative damping on stability are studied through the largest Lyapunov exponent. The corresponding theoretical results are well verified through direct numerical simulations. In addition, the phenomenon of stochastic jump is analyzed for parametric principal resonance responses via finite differential method. The stochastic jump phenomena indicates that the most probable motion is around the larger non-trivial branch of the amplitude response when the intensity of excitation is very small, and the probable motion of amplitude responses will move from the larger non-trivial branch to trivial branch with the increasing of the intensity of excitation. Such stochastic jump can be considered as bifurcation.     

Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter

Our aim in this work was to examine the model underpinning the spread of the Rubella virus using the novel derivative called beta‐derivative. The study of the equilibrium points together with the analysis of the disease free equilibrium points was presented. Due to the complexity of the modified equation, we introduced a new operator based on the Sumudu transform. The properties of this operator were proposed and proved in detail. We made used of this operator together with the idea of perturbation method to derive a special solution of the extended model. The stability of the method for solving this model was presented. The uniqueness of the special solution was presented, and numerical simulations were done. The graphical representations show that the model depends on both parameters and the fractional order. © 2015 Wiley Periodicals, Inc. Complexity 21: 442–451, 2016     

Multiobjective fractional variational calculus in terms of a combined Caputo derivative

The study of fractional variational problems in terms of a combined fractional Caputo derivative is introduced. Necessary optimality conditions of Euler-Lagrange type for the basic, isoperimetric, and Lagrange variational problems are proved, as well as transversality and sufficient optimality conditions. This allows to obtain necessary and sufficient Pareto optimality conditions for multiobjective fractional variational problems.     

Analysis of a system of fractional differential equations

We prove existence and uniqueness theorems for the initial value problem for the system of fractional differential equations , where D^α denotes standard Riemann-Liouville fractional derivative, 0<α<1, and A is a square matrix. The unique solution to this initial value problem turns out to be , where E_α denotes the Mittag-Leffler function generalized for matrix arguments. Further we analyze the system , , 0<α<1, and investigate dependence of the solutions on the initial conditions.     

Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time

We consider the inverse problem of finding the temperature distribution and the heat source whenever the temperatures at the initial time and the final time are given. The problem considered is one dimensional and the unknown heat source is supposed to be space dependent only. The existence and uniqueness results are proved.     

Couette flow of viscoelastic fluid with constitutive relation involving general Caputo-type fractional derivative

In this paper, the general Caputo-type fractional differential operator introduced by Pr. Anatoly N. Kochubei is applied to the linear theory of viscoelasticity. Firstly, using the general Caputo-type derivative, a generalized linear viscoelastic constitutive equation is proposed for the first time. Secondly, the momentum equation for the plane Couette flow of viscoelastic fluid with the constitutive relation is given as an integrodifferential equation and the analytical solution of the equation is established by employing the separation of variables method. Lastly, for special cases of the general constitutive relation, the analytical solutions are obtained in terms of the Mittag-Leffler functions.     

A comparison of numerical solutions of fractional diffusion models in finance

We compare the numerical solutions of three fractional partial differential equations that occur in finance. These fractional partial differential equations fall in the class of Lévy models. They are known as the FMLS (Finite Moment Log Stable), CGMY and KoBol models. Conditions for the convergence of each of these models is obtained.     

On the derivation of fractional diffusion equation with an absorbent term and a linear external force

Recently, the generalized fractional reaction–diffusion equation subject to an external linear force field has been proposed to describe the transport processes in disordered systems. The solution of this generalized model can be formally expressed in closed form through the Fox function. For the sack of completeness, we dedicate this work to construct a neatly derivation of the generalized fractional reaction–diffusion equation. Remarkably, such derivation could in general offer some novel and inspiring inspection to the phenomena of anomalous transport. For instance, there is a strong evidence that the fractional calculus offers some physical insight into the origin of fractional dynamics for a systems which exhibit multiple trapping.     

Analysis of nonlinear systems arise in thermoelasticity using fractional natural decomposition scheme

The pivotal aim of the present study is to employ fractional natural decomposition method (FNDM) to find the solution for a nonlinear system arising in thermoelasticity. The considered coupled system is generalised many physical phenomena associated with the material with elastic characters and its temperature and also which is called a Cauchy problem. We consider the coupled system by incorporating the Caputo fractional operator and investigate three distinct cases for different initial values to illustrate the applicability and efficiency of the FNDM. With respect to fractional order, we capture the behaviour of the achieved solution cited in three different cases and exemplified with the aid of 2D and 3D plots for the particular value of the parameters in the model. Moreover, some interesting behaviours of the projected model are confirms the prominence of the employed fractional operator while analysing the nonlinear coupled equations exemplifying real-world problems and also shows the capability of the considered algorithm.