Regularity results for fractional diffusion equations involving fractional derivative with Mittag–Leffler kernel

This paper studies partial differential equation model with the new general fractional derivatives involving the kernels of the extended Mittag–Leffler type functions. An initial boundary value problem for the anomalous diffusion of fractional order is analyzed and considered. The fractional derivative with Mittag–Leffler kernel or also called Atangana and Baleanu fractional derivative in time is taken in the Caputo sense. We obtain results on the existence, uniqueness, and regularity of the solution.     

Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model

In 2015 Caputo and Fabrizio suggested a new operator with fractional order, this derivative is based on the exponential kernel. Earlier this year 2016 Atangana and Baleanu developed another version which used the generalized Mittag-Leffler function as non-local and non-singular kernel. Both operators show some properties of filter. However the Atangana and Baleanu version has in addition to this, all properties of fractional derivative. In this work, we aimed to represent the model by Allen–Cahn with both derivatives in order to see their difference in a real world problem. Both modified models will be solved numerically via the Crank–Nicholson scheme and their numerical simulations are presented to check the effectiveness of the both kernels.     

Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

In the literature, many researchers have studied Lotka‐Volterra (L‐V) models for different types of studies. In order to continue the study, we consider a fractional‐order L‐V model involving three different species in the Atangana‐Baleanu‐Caputo (ABC) sense of fractional derivative. This new model has potentials for a large number of research‐oriented studies. The first point that arises is whether the new model has a solution or not. Therefore, to answer this question, we consider the existence and uniqueness (EU) of the solutions and then Hyers‐Ulam (HU) stability for the proposed L‐V model.     

Maximum principles for a time–space fractional diffusion equation

In this paper, we focus on maximum principles of a time–space fractional diffusion equation. Maximum principles for classical solution and weak solution are all obtained by using properties of the time fractional derivative operator and the fractional Laplace operator. We deduce maximum principles for a full fractional diffusion equation, other than time-fractional and spatial-integer order diffusion equations.     

Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order

Recently, Atangana and Baleanu proposed a derivative with fractional order to answer some outstanding questions that were posed by many researchers within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. In this paper, we presented further relationship of their derivatives with some integral transform operators. New results are presented. We applied this derivative to a simple nonlinear system. We show in detail the existence and uniqueness of the system solutions of the fractional system. We obtain a chaotic behavior which was not obtained by local derivative.     

Fractional order theory of Cattaneo-type thermoelasticity using new fractional derivatives

Transient thermoelastic interactions between materials and the moving heat sources, i.e. Laser additive manufacturing, Laser-assisted thermotherapy, high speed sliding and rolling contacts, are becoming increasingly important. In this work, a unified fractional thermoelastic theory is developed, and applied to study transient responses caused by a moving heat source. Theoretically, new insights on fractional thermoelasticity are provided by introducing new definitions of fractional derivative, i.e. Caputo-Fabrizio, Atangana–Baleanu and Tempered-Caputo type. Numerically, a semi-infinite medium subjected to a source of heat moving with constant velocity is considered within the present model under two different sets of boundary conditions: stress free and temperature given for the first, displacement fixed and thermally adiabatic for the second. Analytical solutions to all responses are firstly formulated in Laplace domain, and then transformed into time domain through numerical method. The numerical results show that Caputo-Fabrizio and Atangana–Baleanu type models predict smaller transient responses than Caputo type theory, while Tempered-Caputo model may give larger results by increasing the tempered parameter. Meanwhile, the effect of fractional order, tempered parameter of Tempered-Caputo model, and the velocity of heat source on all responses is discussed in detail. The time history of responses shows that: for long-term process, the exponential function of TC definition will make sense, and the temperature from TC model is greatly different from that of C model. This work may provide comprehensive understanding for thermoelastic interactions due to moving heat source, and open up possibly wide applications of such new fractional derivatives.     

New analysis and application of fractional order Schrödinger equation using with Atangana–Batogna numerical scheme

In this work, an analytical approximation to the solution of Schrodinger equation has been provided. The fractional derivative used in this equation is the Caputo derivative. The existence and uniqueness conditions of solutions for the proposed model are derived based on the power law. While solving the fractional order Schrodinger equation, Atangana–Batogna numerical method is presented for fractional order equation. We obtain an efficient recurrence relation for solving these kinds of equations. To illustrate the usefulness of the numerical scheme, the numerical simulations are presented. The results show that the numerical scheme is very effective and simple.     

Maximum principle for the generalized time-fractional diffusion equation

In the paper, a maximum principle for the generalized time-fractional diffusion equation over an open bounded domain is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Caputo-Dzherbashyan fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the generalized time-fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.     

Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders

The solvability of initial value problems for nonlinear Langevin equation involving two fractional orders are discussed in this paper. An existence result for the solution is obtained using the Leray–Schauder nonlinear alternative. In addition, sufficient conditions for unique solution are established under the Banach contraction principle. The existence results for the initial value problems of nonlinear classical Langevin equation follow as a special case of our results.     

分数阶微分方程的一些新结果

第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性（次线性）条件下C310,11正解存在的充分必要条件．最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．     

Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative

A Hamilton–Jacobi equation with Caputo’s time fractional derivative of order less than one is considered. The notion of a viscosity solution is introduced to prove unique existence of a solution to the initial value problem under periodic boundary conditions. For this purpose, comparison principle as well as Perron’s method is established. Stability with respect to the order of derivative as well as the standard one is studied. Regularity of a solution is also discussed. Our results in particular apply to a linear transport equation with time fractional derivatives with variable coe?cients.     

Maximum principle for Hadamard fractional differential equations involving fractional Laplace operator

The purpose of the current study is to investigate IBVP for spatial-time fractional differential equation with Hadamard fractional derivative and fractional Laplace operator(−Δ)^β. A new Hadamard fractional extremum principle is established. Based on the new result, a Hadamard fractional maximum principle is also proposed. Furthermore, the maximum principle is applied to linear and nonlinear Hadamard fractional equations to obtain the uniqueness and continuous dependence of the solution of the IBVP at hand.     

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.     

抽象空间缓增分数阶微分方程的吸引性

研究了抽象空间中缓增分数阶微分方程解的吸引性.建立了Cauchy问题存在全局吸引解的充分条件.揭示了缓增分数阶导数求解分数微分方程解的特征.     

Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.     

On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.     

The Maximum Principle for Time-Fractional Diffusion Equations and Its Application

Using an equivalent reformulation of the definition of the Caputo fractional derivative, we present the stability and convergence analysis for a finite difference scheme that we use to solve a time-fractional diffusion equation. The analysis is based on a maximum principle for such equations.     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.     

The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations

In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.