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.     

Banach空间中一类分数阶微分方程边值问题

研究Banach空间中一类非线性分数阶微分方程边值问题.构建此类方程的Green函数,利用非紧测度和相关的不动点定理,得到了此类方程的mild解存在的几个充分条件,所得结果改进和推广了一些已有的结论.     

Fractional boundary value problems with integral boundary conditions

In this article, we study a type of nonlinear fractional boundary value problem with integral boundary conditions. By constructing an associated Green's function, applying spectral theory and using fixed point theory on cones, we obtain criteria for the existence, multiplicity and nonexistence of positive solutions.     

非线性分数阶微分方程的奇摄动

研究了—类奇摄动非线性分数阶微分方程Cauchy问题．在适当的条件下,首先求出了原问题的外部解,然后利用伸长变量、合成展开法和幂级数展开理论构造出解的初始层项,并由此得到解的形式渐近展开式．最后利用微分不等式理论,讨论了问题解的渐近性态,得到了原问题解的一致有效的渐近估计式．     

分数阶微分方程边值问题研究简介

分数阶微积分是一个古老而又新颖的课题,近30年来,由于在包括分形现象在内的物理、工程等诸多应用学科领域应用的拓展,激发了科研人员对分数阶微积分的巨大热情。分数阶微分方程现在已应用于分数物理学、混沌与湍流、粘弹性力学与非牛顿流体力学、高分子材料的解链、自动控制理论、化学物理、随机过程和反常扩散等许多科学领域。分数阶微分方程边值问题是非线性常微分方程理论研究中一个活跃而成果丰硕的领域。本文讨论了分数阶微分方程边值问题的一些理论,介绍了作者的著作《分数阶微分方程边值问题理论及应用》的基本内容。     

Initial value problems for arbitrary order fractional differential equations with delay

By fixed point theory the nonlinear alternative of Leray–Schauder type, and the properties of absolutely continuous functions space, we study the existence and uniqueness of initial value problems for nonlinear higher fractional equations with delay, and obtain some new results involving local and global solutions.     

Pontryagin maximum principle for fractional ordinary optimal control problems

In the paper, fractional systems with Riemann–Liouville derivatives are studied. A theorem on the existence and uniqueness of a solution of a fractional ordinary Cauchy problem is given. Next, the Pontryagin maximum principle for nonlinear fractional control systems with a nonlinear integral performance index is proved. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability analysis and a numerical scheme for fractional Klein‐Gordon equations

Fractional order nonlinear Klein‐Gordon equations (KGEs) have been widely studied in the fields like; nonlinear optics, solid state physics, and quantum field theory. In this article, with help of the Sumudu decomposition method (SDM), a numerical scheme is developed for the solution of fractional order nonlinear KGEs involving the Caputo's fractional derivative. The coupled method provides us very efficient numerical scheme in terms of convergent series. The iterative scheme is applied to illustrative examples for the demonstration and applications.     

SOLVABILITY FOR FRACTIONAL-ORDER TWO-POINT BOUNDARY VALUE PROBLEM AT RESONANCE

In this paper, we are concerned with the existence of solution to a boundary value problem of nonlinear fractional differential equation at resonance. By means of the coincidence degree theory, the existence of solution is obtained.     

Positive solutions of higher-order nonlinear fractional differential systems with nonlocal boundary conditions

The paper deals with the existence and multiplicity of positive solutions for a system of higher-order nonlinear fractional differential equations with nonlocal boundary conditions. The main tool used in the proof is fixed point index theory in cone. Some limit type conditions for ensuring the existence of positive solutions are given.     

On Time-space Fractional Reaction-diffusion Equations with Nonlocal Initial Conditions

This paper investigates the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. Based on the operator semigroup theory, we transform the time-space fractional reaction-diffusion equation into an abstract evolution equation. The existence and uniqueness of mild solution to the reaction-diffusion equation are obtained by solving the abstract evolution equation. Finally, we verify the Mittag-Leffler-Ulam stabilities of the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. The results in this paper improve and extend some related conclusions to this topic.     

奇摄动分数阶微分方程的渐近解

本文研究了一类奇摄动非线性分数阶微分方程初值问题.利用伸长变量构造出解的形式展开式,并利用微分不等式理论,证明了解的一致有效的渐近式.所得的结果具有较好精度的近似解.     

分数阶微分方程的比较定理

本文给出了非线性Riemann—Liouville分数阶微分方程和Caputo分数阶微分方程与相应的非线性Volterra积分方程的等价性,并在此基础上建立了分数阶微分方程的比较定理．     

Existence of fractional differential chains and factorizations based on transformations

In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B‐transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of positive solutions of nonlinear fractional delay differential equations

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional delay differential equations. Using a nonlinear alternative of Leray-Schauder type, we show the existence of positive solutions for the equations in question.     

Nontrivial solutions for impulsive fractional differential systems through variational methods

Fractional differential equations (FDEs) as a generalization of ordinary differential equations and integration to arbitrary noninteger orders have gained importance due to their numerous applications in many fields of science and engineering. Indeed, there are a large number of phenomena, including fluid flow, diffusive transport akin to diffusion, rheology, probability, and electrical networks, that are modeled by different equations involving fractional order derivatives. This paper deals with multiplicity results of solutions for a class of impulsive fractional differential systems. The approach is based on variational methods and critical point theory. Indeed, we establish existence results for our system under some algebraic conditions on the nonlinear part with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear and the impulsive terms. Moreover by combining two algebraic conditions on the nonlinear term, which guarantee the existence of two weak solutions, applying the mountain pass theorem, we establish the existence of third weak solution for our system.     

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.     

Fractional quasi AKNS‐technique for nonlinear space–time fractional evolution equations

This paper aims to formulate the fractional quasi‐inverse scattering method. Also, we give a positive answer to the following question: can the Ablowitz‐Kaup‐Newell‐Segur (AKNS) method be applied to the space–time fractional nonlinear differential equations? Besides, we derive the Bäcklund transformations for the fractional systems under study. Also, we construct the fractional quasi‐conservation laws for the considered fractional equations from the defined fractional quasi AKNS‐like system. The nonlinear fractional differential equations to be studied are the space–time fractional versions of the Kortweg‐de Vries equation, modified Kortweg‐de Vries equation, the sine‐Gordon equation, the sinh‐Gordon equation, the Liouville equation, the cosh‐Gordon equation, the short pulse equation, and the nonlinear Schrödinger equation.     

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid‐saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time‐fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time‐fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Extremal solutions for a nonlinear impulsive differential equations with multi-orders fractional derivatives

In this paper, by employing the lower and upper solutions method, we give an existence theorem for the extremal solutions for a nonlinear impulsive differential equations with multi-orders fractional derivatives and integral boundary conditions. A new comparison result is also established.