分数阶微分方程边值问题解的存在性

应用Gteen函数将分数阶微分方程边值问题可转化为等价的积分方程．近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性．讨论非线性分数阶微分方程边值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Caratheodory条件,利用非紧性测度的性质和M6nch’s不动点定理证明解的存在性．     

Exact solutions of nonlinear time fractional partial differential equations by sub‐equation method

In this article, the sub‐equation method is presented for finding the exact solutions of a nonlinear fractional partial differential equations. For this, the fractional complex transformation method has been used to convert fractional‐order partial differential equation to ordinary differential equation. The fractional derivatives are described in Jumarie's the modified Riemann–Liouville sense. We apply to this method for the nonlinear time fractional differential equations. With the aid of symbolic computation, a variety of exact solutions for them are obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

一类分数阶微分方程边值问题三重正解的存在性

讨论了非线性分数阶微分方程的两点边值问题.其导数是Riemann-Liouville型分数阶导数,应用推广了的双锥不动点定理,证明其在L(0,1)中存在三重正解.     

Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation

In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Bäcklund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations.     

Numerical Solution for a Variable-Order Fractional Nonlinear Cable Equation via Chebyshev Cardinal Functions

In this paper, a variable-order fractional derivative nonlinear cable equation is considered. It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of class of fractional partial differential equation with variable coefficient of fractional differential equation in various continues functions of spatial and time orders. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

应用(G'/G~2)-展开法求解非线性分数阶微分方程（英文）

In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractional differential equations with Jumarie's modified Riemann-Liouville derivative. And then, time-fractional Burgers equation and space-fractional coupled Konopelchenko-Dubrovsky equations are provided to show that this method is effective in solving nonlinear fractional differential equations.     

Bäcklund transformation of fractional Riccati equation and its applications to the space–time FDEs

In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.     

联合Duffing方程和Van der Pol方程的非线性分数阶微分方程

本文研究了Adomian分解方法在非线性分数阶微分方程求解中的应用. 利用Riemann-Liouville分数阶导数和Adomian分解方法, 将Duffing方程和Van der Pol方程联合在一个分数阶方程中,并获得了此方程的解析近似解.     

Moving collocation methods for time fractional differential equations and simulation of blowup

A moving collocation method is proposed and implemented to solve time fractional differential equations. The method is derived by writing the fractional differential equation into a form of time difference equation. The method is stable and has a third-order convergence in space and first-order convergence in time for either linear or nonlinear equations. In addition, the method is used to simulate the blowup in the nonlinear equations.     

The homotopy perturbation method applied to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations

The fractional derivatives in the sense of Caputo, and the homotopy perturbation method are used to construct approximate solutions for nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equations with respect to time and space fractional derivatives. Also, we apply complex transformation to convert a time and space fractional nonlinear KPP equation to an ordinary differential equation and use the homotopy perturbation method to calculate the approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.     

Efficient analytic method for solving nonlinear fractional differential equations

In this paper, we propose a new approach of the generalized differential transform method (GDTM) for solving nonlinear fractional differential equations. In GDTM, it is a key to derive a recurrence relation of generalized differential transform (GDT) associated with the solution in the given fractional equation. However, the recurrence relations of complex nonlinear functions such as exponential, logarithmic and trigonometry functions have not been derived before in GDTM. We propose new algorithms to construct the recurrence relations of complex nonlinear functions and apply the GDTM with the proposed algorithms to solve nonlinear fractional differential equations. Several illustrative examples are demonstrated to show the effectiveness of the proposed method. It is shown that the proposed technique is robust and accurate for solving fractional differential equations.     

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.     

一致分数阶非线性微分方程的精确解

同时考虑了Kudryashov方法和Khalil一致分数阶变换,构造了求解一致分数阶非线性微分方程精确解的新方法,并将其用于求解时间-空间一致分数阶Whitham-Boroer-Kaup方程,得到了Whitham-Boroer-Kaup方程新的精确解,验证了该方法的有效性和可行性.     

An iterative method for solving nonlinear functional equations

An iterative method for solving nonlinear functional equations, viz. nonlinear Volterra integral equations, algebraic equations and systems of ordinary differential equation, nonlinear algebraic equations and fractional differential equations has been discussed.     

非线性分数阶微分方程三点边值问题解的存在性

讨论了一类非线性分数阶微分方程三点边值问题解的存在性.微分算子是Riemann-Liouville导算子并且非线性项依赖于低阶分数阶导数.通过将所考虑的问题转化为等价的Fredholm型积分方程,利用Schauder不动点定理获得该三点边值问题至少存在一个解.     

Exact solutions of some systems of fractional differential‐difference equations

In this paper, the ‐expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential‐difference equations with the modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential‐difference equation into its differential‐difference equation of integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time‐fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Lie symmetry analysis to the time fractional generalized fifth-order KdV equation

In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional generalized fifth-order KdV equation. It shows that this equation can be reduced to an equation which is related to the Erdélyi–Kober fractional derivative. Of course, this method can also be applied to other nonlinear fractional partial differential equations.     

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.