Existence of solutions for a singular system of nonlinear fractional differential equations

In this paper, we establish the existence of positive solutions for a singular system of nonlinear fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Green’s function and its corresponding properties, we transform the derivative systems into equivalent integral systems. The existence is based on a nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

分数阶非线性方程精确解的新方法

将分数阶复变换方法和[(G/G)]方法相结合得到了一种辅助方程方法,用来求解分数阶非线性微分方程。利用该方法并借助于软件Mathematica的符号计算功能求解了分数阶Calogero KDV方程,得到了该方程新的精确解。     

Multiple solutions for fractional differential equations with nonlinear boundary conditions

In this paper, we study certain fractional differential equations with nonlinear boundary conditions. By means of the Amann theorem and the method of upper and lower solutions, some new results on the multiple solutions are obtained.     

On the global existence of solutions to a class of fractional differential equations

We present two global existence results for an initial value problem associated to a large class of fractional differential equations. Our approach differs substantially from the techniques employed in the recent literature. By introducing an easily verifiable hypothesis, we allow for immediate applications of a general comparison type result from [V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA 69 (2008), 2677–2682].     

Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions

In the this paper, we establish sufficient conditions for the existence and nonexistence of positive solutions to a general class of integral boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann-Liouville sense. Our analysis rely on Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type. As applications, some examples are also provided to illustrate our main results.     

Nonnegative solutions of fractional functional differential equations

In this paper, we discuss the existence and monotone iterative method of nonnegative solutions for fractional functional differential equations. The main conclusion is that the nonnegative solutions can be derived from the monotone iterative method, which starts off with a nonnegative upper solution or the zero function under different conditions. Our approach of obtaining nonnegative solutions is feasible for computational purposes.     

Using the iterative reproducing kernel method for solving a class of nonlinear fractional differential equations

In previous works, we have devoted to using the reproducing kernel methods solving integer order differential equations, based on the review of previous works, in this paper, we mainly present a method for solving a class of higher order fractional differential equations with general boundary value problems by using Taylor formula into reproducing kernel space. Its analytical solution is represented in the form of series. The analytical solution and approximate solution obtained by this method is given and it is uniformly converge to the exact solution and its corresponding derivatives. The numerical examples are studied to demonstrate the accuracy of the present method.     

Explicit solutions of fractional differential equations with uncertainty

We give the explicit solutions of uncertain fractional differential equations (UFDEs) under Riemann–Liouville H-differentiability using Mittag-Leffler functions. To this end, Riemann–Liouville H-differentiability is introduced which is a direct generalization of the concept of Riemann–Liouville differentiability in deterministic sense to the fuzzy context. Moreover, equivalent integral forms of UFDEs are determined which are applied to derive the explicit solutions. Finally, some illustrative examples are given.     

求一类非线性偏微分方程解析解的简洁构造算法

通过引入一个变换,利用齐次平衡原理和选准一个待定函数来构造求解一类非线性偏微分方程解析解的算法.作为实例,我们将该算法应用到了mKdV方程,KdV-Burgers方程和KdV-Burgers-Kuramoto方程.借助符号计算软件Mathematica获得了这些方程的解析解.不难看出,该方法不仅简洁,而且有望进一步扩展.     

Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions

In this paper, we consider the existence of solutions for a class of nonlinear impulsive problems with periodic boundary conditions. By using critical point theory, we obtain some existence theorems of infinitely many solutions for the nonlinear impulsive problem when the impulsive functions are superlinear. We extend and improve some recent results.     

A fractional variational iteration method for solving fractional nonlinear differential equations

Recently, fractional differential equations have been investigated by employing the famous variational iteration method. However, all the previous works avoid the fractional order term and only handle it as a restricted variation. A fractional variational iteration method was first proposed in [G.C. Wu, E.W.M. Lee, Fractional variational iteration method and its application, Phys. Lett. A 374 (2010) 2506–2509] and gave a generalized Lagrange multiplier. In this paper, two fractional differential equations are approximately solved with the fractional variational iteration method.     

An effective method for solving nonlinear fractional differential equations

A new technique based on beta functions is applied to compute the exact formula for the Riemann–Liouville fractional integral of the fractional-order generalized Chelyshkov wavelets. An approximation method based on the wavelets is proposed to effectively solve nonlinear fractional differential equations. Illustrative examples show that the proposed method gives solutions with less errors in comparison with the previous methods.

     

Periodic solutions for a class of nonlinear hyperbolic evolution equations

The existence of periodic solutions is constructively substantiated for a class of nonlinear hyperbolic evolution equations. A priori estimates are obtained. Examples are given to illustrate the results.     

Solution of nonlinear fractional differential equations using an efficient approach

We present an efficient approach for solving nonlinear fractional differential equations. The convergence analysis of the approach is studied. To demonstrate the working of the presented approach, we consider three special cases of nonlinear fractional differential equations. The results of theses examples and comparison with different methods provide confirmation for the validity of the proposed approach.     

Existence and multiplicity of positive periodic solutions for a class of higher-dimension functional differential equations with impulses

This paper deals with the existence of multiple periodic solutions for n-dimensional functional differential equations with impulses. By employing the Krasnoselskii fixed point theorem, we obtain some easily verifiable sufficient criteria which extend previous results.