一类分数阶微分方程初值问题的奇摄动

研究了一类非线性分数阶微分方程加权初值问题的奇异摄动.在适当的条件下,首先求出了原问题的外部解,然后利用边界层函数法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.     

Existence of fractional differential equations

Consider the fractional differential equation

Dαx=f(t,x),     

Theory of fractional functional differential equations

In this paper, the basic theory for the initial value problems for fractional functional differential equations is considered, extending the corresponding theory of ordinary functional differential equations.     

Laplace transform and fractional differential equations

In this paper, we give a sufficient condition to guarantee the rationality of solving constant coefficient fractional differential equations by the Laplace transform method.     

On inference for fractional differential equations

Based on Malliavin calculus tools and approximation results, we show how to compute a maximum likelihood type estimator for a rather general differential equation driven by a fractional Brownian motion with Hurst parameter $H>1/2$ . Rates of convergence for the approximation task are provided, and numerical experiments show that our procedure leads to good results in terms of estimation.     

Boundary problems for fractional differential equations

In this paper, the existence of solutions of fractional differential equations with nonlinear boundary conditions is investigated. The monotone iterative method combined with lower and upper solutions is applied. Fractional differential inequalities are also discussed. Two examples are added to illustrate the results.     

Basic theory of fractional differential equations

In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann–Liouville differential operators is discussed employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.     

Nonsmooth analysis and fractional differential equations

In this paper we study Euler solutions, strong and weak invariance of solutions for fractional differential equations.     

Operational solution of fractional differential equations

The main goal of this paper is to solve fractional differential equations by means of an operational calculus. Our calculus is based on a modified shift operator which acts on an abstract space of formal Laurent series. We adopt Weyl’s definition of derivatives of fractional order.     

Jacobian-predictor-corrector approach for fractional differential equations

We present a novel predictor-corrector method, called Jacobian-predictor-corrector approach, for the numerical solutions of fractional ordinary differential equations, which are based on the polynomial interpolation and the Gauss-Lobatto quadrature w.r.t. the Jacobi-weight function $\omega (s)=(1-s)^{\alpha -1} (1+s)^{0}$ . This method has the computational cost O(N _E ) and the convergent order N _I , where N _E and N _I are, respectively, the total computational steps and the number of used interpolation points. The detailed error analysis is performed, and the extensive numerical experiments confirm the theoretical results and show the robustness of this method.     

A fractional characteristic method for solving fractional partial differential equations

The method of characteristics has played a very important role in mathematical physics. Previously, it has been employed to solve the initial value problem for partial differential equations of first order. In this work, we propose a new fractional characteristic method and use it to solve some fractional partial differential equations.     

Mild solutions for abstract fractional differential equations

We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a,?k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C ₀-semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behaviour of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley–Torvik equations.     

Global attractivity for nonlinear fractional differential equations

We present some results for the global attractivity of solutions for fractional differential equations involving Riemann-Liouville fractional calculus. The results are obtained by employing Krasnoselskii’s fixed point theorem. Similar results for fractional differential equations involving Caputo fractional derivative are also obtained by using the classical Schauder’s fixed point theorem. Several examples are given to illustrate our main results.     

Approximate controllability of Hilfer fractional differential equations

In this paper, the approximate controllability for a class of Hilfer fractional differential equations (FDEs) of order 1<α<2 and type 0 ≤ β ≤ 1 is considered. The existence and uniqueness of mild solutions for these equations are established by applying the Banach contraction principle. Further, we obtain a set of sufficient conditions for the approximate controllability of these equations. Finally, an example is presented to illustrate the obtained results.     

Solvability of hyperbolic fractional partial differential equations

The main purpose of this paper is to study the existence and uniqueness of solutions for the hyperbolic fractional differential equations with integral conditions. Under suitable assumptions, the results are established by using an energy integral method which is based on constructing an appropriate multiplier. Further we find the solution of the hyperbolic fractional differential equations using Adomian decomposition method. Examples are provided to illustrate the theory.     

On the fractional differential equations with uncertainty

This paper is based on the concept of fuzzy differential equations of fractional order introduced by Agarwal et al. [R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal. 72 (2010) 2859-2862]. Using this concept, we prove some results on the existence and uniqueness of solutions of fuzzy fractional differential equations.