Some Remarks on One-dimensional Functions and Their Riemann–Liouville Fractional Calculus

A one-dimensional continuous function of unbounded variation on [0,1] has been constructed.The length of its graph is infnite,while part of this function displays fractal features.The Box dimension of its Riemann–Liouville fractional integral has been calculated.     

Binding numbers for fractional ID-k-factor-critical graphs

LetG be a graph,and k≥2 be a positive integer.A graph G is fractional independentset-deletable k-factor-critical(in short,fractional ID-k-factor-critical),if G I has a fractional k-factor for every independent set I of G.The binding number bind(G)of a graph G is defined as bind(G)=min|NG(X)||X|:=X V(G),NG(X)=V(G).In this paper,it is proved that a graph G is fractional ID-k-factor-critical if n≥6k 9 and bind(G)(3k 1)(n 1)kn 2k+2.     

Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives

In this Letter, we propose to use the Cantor-type cylindrical-coordinate method in order to investigate a family of local fractional differential operators on Cantor sets. Some testing examples are given to illustrate the capability of the proposed method for the heat-conduction equation on a Cantor set and the damped wave equation in fractal strings. It is seen to be a powerful tool to convert differential equations on Cantor sets from Cantorian-coordinate systems to Cantor-type cylindrical-coordinate systems.     

Lattice Boltzmann method for the fractional sub‐diffusion equation

A lattice Boltzmann model for the fractional sub‐diffusion equation is presented. By using the Chapman–Enskog expansion and the multiscale time expansion, several higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales are obtained. Furthermore, the modified partial differential equation of the fractional sub‐diffusion equation with the second‐order truncation error is obtained. In the numerical simulations, comparisons between numerical results of the lattice Boltzmann models and exact solutions are given. The numerical results agree well with the classical ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Fractional Bilinear Stochastic Equations with the Drift in the First Fractional Chaos

Abstract

In the paper we compute the explicit form of the fractional chaos decomposition of the solution of a fractional stochastic bilinear equation with the drift in the fractional chaos of order one and initial condition in a finite fractional chaos. The large deviations principle is also obtained for the one-dimensional distributions of the solution of the equation perturbed by a small noise.     

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical growth

In this paper, we study the following fractional Schrödinger equation with critical or supercritical growth where 0 < s < 1, N > 2s, λ > 0, , , ( ? Δ)^s denotes the fractional Laplacian of order s and f is a continuous superlinear but subcritical function. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 by variational methods. Our main contribution is related to the fact that we are able to deal with the case .     

PDα‐type distributed learning control for nonlinear fractional‐order multiagent systems

This paper considers the consensus tacking problem for nonlinear fractional‐order multiagent systems by presenting a PD^α‐type iterative learning control update law with initial learning mechanisms. The asymptotical convergence of the proposed distributed learning algorithm is strictly proved by using the properties of fractional calculus. A sufficient condition is derived to guarantee the whole multiagent system achieving an asymptotic output consensus. An illustrative example is given to verify the theoretical results.