The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total memory and work, where NT and NS represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel t –1–α on the interval [Δt, T] with a uniform absolute error ε. We give the theoretical analysis to show that the number of exponentials N exp needed is of order for T≫1 or for TH1 for fixed accuracy ε. The resulting algorithm requires only storage and work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

In this paper, we first apply cosine radial basis function neural networks to solve the fractional differential equations with initial value problems or boundary value problems. In the examples, we successfully obtained the numerical solutions for the fractional Riccati equations and fractional Langevin equations. The computer graphics and numerical solutions show that this method is very effective.

The second order weighted and shifted Grünwald difference (WSGD) operators are developed in [Tian, Zhou and Deng, Math. Comput., 84 (2015), pp. 1703–1727] to solve space fractional partial differential equations. Along this direction, we further design a new family of second order WSGD operators; by properly choosing the weighted parameters, they can be effectively used to discretize space (Riemann-Liouville) fractional derivatives. Based on the new second order WSGD operators, we derive a family of difference schemes for the space fractional advection diffusion equation. By von Neumann stability analysis, it is proved that the obtained schemes are unconditionally stable. Finally, extensive numerical experiments are performed to demonstrate the performance of the schemes and confirm the convergence orders.

It is well known that the standard Lipschitz space in Euclidean space, with exponent α ∈ (0, 1), can be characterized by means of the inequality , where is the Poisson integral of the function f. There are two cases: one can either assume that the functions in the space are bounded, or one can not make such an assumption. In the setting of the Ornstein–Uhlenbeck semigroup in ℝn , Gatto and Urbina defined a Lipschitz space by means of a similar inequality for the Ornstein–Uhlenbeck Poisson integral, considering bounded functions. In a preceding paper, the authors characterized that space by means of a Lipschitz-type continuity condition. The present paper defines a Lipschitz space in the same setting in a similar way, but now without the boundedness condition. Our main result says that this space can also be described by a continuity condition. The functions in this space turn out to have at most logarithmic growth at infinity.

Let $E$ be a finite-dimensional normed space and $\unicode[STIX]{x1D6FA}$ a non-empty convex open set in $E$ . We show that the Lipschitz-free space of $\unicode[STIX]{x1D6FA}$ is canonically isometric to the quotient of $L^{1}(\unicode[STIX]{x1D6FA},E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$ .

We introduce the open degree of a compact space, and we show that for every natural number $n$, the separable Rosenthal compact spaces of degree $n$ have a finite basis.

In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, in both the centred and uncentred versions. In particular, we solve a question originally posed by Bober et al. [‘On a discrete version of Tanaka’s theorem for maximal functions’, Proc. Amer. Math. Soc.140 (2012), 1669–1680].

In this paper the fractional Euler-Lagrange equation is considered. The fractional equation with the left and right Caputo derivatives of order α ∈ (0,1] is transformed into its corresponding integral form. Next, we present a numerical solution of the integral form of the considered equation. On the basis of numerical results, the convergence of the proposed method is determined. Examples of numerical solutions of this equation are shown in the final part of this paper.

In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (nonfractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.

An efficient high order numerical method is presented to solve the mobile-immobile advection-dispersion model with the Coimbra time variable-order fractional derivative, which is used to simulate solute transport in watershed catchments and rivers. On establishing an efficient recursive algorithm based on the properties of Jacobi polynomials to approximate the Coimbra variable-order fractional derivative operator, we use spectral collocation method with both temporal and spatial discretisation to solve the time variable-order fractional mobile-immobile advection-dispersion model. Numerical examples then illustrate the effectiveness and high order convergence of our approach.

We identify a close relation between stable distributions and the limiting homomorphisms central to the theory of regular variation. In so doing some simplifications are achieved in the direct analysis of these laws in Pitman and Pitman (2016); stable distributions are themselves linked to homomorphy.

In this paper, we show that the cost of an optimal train journey on level track over a fixed distance is a strictly decreasing and strictly convex function of journey time. The precise structure of the cost–time curves for individual trains is an important consideration in the design of energy-efficient timetables on complex rail networks. The development of optimal timetables for busy metropolitan lines can be considered as a two-stage process. The first stage seeks to find optimal transit times for each individual journey segment subject to the usual trip-time, dwell-time, headway and connection constraints in such a way that the total energy consumption over all proposed journeys is minimized. The second stage adjusts the arrival and departure times for each journey while preserving the individual segment times and the overall journey times, in order to best synchronize the collective movement of trains through the network and thereby maximize recovery of energy from regenerative braking. The precise nature of the cost–time curve is a critical component in the first stage of the optimization.

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann–Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy–Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on ℝd . Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form xPl and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.

We give a new necessary and sufficient condition for an iterated function system to satisfy the deterministic chaos game. As a consequence, we give a new example of an iterated function system which satisfies the deterministic chaos game.

A set is shy or Haar null (in the sense of Christensen) if there exists a Borel set and a Borel probability measure μ on C[0, 1] such that and for all f ∈ C[0, 1]. The complement of a shy set is called a prevalent set. We say that a set is Haar ambivalent if it is neither shy nor prevalent.

The main goal of the paper is to answer the following question: what can we say about the topological properties of the level sets of the prevalent/non-shy many f ∈ C[0, 1]?

The classical Bruckner–Garg theorem characterizes the level sets of the generic (in the sense of Baire category) f ∈ C[0, 1] from the topological point of view. We prove that the functions f ∈ C[0, 1] for which the same characterization holds form a Haar ambivalent set.

In an earlier paper, Balka et al. proved that the functions f ∈ C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set in C[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions f ∈ C[0, 1] for which positively many level sets with respect to the occupation measure λ ◦ f –1 are not perfect form a Haar ambivalent set in C[0, 1].

We show that for the prevalent f ∈ C[0, 1] for the generic y ∈ f([0, 1]) the level set f –1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functions f ∈ C[0, 1] for which there exists a perfect set Pf ⊂ [0, 1] such that fʹ(x) = ∞ for all x ∈ Pf is Haar ambivalent.

High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

The bivariate series defines a partial theta function. For fixed q (∣q∣ < 1), θ(q, ·) is an entire function. For q ∈ (–1, 0) the function θ(q, ·) has infinitely many negative and infinitely many positive real zeros. There exists a sequence of values of q tending to –1+ such that has a double real zero (the rest of its real zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the real negative zeros of (respectively, for k sufficiently large is the second from the left of the real negative zeros of ). For k sufficiently large one has . One has and .

A finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations. Using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed. An approach based on the classical Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Consistency, stability and convergence of the method are established. Numerical experiments illustrating the effectiveness of the theoretical analysis are provided.

In this paper, the (G'/G)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of 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. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.