A new regularization method for a Cauchy problem of the time fractional diffusion equation

In this paper, we consider a Cauchy problem of the time fractional diffusion equation (TFDE). Such problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α (0 < α ≤ 1). We show that the Cauchy problem of TFDE is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates in the interior and on the boundary of solution domain are obtained respectively under different a-priori bound assumptions for the exact solution and suitable choices of regularization parameters. Finally, numerical examples are given to show that the proposed numerical method is effective.     

Quantum trajectories of an oscillator interacting with an electromagnetic field, a classical force, and a heat bath

We consider a solvable problem describing the dynamics of a quantum oscillator interacting with an electromagnetic field, a classical force, and a heat bath. We propose a general method for solving Markovian master equations, the method of quantum trajectories. We construct the stochastic evolution operator involving the stochastic analogue of the Baker-Hausdorff formula and calculate the system density matrix for an arbitrary initial state. As a physical application, we evaluate the influence of the environment at a finite temperature on the accuracy of measuring a weak classical force by the interference method. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 444–459, March, 2009.     

Piecewise-linear,discontinuous Galerkin method for a fractional diffusion equation

We use a piecewise-linear, discontinuous Galerkin method for the time discretization of a fractional diffusion equation involving a parameter in the range − 1 < α < 0. Our analysis shows that, for a time interval (0,T) and a spatial domain Ω, the error in L_￥((0,T);L₂(W))L_\infty\bigr((0,T);L_2(\Omega)\bigr) is of order k ^2 + α , where k denotes the maximum time step. Since derivatives of the solution may be singular at t = 0, our result requires the use of non-uniform time steps. In the limiting case α = 0 we recover the known O(k ²) convergence for the classical diffusion (heat) equation. We also consider a fully-discrete scheme that employs standard (continuous) piecewise-linear finite elements in space, and show that the additional error is of order h ²log(1/k). Numerical experiments indicate that our O(k ^2 + α ) error bound is pessimistic. In practice, we observe O(k ²) convergence even for α close to − 1.     

Fractional integration associated with second order divergence operators on R<Superscript>n</Superscript>

The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (−Δ)^−α/2 are extended to the generalised fractional integrals L ^–α/2 for 0 < α < n, where L=−div A∇ is a uniformly complex elliptic operator with bounded measurable coefficients in ℝⁿ.     

Weighted Integrals and Bloch Spaces of n-Harmonic Functions on the Polydisc

We study anisotropic mixed norm spaces h(p,q,α) consisting of n-harmonic functions on the unit polydisc of by means of fractional integro-differentiation including small 0 < p < 1 and multi-indices α = (α ₁,...,α _n ) with non-positive α _j  ≤ 0. As an application, two different Bloch spaces of n-harmonic functions are characterized.      

Weighted endpoint estimates for commutators of fractional integrals

Given α, 0 < α < n, and b ∈ BMO, we give sufficient conditions on weights for the commutator of the fractional integral operator, [b, I _α ], to satisfy weighted endpoint inequalities on ℝⁿ and on bounded domains. These results extend our earlier work [3], where we considered unweighted inequalities on ℝⁿ.     

On a class of spaces of analytic functions

The spacesb (p, q, λ) (0<p<q⩽∞, 0<λ⩽∞) of functions, analytic in the circle |z|< 1, are introduced, and an unimprovable estimate is obtained for the Taylor coefficients of a functionf∃ b (p, q, λ). It is shown that B(p, q, λ) is the space of fractional derivatives f(α) of order α (−∞<α<1/p−1/q) of a function f of B(s, q, λ), where s=p/(1−αp). Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 141–150, February, 1977.     

Stochastic partial differential equations for some measure-valued diffusions

Summary We consider two classes of measure-valued diffusion processes; measure-valued branching diffusions and Fleming-Viot diffusion models. When the basic space is R ¹, and the drift operator is a fractional Laplacian of order 1<α≦2, we derive stochastic partial differential equations based on a space-time white noise for these two processes. The former is the expected one by Dawson, but the latter is a new type of stochastic partial differential equation.     

Fractional Integral Operators on Anisotropic Hardy Spaces

In this paper, we introduce the fractional integral operator T of degree α of order m with respect to a dilation A for 0 < α < 1 and . First we establish the Hardy-Littlewood-Sobolev inequalities for T on anisotropic Hardy spaces associated with dilation A, which show that T is bounded from H ^p to H ^q , or from H ^p to L ^q , where 0 < p ≤ 1/(1 + α) and 1/q = 1/p − α. Then we give anisotropic Hardy spaces estimates for a class of multilinear operators formed by fractional integrals or Calderón-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of T and a BMO function. Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations

Existence of positive solutions for the nonlinear fractional differential equation D ^αu = f(x,u), 0 < α < 1 has been given (S. Zhang. J. Math. Anal. Appl. 252 (2000), 804–812) where D ^α denotes Riemann–Liouville fractional derivative. In the present work we extend this analysis for n-term non autonomous fractional differential equations. We investigate existence of positive solutions for the following initial value problem

Well-posedness of equations with fractional derivative via the method of sum

We study the well-posedness of the equations with fractional derivative Dαu(t)=Au(t)+f(t)(0 ≤t≤2π),where A is a closed operator in a Banach space X,0α1 and Dα is the fractional derivative in the sense of Weyl.Although this problem is not always well-posed in Lp(0,2π;X) or periodic continuous function spaces Cper([0,2π];X),we show by using the method of sum that it is well-posed in some subspaces of L p(0,2π;X) or C per([0,2π];X).     

Can quantum effects due to a massless conformally coupled field avoid gravitational singularities?

Using quantum corrections from massless fields conformally coupled to gravity, we study the possibility of avoiding singularities that appear in the flat Friedmann-Robertson-Walker model. We assume that the universe contains a barotropic perfect fluid with the state equation p = ωρ, where p is the pressure and ρ is the energy density. We study the dynamics of the model for all values of the parameter ω and also for all values of the conformal anomaly coefficients α and β. We show that singularities can be avoided only in the case where α > 0 and β < 0. To obtain an expanding Friedmann universe at late times with ω > −1 (only a one-parameter family of solutions, but no a general solution, has this behavior at late times), the initial conditions of the nonsingular solutions at early times must be chosen very exactly. These nonsingular solutions consist of a general solution (a two-parameter family) exiting the contracting de Sitter phase and a one-parameter family exiting the contracting Friedmann phase. On the other hand, for ω < −1 (a phantom field), the problem of avoiding singularities is more involved because if we consider an expanding Friedmann phase at early times, then in addition to fine-tuning the initial conditions, we must also fine-tune the parameters α and β to obtain a behavior without future singularities: only a oneparameter family of solutions follows a contracting Friedmann phase at late times, and only a particular solution behaves like a contracting de Sitter universe. The other solutions have future singularities.     

Subsequences of normal sequences

In this paper, we characterize a set of indices τ={τ(0)<τ(1)<…} such that forany normal sequence (α(0), α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ(1)),…) is a normal sequence of the same type. Assume that_n→∞. Then, we prove that τ has this property if and only if the 0–1 sequence (θ _τ (0), whereθ _τ (i)=1 or 0 according asi∈{τ(j);j=0, 1,…} or not, iscompletely deterministic in the sense of B. Weiss.     

Quantum three-body problems

A scheme for dealing with the quantum three-body problem is presented to separate the rotational degrees of freedom completely from the internal ones. In this method, the three-body Schrodinger equation is reduced to a system of coupled partial differential equations, depending only upon three internal variables. For arbitrary total orbital angular momentum / and the parity (− 1) ^l+λ (λ = 0 or 1), the number of the equations in this system isl = 1 −λ. By expanding the wavefunction with respect to a complete set of orthonormal basis functions, the system of equations is further reduced to a system of linear algebraic equations.     

On driftless one-dimensional SDE’s with respect to stable Levy processes

The time-dependent SDE dX _t = b(t, X _t−)dZ _t with X ₀ = x ₀ ∈ ℝ, and a symmetric α-stable process Z, 1 < α ⩽ 2, is considered. We study the existence of nonexploding solutions of the given equation through the existence of solutions of the equation in class of time change processes, where is a symmetric stable process of the same index α as Z. The approach is based on using the time change method, Krylov’s estimates for stable integrals, and properties of monotone convergence. The main existence result extends the results of Pragarauskas and Zanzotto (2000) for 1 < α < 2 and those of T. Senf (1993) for α = 2. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 517–531, October–December, 2007.     

Characterizations of K -Functionals Built from Fractional Powers of Infinitesimal Generators of Semigroups

   Abstract. On a Banach space X consider an equibounded (C_0)-semigroup of linear operators { T(t): t ≥ 0} with infinitesimal generator A . We introduce fractional powers (-A) ^α , α >0 , of A with domain D((-A) ^α )) and characterize the K -functionals with respect to (X,D((-A) ^α )) via fractional differences [I-T(t)] ^α , via appropriate truncated hypersingular integrals and via some type of fractional integral over the resolvent of A . Immediate consequences are an abstract Marchaud-type inequality for moduli of smoothness arising from (semi-) groups of operators as well as optimal and nonoptimal approximation results.     

A limit theorem for random coverings of a circle

LetN _{α, m} equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that lim_α→0 P(N_α,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutel's resultE(N_α,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Euler's constant.     

Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1. For 0 < β < ? we show that a function Φ: (0, T) → ℒ(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H-cylindrical fractional Brownian motion.     

Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation

In this paper, we consider a space-time Riesz–Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β ₁ ∈ (0,1) and β ₂ ∈ (1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.     

Nonnegative solution for singular nonlinear fractional differential equation with coefficient that changes sign

In this paper, we consider the existence of nonnegative solutions of initial value problem for singular nonlinear fractional differential equation