Higher order numerical methods for solving fractional differential equations

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0<α<1. The order of convergence of the numerical method is O(h ^3?α ). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α>0. The order of convergence of the numerical method is O(h ³) for α≥1 and O(h ^1+2α ) for 0<α≤1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.     

Efficient three-level scheme for parabolic equations in cylindrical coordinates in a region with a small hole

An efficient three-level scheme for parabolic equations in cylindrical coordinates is constructed in a region with a small hole. No axial symmetry is assumed. The convergence rate of the scheme is estimated under minimum requirements on the initial data. The estimates are uniform with respect to a small parameter—the inner diameter of the region. The order of convergence is τ + h ², τ^1/2 + h, τ + h, depending on the smoothness of the data.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

Compact finite difference scheme for the solution of time fractional advection-dispersion equation

In this paper, a compact finite difference method is proposed for the solution of time fractional advection-dispersion equation which appears extensively in fluid dynamics. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of order O(τ ^2???α ), 0?<?α?<?1, and spatial derivatives are replaced with a fourth order compact finite difference scheme. We will prove the unconditional stability and solvability of proposed scheme. Also we show that the method is convergence with convergence order O(τ ^2???α ?+?h ⁴). Numerical examples confirm the theoretical results and high accuracy of proposed scheme.     

A boundary parametric approximation to the linearized scalar potential magnetostatic field problem

We consider the linearized scalar potential formulation of the magnetostatic field problem in this paper. Our approach involves a reformulation of the continuous problem as a parametric boundary problem. By the introduction of a spherical interface and the use of spherical harmonics, the infinite boundary conditions can also be satisfied in the parametric framework. That is the field in the exterior of a sphere is expanded in a ‘harmonic series’ of eigenfunctions for the exterior harmonic problem. The approach is essentially a finite element method coupled with a spectral method via a boundary parametric procedure. The reformulated problem is discretized by finite element techniques which leads to a discrete parametric problem which can be solved by well conditioned iteration involving only the solution of decoupled Neumann type elliptic finite element systems and L² projection onto subspaces of spherical harmonics. Error and stability estimates given show exponential convergence in the degree of the spherical harmonics and optimal order convergence with respect to the finite element approximation for the resulting fields in L².     

Regularity and integrability of spherical means

The regularity and integrability of spherical means of functions inL ^p (? ⁿ ),n≥2, are studied. An application is given to convergence of Fourier integrals.     

On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains

The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian (−Δ)^α/2, 0<α<2, in a bounded C^1,1 domain D in Rⁿ. In particular, we introduce a new Kato class K_α(D) and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian.     

Stability and Super Convergence Analysis of ADI-FDTD for the 2D Maxwell Equations in a Lossy Medium

Several new energy identities of the two dimensional(2D) Maxwell equations in a lossy medium in the case of the perfectly electric conducting boundary conditions are proposed and proved. These identities show a new kind of energy conservation in the Maxwell system and provide a new energy method to analyze the alternating direction implicit finite difference time domain method for the 2D Maxwell equations (2D-ADI-FDTD). It is proved that 2D-ADI-FDTD is approximately energy conserved, unconditionally stable and second order convergent in the discrete L² and H¹ norms, which implies that 2D-ADI-FDTD is super convergent. By this super convergence, it is simply proved that the error of the divergence of the solution of 2D-ADI-FDTD is second order accurate. It is also proved that the difference scheme of 2D-ADI-FDTD with respect to time t is second order convergent in the discrete H¹ norm. Experimental results to confirm the theoretical analysis on stability, convergence and energy conservation are presented.     

Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces

It is known from the analysis by Sloan and Woźniakowski that under appropriate conditions on the weights, the optimal rate of convergence for multivariate integration in weighted Korobov spaces is O(n^−α/2+δ) where α>1 is some parameter of the spaces, δ is an arbitrary positive number, and the implied constant in the big O notation depends only on δ, and is independent on the number of variables. Similarly, the optimal rate for weighted Sobolev spaces is O(n^−1+δ). However, their work did not show how to construct rules achieving these rates of convergence. The existing theory of the component-by-component constructions developed by Sloan, Kuo and Joe for the Sobolev case yields the rules achieving O(n^−1/2) error bounds. Here we present theorems which show that those lattice rules constructed by the component-by-component algorithms in fact achieve the optimal rate of convergence under appropriate conditions on the weights.     

On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers

The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For R/r exceeding about 30, where r and R are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large R/r) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for R/r of up to 5 × 10³. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from R/r ～ 30 and is considerably more efficient for large values of R/r. It is also established that the convergence rates of both methods depend little on the stretching coefficient η of circularly rectangular mesh cells in a range of η that is well sufficient for effective use of the multigrid method for arbitrary values of R/r smaller than ～ 5 × 10³.     

On the divergence of multiple Fourier-Haar series

It is proved that for any dimension n ?? 2, L(ln⁺ L) ^n?1 is the widest integral class in which the almost everywhere convergence of spherical partial sums of multiple Fourier-Haar series is provided. Moreover,it is shown that the divergence effects of rectangular and spherical general terms of multiple Fourier-Haar series can be achieved simultaneously on a set of full measure by an appropriate rearrangement of values of arbitrary summable function f not belonging to L(ln⁺ L) ^n?1.     

On lacunary statistical convergence of order α

In this article, we introduce the concept of lacunary statistical convergence of order α of real number sequences and give some inclusion relations between the sets of lacunary statistical convergence of order α and strong N_θ^α(p)-summability. Furthermore, some relations between the spaces N_θ^αS_θ^α are examined.     

On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients

Implicit difference schemes of O(k⁴ + k²h² + h⁴), where k0, h 0 are grid sizes in time and space coordinates respectively, are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and Dirichlet boundary conditions. The proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry. The numerical examples are given to illustrate the fourth order convergence of the methods.     

Projection theorems for hitting probabilities and a theorem of Littlewood

Littlewood (Proc. London Math. Soc. (2), 28 1928, 383–394) showed that a positive superharmonic function u on the unit disc has radial limits a.e. Using techniques due to Doob this result is extended to all rank one symmetric spaces. In addition simplifications are obtained of Doob's (Ann. Inst. Fourier (Grenoble), 15 1965, 113–135) proof of normal convergence a.e. of a positive superharmonic function on a half space. The symmetric space analogue of this half space result is also obtained. The methods used are shown to fail for the potential theory on $\text{R}$ⁿ associated with Δu = αu (α > 4 0). It is an open question as to whether Littlewood's theorem holds in this context.     

Isomorphisms,definable relations,and Scott families for integral domains and commutative semigroups

In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δ _α ⁰ -categorical integral domain (commutative semigroup) which is not relatively Δ _α ⁰ -categorical (i.e., no formally Σ _α ⁰ Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σ _α ⁰ -relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σ _α ⁰ -relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δ _α ⁰ -dimension is equal to n. (4) For every computable successor ordinal α, there exists an integral domain (commutative semigroup) with presentations only in the degrees of sets X such that Δ _α ⁰ (X) is not Δ _α ⁰ . In particular, for every finite n, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not n-low.     

A second order finite difference-spectral method for space fractional diffusion equations

A high order finite difference-spectral method is derived for solving space fractional diffusion equations,by combining the second order finite difference method in time and the spectral Galerkin method in space.The stability and error estimates of the temporal semidiscrete scheme are rigorously discussed,and the convergence order of the proposed method is proved to be O(τ2+Nα-m)in L2-norm,whereτ,N,αand m are the time step size,polynomial degree,fractional derivative index and regularity of the exact solution,respectively.Numerical experiments are carried out to demonstrate the theoretical analysis.     

A highly accurate homogeneous scheme for solving the laplace equation on a rectangular parallelepiped with boundary values in C k, 1

In this paper, a homogeneous scheme with 26-point averaging operator for the solution of Dirichlet problem for Laplace??s equation on rectangular parallelepiped is analyzed. It is proved that the order of convergence is O(h ⁴), where h is the mesh step, when the boundary functions are from C ^{3, 1}, and the compatibility condition, which results from the Laplace equation, for the second order derivatives on the adjacent faces is satisfied on the edges. Futhermore, it is proved that the order of convergence is O(h ⁶(|lnh| + 1)), when the boundary functions are from C ^{5, 1}, and the compatibility condition for the fourth order derivatives is satisfied. These estimations can be used to justify different versions of domain decomposition methods.     

Lopsided PMHSS iteration method for a class of complex symmetric linear systems

Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, we introduce a lopsided PMHSS (LPMHSS) iteration method for solving a broad class of complex symmetric linear systems. The convergence properties of the LPMHSS method are analyzed, which show that, under a loose restriction on parameter α, the iterative sequence produced by LPMHSS method is convergent to the unique solution of the linear system for any initial guess. Furthermore, we derive an upper bound for the spectral radius of the LPMHSS iteration matrix, and the quasi-optimal parameter α ^? which minimizes the above upper bound is also obtained. Both theoretical and numerical results indicate that the LPMHSS method outperforms the PMHSS method when the real part of the coefficient matrix is dominant.     

Numerical evaluation of various levels of singular integrals, arising in BEM and its application in hydrofoil analysis

The sound implementation of the boundary element method (BEM) is highly dependent on an accurate numerical integration of singular integrals. In this paper, a set of various types of singular domain integrals with three-dimensional boundary element discretization is evaluated based on a transformation integration technique. In the BEM, the integration domain (body surface) needs to be discretized into small elements. For each element, the integral I(x^p, x) is calculated on the domain dS. Several types of integrals IB_α and IC_α are numerically and analytically computed and compared with the relative error. The method is extended to evaluate singular integrals which arise in the solution of the three-dimensional Laplace’s equation. An example of the elliptic hydrofoil is performed to study the physical accuracy. The results obtained using both numerical and analytical methods are shown in good agreement with the experimental data.