Finite element method for space-time fractional diffusion equation

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order β (1 < β ≤ 2), and the first order time derivative by a Caputo fractional derivative of order γ (0 < γ ≤ 1). For the 0 < γ < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(τ^2?γ + h²) and O(τ² + h²), respectively, in which τ is the time step size and h is the space step size. And for the case γ = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(τ² + h²) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.     

High accuracy analysis of the lowest order <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-Galerkin mixed finite element method for nonlinear sine-Gordon equations

The lowest order H¹-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart-Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h²)/O(h² + τ²) in H¹-norm and H(div;Ω)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively, which improve the results in the previous literature.     

Unconditional and optimal <Emphasis Type="Bold">H</Emphasis><Superscript>2</Superscript>-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions

The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrödinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H ²-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h ² + τ ²) with mesh-size h and time step τ in the discrete H ²-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.     

Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations

In this paper, a fully discrete local discontinuous Galerkin method for a class of multi-term time fractional diffusion equations is proposed and analyzed. Using local discontinuous Galerkin method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established. By choosing the numerical flux carefully, we prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^2?α ), where k, h, and Δt are the degree of piecewise polynomial, the space, and time step sizes, respectively. Numerical examples are carried out to illustrate the effectiveness of the numerical scheme.     

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

Non-polynomial spline method for the solution of two-dimensional linear wave equations with a nonlinear source term

In this paper, two classes of methods are developed for the solution of two space dimensional wave equations with a nonlinear source term. We have used non-polynomial cubic spline function approximations in both space directions. The methods involve some parameters, by suitable choices of the parameters, a new high accuracy three time level scheme of order O(h ⁴ + k ⁴ + τ ² + τ ² h ² + τ ² k ²) has been obtained. Stability analysis of the methods have been carried out. The results of some test problems are included to demonstrate the practical usefulness of the proposed methods. The numerical results for the solution of two dimensional sine-Gordon equation are compared with those already available in literature.     

Locally One-Dimensional Difference Scheme for the Third Boundary Value Problem for a Parabolic Equation of the General Form with a Nonlocal Source

We consider a locally one-dimensional scheme for an equation of parabolic type of the general form in a p-dimensional parallelepiped, obtain an a priori estimate for its solution, and prove that the solutions of this scheme converge to a solution of the equation at the rate O(|h|² + τ), where |h|² = h ₁ ² + · · · + h _p ² and p_α, α = 1,..., p, and τ are the steps in the space and time variables. We do not assume that the operator in the leading part of the equation is sign definite.     

Analysis of a new finite difference/local discontinuous Galerkin method for the fractional Cattaneo equation

In this paper, we first present a new finite difference scheme to approximate the time fractional derivatives, which is defined in the sense of Caputo, and give a semidiscrete scheme in time with the truncation error O((Δt)^3?α ), where Δt is the time step size. Then a fully discrete scheme based on the semidiscrete scheme for the fractional Cattaneo equation in which the space direction is approximated by a local discontinuous Galerkin method is presented and analyzed. We prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^3?α ), where k is the degree of piecewise polynomial. Numerical examples are also given to confirm the theoretical analysis.     

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation(NLDE) in the nonrelativistic limit regime, involving a small dimensionless parameter 0 ε≤ 1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O(ε~2) and O(1) in time and space,respectively. We begin with the conservative Crank-Nicolson finite difference(CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ as well as the small parameter 0 ε≤ 1. Based on the error bound, in order to obtain ‘correct' numerical solutions in the nonrelativistic limit regime, i.e., 0 ε■ 1, the CNFD method requests the ε-scalability: τ = O(ε~3) and h= O(ε~(1/2)). Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and timesplitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O(ε~2) and h = O(1) when 0 ε■1. Extensive numerical results are reported to confirm our error estimates.     

Pointwise estimates of polynomials orthogonal on a circle with respect to a weight not belonging to the spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript> (<Emphasis Type="Italic">r</Emphasis> &gt; 1)

Two-sided pointwise estimates are established for polynomials that are orthogonal on the circle |z| = 1 with respect to the weight ?(τ): = h(τ)|sin(τ/2)|^?1 g(|sin(τ/2)|) (τ ∈ ?), where g(t) is a concave modulus of continuity slowly changing at zero such that t ^?1 g(t) ∈ L ¹[0, 1] and h(τ) is a positive function from the class C _2π with a modulus of continuity satisfying the integral Dini condition. The obtained estimates are applied to find the order of the distance from the point t = 1 to the greatest zero of a polynomial orthogonal on the segment [?1, 1].     

One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions

We consider integrals of the form

$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$

, where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? ⁿ , and θ ∈ ? ^k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.

     

Numerical approximation of solution for the coupled nonlinear Schrödinger equations

In this article, a compact finite difference scheme for the coupled nonlinear Schrödinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ² + h⁴) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.     

The Relaxed Edge-Coloring Game and <Emphasis Type="Italic">k</Emphasis>-Degenerate Graphs

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with Δ(F)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ?j and d≥2j+2 for 0≤j≤Δ?1. This both improves and generalizes the result for trees in Dunn, C. (Discret. Math. 307, 1767–1775, 2007). More broadly, we generalize the main result in Dunn, C. (Discret. Math. 307, 1767–1775, 2007) by showing that if G is k-degenerate with Δ(G)=Δ and j∈[Δ+k?1], then there exists a function h(k,j) such that Alice has a winning strategy for this game with r=Δ+k?j and d≥h(k,j).     

On using cubic spline for the solution of problems in calculus of variations

Two different approaches based on cubic B-spline are developed to approximate the solution of problems in calculus of variations. Both direct and indirect methods will be described. It is known that, when using cubic spline for interpolating a function g∈C⁴[a,b] on a uniform partition with the step size h, the optimal order of convergence derived is O(h⁴). In Zarebnia and Birjandi (J. Appl. Math. 1–10, 2012) a non-optimal O(h²) method based on cubic B-spline has been used to solve the problems in calculus of variations. In this paper at first we will obtain an optimal O(h⁴) indirect method using cubic B-spline to approximate the solution. The convergence analysis will be discussed in details. Also a locally superconvergent O(h⁶) indirect approximation will be describe. Finally the direct method based on cubic spline will be developed. Some test problems are given to demonstrate the efficiency and applicability of the numerical methods.     

Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of O(h) order in energy norm and of O(h²) order in L² norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O(h²) order in energy norm, and the convergence rate in L² norm is still of O(h²) order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.     

A new infeasible-interior-point algorithm for linear programming over symmetric cones

In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N(τ₁, τ₂, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r^1.5logε^?1) for the Nesterov-Todd (NT) direction, and O(r²logε^?1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and ε > 0 is the required precision. If starting with a feasible point (x⁰, y⁰, s⁰) in N(τ₁, τ₂, η), the complexity bound is \(O\left( {\sqrt r \log {\varepsilon ^{ - 1}}} \right)\) for the NT direction, and O(rlogε^?1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.     

Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n ? 2)l = ω ₁. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u ₀ ≥ 0.     

On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay⁺(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay⁺(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z₃ and Zn₂ ⁿ, n ≥ 1, where Z_k is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.     

Oscillation criteria for quasi-linear functional dynamic equations on time scales

This paper is concerned with oscillation of the second-order quasilinear functional dynamic equation

$$(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$$

on a time scale \(\mathbb{T}\) where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on \(\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}\) and \(\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty \). We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.

     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.