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.     

Recursive polynomial interpolation algorithm (RPIA)

Two difference schemes are derived for numerically solving the one-dimensional time distributed-order fractional wave equations. It is proved that the schemes are unconditionally stable and convergent in the \(L^{\infty }\) norm with the convergence orders O(τ ² + h ²+Δγ ²) and O(τ ² + h ⁴+Δγ ⁴), respectively, where τ,h, and Δγ are the step sizes in time, space, and distributed order. A numerical example is implemented to confirm the theoretical results.     

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.     

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.     

Cardinal interpolation with general multiquadrics: convergence rates

This article pertains to interpolation of Sobolev functions at shrinking lattices \(h\mathbb {Z}^{d}\) from L _p shift-invariant spaces associated with cardinal functions related to general multiquadrics, ? _{α, c} (x) := (|x|² + c ²) ^α . The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, L _p error estimates in terms of the dilation h are considered for the associated cardinal interpolation scheme. This analysis expands the range of α values which were previously known to give such convergence rates (i.e. O(h ^k ) for functions with derivatives of order up to k in L _p , \(1<p<\infty \)). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.     

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.     

Grid analogs of two functions harmonic on the plane with a cut

For the Laplace equation in an unbounded domain (in the first quadrant, upper half-plane, plane with a cut), the Dirichlet and Neumann problems whose solutions are the imaginary and real parts of the complex function z ²lnz, respectively, are considered. Both problems are approximated on a square grid using the classic five-point difference scheme. The grid Fourier transform is applied to represent the solutions to the aforementioned grid problems in an integral form and obtain their asymptotic decompositions. It follows from the results that the accuracy of these grid solutions in the L _∞ ^h norm is O(h ²), where h is the grid step.     

Estimates of reachable sets of control systems with nonlinearity and parametric perturbations

In the Banach space L₁(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L₂(M, τ) is introduced, and its main properties are established. A convergence criterion in L₂(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L₁(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖₁ ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L₁(M, τ) is complemented. The convergence in L₂(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained.     

Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

Let O ? R ^d be a bounded domain of class C ^1,1. Let 0 < ε - 1. In L ₂(O;C ⁿ ) we consider a positive definite strongly elliptic second-order operator B _D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B _D,ε ? ζQ ₀(·/ε))^?1 as ε → 0. Here the matrix-valued function Q ₀ is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L ₂(O;C ⁿ )-operator norm and in the norm of operators acting from L ₂(O;C ⁿ ) to the Sobolev space H ¹(O;C ⁿ ) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q ₀(x/ε)? _t v _ε (x, t) = ?(B _D,ε v _ε )(x, t).     

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.     

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.     

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].     

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).     

Nikol’skii inequality between the uniform norm and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) ^α (1 + x) ^β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L _q ^(α+1,,β) with the Jacobi weight ? ^(α+1,β)(x) = (1?x) ^α+1(1+x) ^β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L _q ^(α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained.     

Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes that converge uniformly with respect to the perturbation parameter ?, ? ∈ (0, 1]. The approach is based on the decomposition of a discrete solution into regular and singular components, which are solutions of discrete subproblems on uniform grids. Using the asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges ?-uniformly in the maximum norm at the rate O (N ^?2 ln² N), where N + 1 is the number of nodes in the grid used; for fixed values of the parameter ?, the scheme converges at the rate O(N ^?2). Using the Richardson technique, an improved scheme of the solution decomposition method is constructed, which converges ?-uniformly in the maximum norm at the rate O(N ^?4 ln⁴ N).     

Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems

We consider a coupled system of first-order singularly perturbed quasilinear differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. The quasilinear system is discretized by using first and second order accurate finite difference schemes for which we derive general error estimates in the discrete maximum norm. As consequences of these error estimates we establish nodal convergence of O((N ^?1 lnN) ^p ),p=1,2, on the Shishkin mesh and O(N ^?p ),p=1,2, on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical computations are included which confirm the theoretical results.     

Numerical analysis of a Stokes interface problem based on formulation using the characteristic function

Numerical analysis of a model Stokes interface problem with the homogeneous Dirichlet boundary condition is considered. The interface condition is interpreted as an additional singular force field to the Stokes equations using the characteristic function. The finite element method is applied after introducing a regularization of the singular source term. Consequently, the error is divided into the regularization and discretization parts which are studied separately. As a result, error estimates of order h^1/2 in H¹ × L² norm for the velocity and pressure, and of order h in L² norm for the velocity are derived. Those theoretical results are also verified by numerical examples.     

The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

Jacobi translation and the inequality of different metrics for algebraic polynomials on an interval

The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with Jacobi weight ?^(α,β)(x) = (1 ? x)^α(1 + x)^β α ≥ β > ?1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space L _q ^(α,β) , 1 ≤ q < ∞, \(\alpha > \beta \geqslant - \frac{1}{2}\), is attained.     

Spatial High Accuracy Analysis of FEM for Two-dimensional Multi-term Time-fractional Diffusion-wave Equations

In this paper, high-order numerical analysis of finite element method (FEM) is presented for two-dimensional multi-term time-fractional diffusion-wave equation (TFDWE). First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H¹?norm and temporal convergence in L²-norm with order \(O(h^{2}+\tau^{3-\alpha})\), where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H¹-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.