Third-order Difference Research Articles

Explicit bounds for third-order difference equations

AbstractThis paper gives explicit, applicable bounds for solutions of a wide class of third-order difference equations with nonconstant coefficients. The techniques used are readily adaptable for higher-order equations. The results extend recent work of the authors for second-order equations.

On the oscillation of certain third-order difference equations

We establish some new criteria for the oscillation of third-order difference equations of the form , where Δ is the forward difference operator defined by Δx(n) = x(n+1) - x(n).

Global asymptotic properties of third-order difference equations

The third-order nonlinear difference equations Δ ( p n Δ ( r n Δ x n ) ) + q n f ( x n + p ) = 0 , p ∈ { 0 , 1 , 2 } , ( E p ) where ( P n ), ( r n ), and ( q n ) are sequences of positive real numbers for n ε ℕ, ƒ : ℝ → ℝ is a continuous function such that ƒ (u)u > 0 for u ≠ 0, are investigated. All nonoscillatory solutions of these equations are classified according to the sign of their quasidifferences to classes N i, i = 0, 1, 2, 3, and sufficient conditions ensuring N i = , i ε {1,2,3} are given. Special attention is paid to equation ( E 1) for which the generalized zeros of solutions are studied and an energy function F is introduced. The relation between the class N 0 and solutions for which F n < 0 for n ε ℕ is established.

The EPQ Code System for Simulating the Thermal Response of Plasma-Facing Components to High-Energy Electron Impact

The generation of runaway electrons during a thermal plasma disruption is a concern for the safe and economical operation of a tokamak power system. Runaway electrons have high energy, 10 to 300 MeV, and may potentially cause extensive damage to plasma-facing components (PFCs) through large temperature increases, melting of metallic components, surface erosion, and possible burnout of coolant tubes. The EPQ code system was developed to simulate the thermal response of PFCs to a runaway electron impact. The EPQ code system consists of several parts: UNIX scripts that control the operation of an electron-photon Monte Carlo code to calculate the interaction of the runaway electrons with the plasma-facing materials; a finite difference code to calculate the thermal response, melting, and surface erosion of the materials; a code to process, scale, transform, and convert the electron Monte Carlo data to volumetric heating rates for use in the thermal code; and several minor and auxiliary codes for the manipulation and postprocessing of the data. The electron-photon Monte Carlo code used was Electron-Gamma-Shower (EGS), developed and maintained by the National Research Center of Canada. The Quick-Therm-Two-Dimensional-Nonlinear (QTTN) thermal code solves the two-dimensional cylindrical modified heat conduction equation using the Quickest third-order accurate and stable explicit finite difference method and is capable of tracking melting or surface erosion. The EPQ code system is validated using a series of analytical solutions and simulations of experiments. The verification of the QTTN thermal code with analytical solutions shows that the code with the Quickest method is better than 99.9% accurate. The benchmarking of the EPQ code system and QTTN versus experiments showed that QTTN’s erosion tracking method is accurate within 30% and that EPQ is able to predict the occurrence of melting within the proper time constraints. QTTN and EPQ are verified and validated as able to calculate the temperature distribution, phase change, and surface erosion successfully.

Positive solutions of boundary value problems for third-order functional difference equations

The existence of positive solutions are established for the third-order functional difference equation Δ 3 u( n) + a( n) f( n, u( w( n))) = 0, 0 ≤ n ≤ T, satisfying u( n) = φ( n), n 1 ≤ n ≤ 1, and u( n) = ψ( n), T + 3 ≤ n ≤ n 2, with φ(0) = φ(1) = ψ( T + 3) = 0. The results in this paper generalize and substantially improve recent work by Agarwal and Henderson on boundary value problems related to third-order difference equations.

An approach to identification of variances for radar tracking systems

An approach for the identification of the variances of the process noise W( k) and the measurement noise V( k) for radar tracking systems is presented. In the proposed algorithm, the variances are expressed as a linear combination of the covariances of the third-order difference of the measurement data. The variances can then be estimated by estimating the covariances of the third-order difference and solving the matrix equation. The two cases for the measurement noise being white and color are considered in this algorithm. Simulation results show that for both cases a steady estimation is reached after 20 samples.

Oscillation of certain third-order difference equations

Some new criteria for the oscillation of third-order difference equations of the form ▪ are presented.

Positive solutions and nonlinear eigenvalue problems for third-order difference equations

First, solutions that lie in a cone intersected with an annular type region are obtained for the third-order difference equation Δ 3 u( t) + a( t) f( u( t)) = 0, 2 ≤ t ≤ T + 2, satisfying the boundary conditions u(0) = u(1) = u( T + 3) = 0, in the cases that f is either superlinear or sublinear. The second part is devoted to determining values of λ for which there exist positive solutions of Δ 3 u( t) + λa( t) f( u( t)) = 0, 2 ≤ t ≤ T+ 2, satisfying the boundary conditions u(0) = u(1) = u( T + 3) = 0, when both lim x→0 +( f( x) x) and lim x→∞ ( f( x) x) exist as positive real numbers.

Two discretisations of the Ermakov-Pinney equation

We propose two candidates for discrete analogues to the nonlinear Ermakov-Pinney equation. The first one based on an association with a two-dimensional conformal mapping defines a second-degree difference scheme. It possesses the same features as in the continuum: a nonlinear superposition principle relating its general solution to a second-order linear difference equation and by direct linearisation a relationship with a third-order difference equation. The second form, which is new, is obtained from a slight improvement of the superposition principle. It has the advantage of leading to a first degree difference scheme and preserves all the nice properties of its linearisation.

An explicit solution of third-order difference equations

The solutions of homogeneous and nonhomogeneous third-order linear difference equations are obtained.

The piecewise-parabolic method in curvilinear coordinates

We derive interpolation formulae for a third-order finite difference method in curvilinear, orthogonal coordinate systems. These formulae serve as a supplement to Colella and Woodward's PPM scheme for problems where the coordinate origin is included in the computational domain. Numerical examples of the improved accuracy of the advection scheme near coordinate singularities are shown.

Numerical modelling of viscoelastic liquids using a finite-volume method

A new staggered-grid, finite-volume method for the numerical simulation of isothermal viscoelastic liquids is presented. The main features of this method are the use of a primitive variable formulation, the location of the velocities, pressures and stresses on different staggered grids and the use of a third-order difference scheme for the discretization of the constitutive equations. The method is applied to a sudden-expansion, viscoelastic-flow problem for a range of Weissenberg numbers. For one case the mesh has been refined to demonstrate that the method is robust and viscoelastic effects are not filtered out as the mesh size decreases. All the computations have been performed on a PC equipped with a floating-point coprocessor. Although the method has been defined for, and applied to, two-dimensional problems, the technique is readily extendable to three dimensions without excessive cost.

Linear Third-Order Difference Equations: Oscillatory and Asymptotic Behavior

Linear Third-Order Difference Equations: Oscillatory and Asymptotic Behavior

A study of upstream‐weighted high‐order differencing for approximation to flow convection

AbstractThis paper is concerned with a number of upstream‐weighted second‐ and third‐order difference schemes. Also considered are the conventional upwind and central difference schemes for comparison. It commences with a general difference equation which unifies all the given first‐, second‐ and third‐order schemes. The various schemes are evaluated through the use of the general equation. The unboundedness and accuracy of the solutions by the difference schemes are assessed via various analyses: examination of the coefficients of the difference equation, Taylor series truncation error analysis, study of the upstream connection to numerical diffusion, single‐cell analysis. Finally, the difference schemes are tested on one‐ and two‐dimensional model problems. It is shown that the high‐order schemes suffer less from the problem of numerical diffusion than the first‐order upwind difference scheme. However, unboundedness cannot be avoided in the solutions by these schemes. Among them the linear upwind difference scheme presents the best compromise between numerical diffusion and solution unboundedness.

New technique for detection of changes in QRS morphology of ECG signals.

With the use of the electrocardiogram (ECG) as a prototype signal, a new technique was devised for detecting signals embedded in noise. Averaged "normal" digitized ECG signals formed a template to which subsequent ECG QRS complexes were compared. The difference between the averaged template signals and subsequent normal beats was white noise, whereas the difference between the template and ectopic beats consisted of nonrandom signal variation. The template to new signal comparison for the zero-, first-, second-, and third-order differences utilized an approximate F test. Accurate detection of abnormal signals associated with high- and low-frequency noise is accomplished with this method, and the practical clinical utility of the method is under study.

An Implicit Compact Fourth-Order Algorithm for Solving the Shallow-Water Equations in Conservation-Law Form

Abstract An alternating-direction implicit finite-difference scheme is developed for solving the nonlinear shallow-water equations in conservation-law form. The algorithm is second-order time accurate, while fourth-order compact differencing is implemented in a spatially factored form. The application of the higher order compact Pade differencing scheme requires only the solution of either block-tridiagonal or cyclic block-tridiagonal coefficient matrices, and thus permits the use of economical block-tridiagonal algorithms. The integral invariants of the shallow-water equations, i.e., mass, total energy and enstrophy, are well conserved during the numerical integration, ensuring that a realistic nonlinear structure is obtained. Largely in an experimental way, two methods are investigated for determining stable approximations for the extraneous boundary conditions required by the fourth-order method. In both methods, third-order uncentered differences at the boundaries are utilized, and both preserve the o...

The Possible Existence of a Constant Third-Order Difference among the Nuclear Magic Numbers

The Possible Existence of a Constant Third-Order Difference among the Nuclear Magic Numbers

The Possible Existence of a Constant Third-Order Difference among the Nuclear Magic Numbers

The Possible Existence of a Constant Third-Order Difference among the Nuclear Magic Numbers