Second-order Discrete Equations Research Articles

Elementary exact calculations of degree growth and entropy for discrete equations.

Second-order discrete equations are studied over the field of rational functions , where z is a variable not appearing in the equation. The exact degree of each iterate as a function of z can be calculated easily using the standard calculations that arise in singularity confinement analysis, even when the singularities are not confined. This produces elementary yet rigorous entropy calculations.

On the dependence on parameters for second order discrete boundary value problems with the p(k)-Laplacian

In this paper we study the existence and the nonexistence of solutions for the boundary value problems of a class of nonlinear second-order discrete equations depending on a parameter. Variational (the mountain pass technique) and non-variational methods are applied.

The method of upper and lower solutions for discrete BVP on infinite intervals

In this paper, we study the existence of multiple solutions of boundary value problems for second-order discrete equations , , , bounded on . The proofs are based on the upper and lower solutions, degree theory and sequential argument.

Finite element formulation of the first- and second-order discrete ordinates equations for radiative heat transfer calculation in three-dimensional participating media

Two finite element methods (FEMs), FEDOM1 and FEDOM2 (standing for the first and the second finite element discrete ordinates methods, respectively), are formulated and numerically tested. The reference second-order discrete equation is modified in its scattering terms and is applied to the problems of absorbing/emitting and anisotropically scattering media by using the FEM. Numerical features of the developed FEMs are compared with one of the discrete ordinates interpolation method (DOIM), which uses a finite difference scheme. Prediction results of radiative heat transfer by these two FEMs are compared with reference solutions and verified in three-dimensional enclosures containing participating media. The results of FEDOM1 and FEDOM2 agree well with exact solutions for the problem of absorbing/emitting medium with various range of optical thickness. Generally, the two FEMs show more accurate results than DOIM. And FEDOM1 shows more accurate results than FEDOM2 in most of the test problems. Both of the developed FEMs show reasonable results compared with published Monte Carlo solutions for the tested absorbing/emitting and anisotropically scattering media. Although the FEDOM2 is not as accurate as the FEDOM1, it shows its own advantages that it reduces CPU time and memory space of dependent variable to half.

Multiple positive solutions of boundary value problems for second-order discrete equations on the half-line

In this paper, we study the existence of multiple positive solutions of boundary value problems for second-order discrete equations Δ2 x(n − 1) − pΔx(n − 1) − qx(n − 1)+f(n, x(n)) = 0, n ∈ {1,2,…}, αx(0) − βΔx(0) = 0, x(∞) = 0. The proofs are based on the fixed point theorem in Fréchet space (see Agarwal and O'Regan, 2001, Cone compression and expansion and fixed point theorems in Fréchet spaces with application, Journal of Differential Equations, 171, 412–42).

Three-point boundary value problems for second-order discrete equations

We formulate existence results for solutions to discrete equations which approximate three-point boundary value problems for second-order ordinary differential equations.

Accelerated monotone iterative methods for a boundary value problem of second-order discrete equations

An accelerated monotone iterative method for a boundary value problem of second-order discrete equations is presented. This method leads to an existence-comparison theorem as well as a computational algorithm for the solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Some numerical results are presented to illustrate the monotone convergence of the iterative sequences and the rate of convergence of the iterations.