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This paper is concerned with a compact finite difference method with non-isotropic mesh sizes for a two-dimensional fourth-order nonlinear elliptic boundary value problem. By the discrete energy analysis, the optimal error estimates in the discrete L 2 , H 1 and L ∞ norms are obtained without any constraint on the mesh sizes. The error estimates show that the compact finite difference method converges with the convergence rate of fourth-order. Based on a high-order approximation of the solution, a Richardson extrapolation algorithm is developed to make the final computed solution sixth-order accurate. Numerical results demonstrate the high-order accuracy of the compact finite difference method and its extrapolation algorithm in the discrete L 2 , H 1 and L ∞ norms.

This paper is concerned with some numerical methods for a fourth-order semilinear elliptic boundary value problem with nonlocal boundary condition. The fourth-order equation is formulated as a coupled system of two second-order equations which are discretized by the finite difference method. Three monotone iterative schemes are presented for the coupled finite difference system using either an upper solution or a lower solution as the initial iteration. These sequences of monotone iterations, called maximal sequence and minimal sequence respectively, yield not only useful computational algorithms but also the existence of a maximal solution and a minimal solution of the finite difference system. Also given is a sufficient condition for the uniqueness of the solution. This uniqueness property and the monotone convergence of the maximal and minimal sequences lead to a reliable and easy to use error estimate for the computed solution. Moreover, the monotone convergence property of the maximal and minimal sequences is used to show the convergence of the maximal and minimal finite difference solutions to the corresponding maximal and minimal solutions of the original continuous system as the mesh size tends to zero. Three numerical examples with different types of nonlinear reaction functions are given. In each example, the true continuous solution is constructed and is used to compare with the computed solution to demonstrate the accuracy and reliability of the monotone iterative schemes.

This paper is concerned with numerical methods for a class of two-dimensional quasi-linear elliptic boundary value problems. A compact finite difference method with a nonisotropic mesh is proposed for the problems. The existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions, and two sufficient conditions for the uniqueness of the solution are also given. The optimal error estimate in the discrete $L^{\infty}$ norm is obtained under certain conditions. The error estimate shows the fourth-order accuracy of the proposed method when two spatial mesh sizes are proportional. By using an upper solution or a lower solution as the initial iteration, an “almost optimal” Picard type of monotone iterative algorithm is presented for solving the resulting nonlinear discrete system efficiently. Applications using two model problems give numerical results that confirm our theoretical analysis.

This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) [3], for solving systems of two-dimensional reaction–diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method.

The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourth-order Numerov's method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov's method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach.

This paper is concerned with finite difference solutions of a coupled system of nonlinear reaction-diffusion equations. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. The existence and uniqueness of a non-negative finite difference solution and three monotone iterative algorithms for the computation of the solutions are given. It is shown that the time-dependent problem has a unique non-negative solution, whereas the steady-state problem may have multiple non-negative solutions depending on the parameters in the problem. The different non-negative steady-state solutions can be computed from the monotone iterative algorithms by choosing different initial iterations. Also discussed is the asymptotic behaviour of the time-dependent solution in relation to the steady-state solutions. The asymptotic behaviour result gives some conditions ensuring the convergence of the time-dependent solution to a positive or semitrivial non-negative steady-state solution. Numerical results are given to demonstrate the theoretical analysis results.

This paper is concerned with the existence and uniqueness of a solution for a class of 2nth-order nonlinear multi-point boundary value problems. The existence of a solution is proven by the method of upper and lower solutions without any monotone condition on the nonlinear function. A sufficient condition for the uniqueness of a solution is given. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. Two examples are presented to illustrate the results.

This paper is concerned with the computational algorithms for finite difference discretizations of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A higher-order monotone iterative method is presented for solving the finite difference discretizations of both the time-dependent problem and the corresponding steady-state problem. This method leads to an efficient linear iterative algorithm which yields two sequences of iterations that converge monotonically to a unique solution of the system. The monotone property of the iterations gives concurrently improved upper and lower bounds of the solution in each iteration. It is shown that the rate of convergence for the sum of the two produced sequences is of order p+2, where p>=1 is a positive integer depending on the construction of the method, and under an additional requirement, the higher-order rate of convergence is attained for one of these two sequences. An application is given to an enzyme-substrate reaction-diffusion problem, and some numerical results are presented to illustrate the effectiveness of the proposed method.

This paper is concerned with finite difference solutions of a class of fourth-order nonlinear elliptic boundary value problems. The nonlinear function is not necessarily monotone. A new monotone iterative technique is developed, and three basic monotone iterative processes for the finite difference system are constructed. Several theoretical comparison results among the various monotone sequences are given. A simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. Numerical results for a model problem with known analytical solution are given.

This paper is concerned with the extrapolation algorithm of Numerov's scheme for semilinear and strongly nonlinear two-point boundary value problems. The asymptotic error expansion of the solution of Numerov's scheme is obtained. Based on the asymptotic error expansion, Richardson's extrapolation is constructed, and so the accuracy of the numerical solution is greatly increased. Numerical results are presented to demonstrate the efficiency of the extrapolation algorithm.

This paper is concerned with the fourth-order elliptic boundary value problems with nonmonotone nonlinear function. The existence and uniqueness of a solution is proven by the method of upper and lower solutions. A monotone iteration is developed so that the iteration sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution.

This paper is concerned with the asymptotic behavior of the finite difference solutions of a class of nonlinear reaction diffusion equations with time delay. By introducing a pair of coupled upper and lower solutions, an existence result of the solution is given and an attractor of the solution is obtained without monotonicity assumptions on the nonlinear reaction function. This attractor is a sector between two coupled quasi-solutions of the corresponding “steady-state" problem, which are obtained from a monotone iteration process. A sufficient condition, ensuring that two coupled quasi-solutions coincide, is given. Also given is the application to a nonlinear reaction diffusion problem with time delay for three different types of reaction functions, including some numerical results which validate the theoretical analysis.

By a proper transformation, a class of strongly nonlinear two-point boundary value problems are transformed into semilinear problems so that the well-known Numerov's method can be applied. Some applications and numerical results are presented to demonstrate the efficiency of the approach.

Monotone finite difference schemes are proposed for nonlinear systems with mixed quasi-monotonicity. Two monotone iteration processes for the corresponding discrete problems are presented, which converge monotonically to the quasi-solutions of the discrete problems. The limits are the exact solutions under some conditions. A monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed. The numerical results coincide with theoretical analysis.

This paper is concerned with the asymptotic behavior of the finite difference solutions of a coupled system of nonlinear integrodifferential reaction–diffusion equations. The existence of the finite difference solution and the monotone iteration process for solving the finite difference system are given. This includes an existence-uniqueness-comparison theorem. From the monotone iteration process, an attractor of the numerical time-dependent solution is obtained. This attractor is a sector between the pair of coupled quasisolutions of the corresponding numerical steady-state problem, which are obtained from a monotone iteration process. A sufficient condition, ensuring that the two coupled quasisolutions coincide, is given. Also given is the application to a reaction–diffusion problem with three different types of reaction functions, including some numerical results which validate the theory analysis.

Petrov–Galerkin method is investigated for solving nonlinear systems without monotonicity. A new concept of ordered pair of supersolution and subsolution is proposed. The existence of numerical solutions is studied. A new monotone iteration is provided for solving the resulting problem. Two conditions are given, ensuring the geometric convergence rate. The numerical results show the advantages of this method.

A parallel matrix multisplitting iterative method is set up for a class of systems of weakly nonlinear equations without isotone mapping. The two-sided monotone convergence is shown. A sufficient condition is given to ensure the monotone convergence of the new method to the unique solution of the system in some sector. The influences of different multisplittings on the convergence rate are investigated in the sense of monotonicity. A numerical example is given.AMS classification: 65F10; 65H20

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.

Almost monotone approximation is proposed for nonlinear two‐points problem. A general framework is given for studying the existence and uniqueness of numerical solutions. A discrete approximation with high accuracy is constructed. Nonlinear Jacobi iteration and Gauss–Seidel iteration are introduced to save work. The continuous approximation is also considered. The numerical results show the advantages of such an approach.

A compact finite difference method with non-isotropic mesh is proposed for a two-dimensional fourth-order nonlinear elliptic boundary value problem. The existence and uniqueness of its solutions are investigated by the method of upper and lower solutions, without any requirement of the monotonicity of the nonlinear term. Three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently. The convergence and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.

The aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth-order elliptic Lidstone boundary value problems where the nonlinear functions depend on the higher-order derivatives. Sufficient conditions are given for the existence and uniqueness of a solution. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. The approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions. All the results are directly applicable to 2nth-order two-point Lidstone boundary value problems.