Search

This paper is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t = 0 . A generalized Crank-Nicolson-type scheme for the time discretization is proposed by designing a product integration rule for the integral term, and a compact difference approximation is used for the space discretization. The proposed method is constructed on the uniform time mesh, but it can still achieve the second-order convergence in time for weakly singular solutions. The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L 2 -norm is obtained. The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space. The extension of the method to two-dimensional problems is also discussed. A simple comparison is made with several existing methods. Numerical results confirm the theoretical analysis result and show the effectiveness of the proposed method.

This paper is concerned with numerical methods for a class of nonhomogeneous Neumann problems of time-fractional reaction–diffusion equations with variable coefficients. The solutions of this kind of problems often have weak singularity at the initial time. This makes the existing numerical methods with uniform time mesh often lose accuracy. In this paper, we propose and analyze a high-order compact finite difference method with nonuniform time mesh. The time-fractional derivative is approximated by Alikhanov’s high-order approximation on a class of fitted time meshes. For the spatial variable coefficient differential operator, a new fourth-order boundary discretization is developed under the nonhomogeneous Neumann boundary condition, and then a new fourth-order compact finite difference approximation on a space uniform mesh is obtained. Under the assumption of the weak initial singularity of solution, we prove that for the general case of the variable coefficients, the proposed method is unconditionally stable and the numerical solution converges to the solution of the problem under consideration. The convergence result also gives an optimal error estimate of the numerical solution in the discrete L 2 -norm, which shows that the method has the spatial fourth-order convergence, while it attains the temporal optimal second-order convergence provided a proper mesh grading parameter is employed. Numerical results that confirm the sharpness of the error analysis are presented.

β-Catenin is an important co-factor which binds multiple transcriptional molecules and mediates fibrogenic signaling pathways. Its role in kidney transplantation is unknown. We quantified binding of β-catenin within renal tubular epithelial cells to transcription factors, TCF1 and FoxO1, using a proximity ligation assay in 240 transplanted kidneys, and evaluated their pathological and clinical outcomes. β-Catenin/FoxO1 binding in 1-month protocol biopsies inversely correlated with contemporaneous chronic fibrosis, subsequent inflammation. and inflammatory fibrosis (P < .001). The relative binding of β-catenin/TCF1 versus β-catenin/FoxO1 (TF ratio) was the optimal biomarker, and abnormal in diverse fibrotic transplant diseases. A high 1-month TF ratio was followed by greater tubular atrophy and interstitial fibrosis scores, cortical inflammation, renal impairment, and proteinuria at 1 year (n = 131, all P < .001). The TF ratio was associated with reduced eGFR (AUC 0.817), mild fibrosis (AUC 0.717), and moderate fibrosis (AUC 0.769) using receiver operating characteristic analysis. An independent validation cohort (n = 76) confirmed 1-month TF was associated with 12-month moderate fibrosis (15.8% vs. 2.6%, P = .047), however, not with other outcomes or 10-year graft survival, which limits generalizabilty of these findings. In summary, differential binding of β-catenin to TCF1 rather than FoxO1 in renal tubular cells was associated with the fibrogenic response in transplanted kidneys.

This paper is concerned with the construction and analysis of a high-order compact finite difference method for a class of time-fractional sub-diffusion equations under the Robin boundary condition. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ ( 0 , 1 ) . A ( 3 − α ) th-order numerical formula (called the L2 formula here) without any sub-stepping scheme for the approximation at the first-time level is applied to the discretization of the Caputo time-fractional derivative. A new fourth-order compact finite difference operator is constructed to approximate the variable coefficient spatial differential operator under the Robin boundary condition. By developing a technique of discrete energy analysis, the unconditional stability of the proposed method and its convergence of ( 3 − α ) th-order in time and fourth-order in space are rigorously proved for the general case of variable coefficient and for all α ∈ ( 0 , 1 ) . Further approximations are considered for enlarging the applicability of the method while preserving its high-order accuracy. Numerical results are provided to demonstrate the theoretical analysis results.

A high-order compact finite difference method is proposed for solving a class of time-fractional sub-diffusion equations. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ (0, 1). The Caputo time-fractional derivative is discretized by a ( 3 − α ) th-order numerical formula (called the L2 formula here) which is constructed by piecewise quadratic interpolating polynomials but does not require any sub-stepping scheme for the approximation at the first-time level. The variable coefficient spatial differential operator is approximated by a fourth-order compact finite difference operator. By developing a technique of discrete energy analysis, a full theoretical analysis of the stability and convergence of the method is carried out for the general case of variable coefficient and for all α ∈ (0, 1). The optimal error estimate is obtained in the L2 norm and shows that the proposed method has the temporal ( 3 − α ) th-order accuracy and the spatial fourth-order accuracy. Further approximations are also considered for enlarging the applicability of the method while preserving its high-order accuracy. Applications are given to three model problems, and numerical results are presented to demonstrate the theoretical analysis results.

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted L 2 L 2 - and L ∞ L ∞ -norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order min{2−α,1+α} min { 2 − α , 1 + α } , where α∈(0,1) α ∈ ( 0 , 1 ) is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.

This paper is concerned with numerical methods for a class of time-fractional convection-reaction-diffusion equations. The convection and reaction coefficients of the equation may be spatially variable. Based on the weighted and shifted Grunwald–Letnikov formula for the time-fractional derivative and a compact finite difference approximation for the spatial derivative, we establish an unconditionally stable compact difference method. The local truncation error and the solvability of the resulting scheme are discussed in detail. The stability of the method and its convergence of third-order in time and fourth-order in space are rigorously proved by the discrete energy method. Combining this method with a Richardson extrapolation, we present an extrapolated compact difference method which is fourth-order accurate in both time and space. A rigorous proof for the convergence of the extrapolation method is given. Numerical results confirm our theoretical analysis, and demonstrate the accuracy of the compact difference method and the effectiveness of the extrapolated compact difference method.

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.

A Crank-Nicolson-type compact locally one-dimensional (LOD) finite difference method is proposed for a class of two-dimensional modified anomalous fractional sub-diffusion equations with two time Riemann-Liouville fractional derivatives of orders ( 1 - α ) and ( 1 - β ) ( 0 < α , β < 1 ) . The resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one-dimensional problems. This property evidently enhances the simplicity of programming and makes the computations more easy. The unconditional stability and convergence of the scheme are rigorously proved. The error estimates in the standard H 1 - and L 2 -norms and the weighted L ∞ -norm are obtained and show that the proposed compact LOD method has the accuracy of the order 2 min { α , β } in time and 4 in space. A Richardson extrapolation algorithm is presented to increase the temporal accuracy to the order min { α + β , 4 min { α , β } } if α ? β and min { 1 + α , 4 α } if α = β . A comparison study of the compact LOD method with the other existing methods is given to show its superiority. Numerical results confirm our theoretical analysis, and demonstrate the accuracy and the effectiveness of the compact LOD method and the extrapolation algorithm.

This paper is concerned with a numerical method for a class of one-dimensional multi-term time-fractional convection-reaction-diffusion problems, where the differential equation contains a sum of the Caputo time-fractional derivatives of different orders between 0 and 1. In general the solutions of such problems typically exhibit a weak singularity at the initial time. A compact exponential finite difference method, using the well-known L1 formula for each time-fractional derivative and a fourth-order compact exponential difference approximation for the spatial discretization, is proposed on a mesh that is generally nonuniform in time and uniform in space. Taking into account the initial weak singularity of the solution, the stability and convergence of the method is proved and the optimal error estimate in the discrete L2-norm is obtained by developing a discrete energy analysis technique which enables us to overcome the difficulties caused by the nonsymmetric discretization matrices. The error estimate shows that the method has the spatial fourth-order convergence, and reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence. The extension of the method to two-dimensional problems is also discussed. Numerical results confirm the theoretical convergence result, and show the applicability of the method to convection dominated problems.

This paper is concerned with a numerical method for the time-fractional Benjamin–Bona–Mahony (BBM) equation whose solution typically exhibits a weak singularity at the initial time. Lyu and Vong (2019) presented a linearized difference method of second-order in space and third-order in time. We improve their result by proposing a linearized and compact difference method which is fourth-order in space while keeping third-order in time. By using discrete energy analysis, the unconditional convergence of the proposed method is rigorously proved and the optimal H 1 -norm error estimate is obtained. Numerical results confirm the theoretical convergence result.

In this paper, we study numerical methods for a class of two-dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains. A compact finite difference method with an anisotropic 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 ∞ 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 developed for solving the resulting nonlinear discrete system efficiently. Numerical results are presented to confirm our theoretical analysis.

Beidou is the regional satellite navigation system in China, consisting of three kinds of orbiting satellites, MEO, GEO and IGSO, with the orbital altitudes of 21500–36000 km. For improving the accuracy of satellites orbit determination, calibrating microwave measuring techniques and providing better navigation service, all Beidou satellites are equipped with laser retro-reflector arrays (LRAs) to implement high precision laser ranging. The paper presents the design of LRAs for Beidou navigation satellites and the method of inclined installation of LRAs for GEO satellites to increase the effective reflective areas for the regional ground stations. By using the SLR system, the observations for Beidou satellites demonstrated a precision of centimeters. The performances of these LRAs on Beidou satellites are very excellent.

A higher-order compact locally one-dimensional (LOD) finite difference method for two-dimensional nonhomogeneous parabolic differential equations is proposed. The resulting scheme consists of two one-dimensional tridiagonal systems, and all computations are implemented completely in one spatial direction as for one-dimensional problems. The solvability and the stability of the scheme are proved almost unconditionally. The error estimates are obtained in the discrete H 1 , L 2 and L ∞ norms, and show that the proposed compact LOD method has the accuracy of the second-order in time and the fourth-order in space. Two Richardson extrapolation algorithms are presented to increase the accuracy to the fourth-order and the sixth-order in both time and space when the time step is proportional to the spatial mesh size. Numerical results demonstrate the accuracy of the compact LOD method and the high efficiency of its extrapolation algorithms.

High-precision time synchronization between satellites and ground stations plays the vital role in satellite navigation system. Laser time transfer (LTT) technology is widely recognized as the highest accuracy way to achieve time synchronization derived from satellite laser ranging (SLR) technology. Onboard LTT payload has been designed and developed by Shanghai Astronomical Observatory, and successfully applied to Chinese Beidou navigation satellites. By using the SLR system, with strictly controlling laser firing time and developing LTT data processing system on ground, the high precise onboard laser time transfer experiment has been first performed for satellite navigation system in the world. The clock difference and relative frequency difference between the ground hydrogen maser and space rubidium clocks have been obtained, with the precision of approximately 300 ps and relative frequency stability of 10E−14. This article describes the development of onboard LTT payload, introduces the principle, system composition, applications and LTT measuring results for Chinese satellite navigation system.

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.

A fourth-order compact finite difference method is proposed for a class of nonlinear 2nth-order multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as the three- or four-point boundary condition, (n+2)-point boundary condition and 2(n−m)-point boundary condition. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. The convergence and the fourth-order accuracy of the method are proved. An efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems. Various sufficient conditions for the construction of upper and lower solutions are obtained. Some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach.

This paper is concerned with a time-delayed Lotka-Volterra competition reaction-diffusion system with homogeneous Neumann boundary conditions. Some explicit and easily verifiable conditions are obtained for the global asymptotic stability of all forms of nonnegative semitrivial constant steady-state solutions. These conditions involve only the competing rate constants and are independent of the diffusion-convection and time delays. The result of global asymptotic stability implies the nonexistence of positive steady-state solutions, and gives some extinction results of the competing species in the ecological sense. The instability of the trivial steady-state solution is also shown.

This paper is concerned with a compact locally one-dimensional (LOD) finite difference method for solving two-dimensional nonhomogeneous parabolic differential equations. An explicit error estimate for the finite difference solution is given in the discrete infinity norm. It is shown that the method has the accuracy of the second-order in time and the fourth-order in space with respect to the discrete infinity norm. A Richardson extrapolation algorithm is developed to make the final computed solution fourth-order accurate in both time and space when the time step equals the spatial mesh size. Numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm.

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 a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of Rn. Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.

The aim of this paper is to investigate the existence of iterative solutions for a class of 2 n th-order nonlinear multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as three- or four-point boundary condition, ( n + 2)-point boundary condition and 2( n − m )-point boundary condition. The existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique. A monotone iteration is developed so that the iterative 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. Two examples are presented to illustrate the results.

First-principles calculations based on density functional theory have been performed to investigate the behaviors of He in hcp-type Ti. The most favorable interstitial site for He is not an ordinary octahedral or tetrahedral site, but a novel interstitial site (called FC) with a formation energy as low as 2.67 eV, locating the center of the face shared by two adjacent octahedrons. The origin was further analyzed by composition of formation energy of interstitial He defects and charge density of defect-free hcp Ti. It has also been found that an interstitial He atom can easily migrate along (0 0 1) direction with an activation energy of 0.34 eV and be trapped by another interstitial He atom with a high binding energy of 0.66 eV. In addition, the small He clusters with/without Ti vacancy have been compared in details and the formation energies of He(n)V clusters with a pre-existing Ti vacancy are even higher than those of He clusters until n >= 3. (C) 2010 Elsevier B.V. All rights reserved.

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 to investigate the asymptotic behavior of solutions for a time-delayed Lotka–Volterra N-species mutualism reaction–diffusion system with homogeneous Neumann boundary condition. It is shown, under a simple condition on the reaction rates, that the system has a unique bounded time-dependent solution and a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the constant positive steady-state solution as time tends to infinity. This convergence result implies that the trivial steady-state solution and all forms of semitrivial steady-state solutions are unstable, and moreover, the system has no nonconstant positive steady-state solution. A condition ensuring the convergence of the time-dependent solution to one of nonnegative semitrivial steady-state solutions is also given.

This paper is concerned with finite difference solutions of a Michaelis-Menten-type ratio-dependent predator-prey system with diffusion. The system is discretized by the finite difference method, and the investigation is devoted to the finite difference system for the time-dependent solution and its asymptotic behavior in relation to the various steady-state solutions. Three monotone iterative schemes for the computation of the time-dependent solution are presented, and the sequences of iterations are shown to converge monotonically to a unique positive solution. A simple and easily verifiable condition on the rate constants is obtained, which ensures that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges either to a unique positive steady-state solution or to a semitrivial steady-state solution. The above results lead to computational algorithms for the solution as well as the global asymptotic stability of the system. Some numerical results are given. All the conclusions are directly applicable to the finite difference solution of the corresponding ordinary differential system.

This paper is concerned with finite difference solutions of a Lotka–Volterra reaction–diffusion system with three-competing species. The reaction–diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference system for the time-dependent solution and its asymptotic behavior in relation to the corresponding steady-state problem. Three monotone iterative schemes for the computation of the time-dependent solution are presented, and the sequences of iterations are shown to converge monotonically to a unique positive solution. Also discussed is the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions. A simple condition on the competing rate constants is obtained, which ensures that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges either to a unique positive steady-state solution or to one of the semitrivial steady-state solutions. The above results lead to the coexistence and permanence of the competing system as well as computational algorithms for numerical solutions. Some numerical results from these computational algorithms are given. All the conclusions for the reaction–diffusion equations are directly applicable to the finite difference solution of the corresponding ordinary differential system.

A compact finite difference method is proposed for a general class of 2nth-order Lidstone boundary value problems. The existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. A monotone iteration process is provided for solving the resulting discrete system efficiently, and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. The convergence of the finite difference solution and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.

This paper is concerned with three 3-species time-delayed Lotka–Volterra reaction–diffusion models with homogeneous Neumann boundary condition. Some simple conditions are obtained for the global asymptotic stability of the nonnegative semitrivial constant steady-state solutions. These conditions are explicit and easily verifiable, and they involve only the reaction rate constants and are independent of the diffusion and time delays. The result of global asymptotic stability not only implies the nonexistence of positive steady-state solution but also gives some extinction results of the models in the ecological sense. The instability of some nonnegative semitrivial constant steady-state solutions is also shown. The conclusions for the reaction–diffusion systems are directly applicable to the corresponding ordinary differential systems.

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.

The aim of this paper is to investigate the asymptotic behavior of solutions for a class of three-species predator–prey reaction–diffusion systems with time delays under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants of the reaction functions to ensure the convergence of the time-dependent solution to a constant steady-state solution. The conditions for the convergence are independent of diffusion coefficients and time delays, and the conclusions are directly applicable to the corresponding parabolic-ordinary differential system and to the corresponding system without time delays.

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 computational algorithms for finite difference solutions of a class of semilinear elliptic boundary value problems. An accelerated monotone iterative scheme is presented by using the method of upper and lower solutions. The rate of convergence of the iterations is estimated by the infinity norm, and the rate of convergence is quadratic for a larger class of nonlinear functions, including monotone nonincreasing functions. An application is given to a logistic model problem in ecology.

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.

This paper is a survey of some recent developments of Numerov's method for solving nonlinear two-point boundary value problems. The survey consists of three different parts: the existence-uniqueness of a solution, computational algorithm for computing a solution, and some extensions of Numerov's method. The sufficient conditions for the existence and uniqueness of a solution are presented. Some of them are best possible. Various iterative methods are reviewed, including Picard's iterative method, modified Newton's iterative method, monotone iterative method, and accelerated monotone iterative method. In particular, two more direct monotone iterative methods are presented to save computational work. Each of these iterative methods not only gives a computational algorithm for computing a solution, but also leads to an existence (and uniqueness) theorem. The estimate on the rate of convergence of the iterative sequence is given. The extensions of Numerov's method to a coupled problem and a general problem are addressed. The numerical results are presented to validate the theoretical 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.