A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equations

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 finite element first-order equation formulation for the small-disturbance transonic flow problem

The nonlinear, mixed elliptic hyperbolic equation describing a steady transonic flow is considered. The original equation is replaced by a system of first-order equations that are hyperbolic in time and defined in terms of velocity components. Parabolic regularization terms are added to capture shock wave solutions and to damp iterative solution algorithms. A finite element Galerkin method in space and a Crank-Nicolson finite difference method in iterative time are used to reduce the problem to the solution of a system of algebraic equations. Stability and convergence characteristics of the iterative method are discussed. The numerical implementation of the method is explained, and numerical results are presented.     

Monotone Iterates for Solving Nonlinear Monotone Difference Schemes

This paper is concerned with monotone iterative algorithms for solving nonlinear monotone difference schemes of elliptic type. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing the nonlinear monotone difference schemes in the canonical form. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear monotone difference schemes. Numerical experiments are presented.     

A strong convergence theorem for a general split equality problem with applications to optimization and equilibrium problem

The purpose of this paper is to introduce and study an iterative algorithm for solving a general split equality problem. The problem consists of finding a common element of the set of common zero points for a finite family of maximal monotone operators, the set of common fixed points for a finite family of demimetric mappings and the set of common solutions of variational inequality problems for a finite family of inverse strongly monotone mappings in the setting of infinite-dimensional Hilbert spaces. Using our iterative algorithm, we state and prove a strong convergence theorem for approximating a solution of the split equality problem. As special cases, we shall utilize our results to study the split equality equilibrium problems and the split equality optimization problems. Our result complements and extends some related results in literature.     

Monotone methods for a finite difference system of reaction diffusion equation with time delay

The aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction-diffusion equations with time delays. The iterative schemes lead to computational algorithms as well as existence, uniqueness, and upper and lower bounds of the solution. An application to a diffusive logistic equation with time delay is given.     

A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations

A fast finite difference method based on the monotone iterative method and the fast Poisson solver on irregular domains for a 2D nonlinear Poisson–Boltzmann equation is proposed and analyzed in this paper. Each iteration of the monotone method involves the solution of a linear equation in an exterior domain with an arbitrary interior boundary. A fast immersed interface method for generalized Helmholtz equations on exterior irregular domains is used to solve the linear equation. The monotone iterative method leads to a sequence which converges monotonically from either above or below to a unique solution of the problem. This monotone convergence guarantees the existence and uniqueness of a solution as well as the convergence of the finite difference solution to the continuous solution. A comparison of the numerical results against the exact solution in an example indicates that our method is second order accurate. We also compare our results with available data in the literature to validate the numerical method. Our method is efficient in terms of accuracy, speed, and flexibility in dealing with the geometry of the domain     

Lagged diffusivity fixed point iteration for solving steady-state reaction diffusion problems

The paper concerns with the computational algorithms for a steady-state reaction diffusion problem. A lagged diffusivity iterative algorithm is proposed for solving resulting system of quasilinear equations from a finite difference discretization. The convergence of the algorithm is discussed and the numerical results show the efficiency of this algorithm.     

Numerical methods for nonlinear integro-parabolic equations of fredholm type

This paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction-diffusion-convection equations under either linear or nonlinear boundary conditions. The discrete approximation of the problem is based on the finite-difference method, and the computation of the finite-difference solution is by the method of upper and lower solutions. Three types of quasi-monotone reaction functions are considered and for each type, a monotone iterative scheme is obtained. Each of these iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite-difference system. This monotone convergence leads to an existence-uniqueness theorem as well as a computational algorithm for the computation of the solution. An error estimate between the computed approximations and the true finite-difference solution is obtained for each iterative scheme. These error estimates are given in terms of the strength of the reaction function and the effect of diffusion-convection, and are independent of the true solution. Applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm.     

Parallel monotone iterative relaxation methods for a class of two-dimensional discrete boundary value problems

A parallel monotone iterative relaxation method for a class of two-dimensional discrete boundary value problems is established, and the sequence of iterations is shown to converge monotonically either from above or below to a solution of the problem. This monotone convergence results yields a parallel computational algorithm as well as an existence-comparison result for the solutions. To compute the sequence of iterations, the Thomas algorithm can be used in the same fashion as for one-dimensional problem. The existence and comparison results of the upper and lower solutions are given. The local as well as global existence-uniqueness of the solution are obtained. The global convergence of the iterations is investigated, and the influence of the parameters on the rate of convergence of the iterations is analyzed. Numerical results are given to corroborate the analytical results.     

Coupled Path Planning,Region Optimization,and Applications in Intensity-modulated Radiation Therapy

In this paper, we consider an optimization problem in discrete geometry, called coupled path planning (CPP). Given a finite rectangular grid and a non-negative function f defined on the horizontal axis of the grid, we seek two non-crossing monotone paths in the grid, such that the vertical difference between the two paths approximates f in the best possible way. This problem arises in intensity-modulated radiation therapy (IMRT), where f represents an ideal radiation dose distribution and the two coupled paths represent the motion trajectories (or control sequence) of two opposing metal leaves of a delivery device for controlling the area exposed to the radiation source. By finding an optimal control sequence, the CPP problem aims to deliver precisely a prescribed radiation dose, while minimizing the side-effects on the surrounding normal tissue. We present efficient algorithms for different versions of the CPP problems. Our results are based on several new ideas and geometric observations, and substantially improve the solutions based on standard techniques. Implementation results show that our CPP algorithms run fast and produce better quality clinical treatment plans than the previous methods.     

Normalized factorization procedures for the solution of self-adjoint Elliptic partial differential equations in three-space dimensions

New normalized factorization procedures are presented for the coefficient matrix derived from the finite difference discretization of a self-adjoint elliptic 3D-P.D.E. leading to improved iterative schemes of solution. The derived algorithms are shown to be both competitive and computationally efficient in comparison with the existing schemes. Experimental results for a non-linear 3D magneto-hydrodynamic problem are given.     

Monotone iterates for solving coupled systems of nonlinear parabolic equations

This paper deals with numerical solutions of coupled nonlinear parabolic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear parabolic equations. This monotone convergence leads to existence-uniqueness theorems. An analysis of convergence rates of the monotone iterative method is given. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating is proposed. A convergence analysis of the monotone domain decomposition algorithm is presented. An application to a gas–liquid interaction model is given.     

A general iterative method with strongly positive operators for general variational inequalities

In this paper, we introduce and study a general iterative method with strongly positive operators for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. The explicit and implicit iterative algorithms are proposed by virtue of the general iterative method with strongly positive operators. Under two sets of quite mild conditions, we prove the strong convergence of these explicit and implicit iterative algorithms to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively.     

Enclosure of solutions of weakly nonlinear elliptic boundary value problems and their computation

An algorithm is presented to compute approximations as well as continuous bounds for solutions of weakly nonlinear elliptic boundary value problems. The given problem is majorized in some sense and the obtained new problem is solved by a finite element method. The finite element solution is computed by a monotone iteration process and at last transformed to a continuous (lower) bound for a solution. Convergence is proved and mesh refinement effects are discussed. Two numerical examples are given.     

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.     

A.D.I. Methods for the solution of diffusion problems in r−θ geometry

In this paper the application of a particular class of iterative methods, the alternating direction implicit methods applied to the solution of time dependent and steady state diffusion problems in regions involving circular symmetry, is presented.A comparison between the convergence rates for commutative and non-commutative versions of the finite difference equations and a peripheral (azimuthal) block over-relaxation method for the steady-state model problem is given.     

A Monotone Iterative Technique for Nonlinear Fourth Order Elliptic Equations with Nonlocal Boundary Conditions

This paper proposes an accelerated iterative procedure for a nonlinear fourth order elliptic equation with nonlocal boundary conditions. First, an existence and uniqueness theorem is proved for the fourth order elliptic equation via the accelerated iterative procedure. To solve this problem numerically, a finite difference based numerical scheme is also developed in view of the main theorem. Theoretically, the monotone property as well as the convergence analysis are proved for both the continuous and discretized cases. The main result also supplements several algorithms for computing the solution of the fourth order elliptic integro-partial differential equation. The proposed scheme not only accelerates the scheme in the literature but also provides a greater flexibility in choosing the initial guess. The efficacy of the proposed scheme is demonstrated through a comparative numerical study with the recent literature. The numerical simulation confirms the theoretical claims too.     

Hybrid Flow Shop with Setup Times Scheduling Problem

The two-stage hybrid flow shop problem under setup times is addressed in this paper. This problem is NP-Hard. on the other hand, the studied problem is modeling different real-life applications especially in manufacturing and high performance-computing. Tackling this kind of problem requires the development of adapted algorithms. In this context, a metaheuristic using the genetic algorithm and three heuristics are proposed in this paper. These approximate solutions are using the optimal solution of the parallel machines under release and delivery times. Indeed, these solutions are iterative procedures focusing each time on a particular stage where a parallel machines problem is called to be solved. The general solution is then a concatenation of all the solutions in each stage. In addition, three lower bounds based on the relaxation method are provided. These lower bounds present a means to evaluate the efficiency of the developed algorithms throughout the measurement of the relative gap. An experimental result is discussed to evaluate the performance of the developed algorithms. In total, 8960 instances are implemented and tested to show the results given by the proposed lower bounds and heuristics. Several indicators are given to compare between algorithms. The results illustrated in this paper show the performance of the developed algorithms in terms of gap and running time.     

Existence conditions and properties for the maximal periodic solution of periodic Riccati difference equations

The existence and properties of the maximal symmetric periodic solution of the periodic Riccati difference equation, is analysed for the optimal filtering problem of linear periodic discrete-time systems. Special emphasis is given to systems not necessarily reversible and subject only to a detectability assumption. Necessary and sufficient conditions for the existence and uniqueness of periodic non-negative definite solutions of the periodic Riccati difference equation which gives rise to a stable filter are also established. Furthermore, the convergence of non-negative definite solutions of the Riccati equation is investigated.     

Nonnegative solutions of fractional functional differential equations

In this paper, we discuss the existence and monotone iterative method of nonnegative solutions for fractional functional differential equations. The main conclusion is that the nonnegative solutions can be derived from the monotone iterative method, which starts off with a nonnegative upper solution or the zero function under different conditions. Our approach of obtaining nonnegative solutions is feasible for computational purposes.