Monotone Iterative Scheme Research Articles

Monotone enclosure for a class of discrete boundary value problems without monotone nonlinearities

A new concept of a pair of upper and lower solutions is introduced for a class of discrete boundary value problems without monotone nonlinearities. Some comparison results are established. An existence and enclosing theorem for the solutions is given in terms of upper and lower solutions. A new monotone iterative scheme is proposed. The numerical example is given.

Maximum principles for periodic impulsive first order problems

We present some comparison results for the periodic boundary value problem for first order ordinary differential equations with impulses at fixed moments. Then, the upper and lower solution method and the monotone iterative scheme are presented.

Combined iterative methods for numerical solutions of parabolic problems with time delays

We study monotone iterative schemes for finite-difference solutions of reaction-diffusion systems with time delays. In this paper we derive two modified iterative schemes by combing the method of upper-lower solutions and the Jacobi method or the Gauss-Seidel method. The convergence and stability of the monotone iterative schemes are also discussed.

Boundary element monotone iteration scheme for semilinear elliptic partial differential equations

The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin δ \delta for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the H r ( Ω ) H^{r}(\Omega ) , 0 ≤ r ≤ 2 0\le r \le 2 , Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, “higher than optimal order” error estimates can be obtained with respect to the mesh parameter h h . Several examples of semilinear elliptic partial differential equations featuring different situations of existence/nonexistence, uniqueness/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.

Block monotone iterative methods for numerical solutions of nonlinear elliptic equations

Two block monotone iterative schemes for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear elliptic boundary-value problem, are presented and are shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a computational algorithm for numerical solutions as well as an existence-comparison theorem of the system, including a sufficient condition for the uniqueness of the solution. An advantage of the block iterative schemes is that the Thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one-dimensional problems. The block iterative schemes are compared with the point monotone iterative schemes of Picard, Jacobi and Gauss-Seidel, and various theoretical comparison results among these monotone iterative schemes are given. These comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes. Application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given.

Monotone method and convergence acceleration for finite‐difference solutions of parabolic problems with time delays

AbstractIn this article we study a finite‐difference system, which is a discrete version of a class of nonlinear reaction‐diffusion systems with time delays. Existence‐comparison and uniqueness theorem are first established for the discretized problem by the method of upper and lower solutions. A monotone iterative scheme is also developed for the solution of the finite‐difference system. Under a convergence acceleration scheme, it is shown that the monotone sequences converge quadratically to the solution of the finite‐difference system. At last, numerical results on some model problems are demonstrated to substantiate our theorems. © 1995 John Wiley & Sons, Inc.

Finite difference reaction diffusion equations with nonlinear boundary conditions

AbstractThis article is concerned with numerical solutions of finite difference systems of reaction diffusion equations with nonlinear internal and boundary reaction functions. The nonlinear reaction functions are of general form and the finite difference systems are for both time‐dependent and steady‐state problems. For each problem a unified system of nonlinear equations is treated by the method of upper and lower solutions and its associated monotone iterations. This method leads to a monotone iterative scheme for the computation of numerical solutions as well as an existence‐comparison theorem for the corresponding finite difference system. Special attention is given to the dynamical property of the time‐dependent solution in relation to the steady‐state solutions. Application is given to a heat‐conduction problem where a nonlinear radiation boundary condition obeying the Boltzmann law of cooling is considered. This application demonstrates a bifurcation property of two steady‐state solutions, and determines the dynamic behavior of the time‐dependent solution. Numerical results for the heat‐conduction problem, including a test problem with known analytical solution, are presented to illustrate the various theoretical conclusions. © 1995 John Wiley & Sons, Inc.

Monotone iterative methods for a general class of discrete boundary value problems

In this paper we shall offer comparison results as well as monotone iterative schemes for the construction of solutions to a very general class of discrete boundary value problems. The discrete system we consider includes in particular the n th order prototype systems, finite as well as infinite discrete delay equations, and discrete integral equations. Further, the boundary conditions we consider include the initial, terminal, periodic and transport type problems. Numerical examples illustrating the usefulness of the proposed schemes to a variety of boundary value problems are also included.

Finite difference approximations to time-dependent multigroup transport equations in slab geometry

Abstract In this article, we consider a monotone iterative scheme to find the solution of a system of finite difference equations approximation to the time-dependent multigroup transport equations in slab geometry. The iterative scheme, coupled with the upper and lower solutions generates two sequences which converge monotonically to the solution of the finite difference equations. It is shown that this monotone convergence leads to the convergence of the solutions of finite difference equations to the solutions of the continuous equations as the mesh length of the space, angle and time variable approaches zero. Numerical results are given for the time-dependent two group transport equations.

Existence of Weak Solutions for Semilinear Elliptic-Parabolic Equations by a Monotone Iteration Scheme

Existence of Weak Solutions for Semilinear Elliptic-Parabolic Equations by a Monotone Iteration Scheme

Monotone iterations for numerical solutions of nonlinear elliptic partial differential equations

Employing a weighted difference scheme and the method of upper-lower solutions, we develop a monotone iterative scheme for the computation of solutions of nonlinear elliptic partial differential equations. In each iteration, we need only solve a linear system of which the coefficient matrix is an M-matrix. Generally, this matrix is nonsymmetric in contrast to the midly nonlinear problems. However, the squared conjugate gradient (CGS) method together with the diagonal preconditioning technique are proven to be very effective. Numerical results that are given for the nonlinear Burger's equation clearly demonstrate the advantages of our method.

Variational approach for competitive systems with obstacles

We prove existence and uniqueness results for competitive systems consisting of two parabolic equations supplemented with positivity constraints. The coupling terms include local and nonlocal interaction terms. Monotone iteration schemes are used to obtain solutions in a variational approach. The solutions of the system are compared with the solution of a simplified ordinary differential system.

Approximating optimal controls for elliptic obstacle problem by monotone iteration schemes

Approximating optimal controls for elliptic obstacle problem by monotone iteration schemes

An Enclosing Theorem and a Monotone Iterative Scheme for Elliptic Systems Having Nonmonotone Nonlinearities

AbstractThis paper presents an existence and enclosure result for solutions of semilinear elliptic systems in case that the nonlinearities do not possess any monotonicity properties. Moreover, it is shown that even in this case one is able to establish a monotone iterative scheme for constructing the solution, provided that a certain matrix, which is related to the spectrum of an associated linear differential operator, is positive definite.

The method of upper, lower solutions and monotone iterative scheme for higher order hyperbolic partial differential equations

AbstractUniformly monotone convergent iterative methods for obtaining multiple solutions of (n + m)th order hyperbolic partial differential equations together with initial conditions are discussed. Appropriate partial differential inequalities which connect upper and lower solutions, and variation of parameters formula is developed.

A monotone iterative scheme for nonlinear reaction-diffusion systems having nonmonotone reaction terms

A monotone scheme is proposed for the solution of weakly coupled systems of reaction-diffusion equations without any monotonicity property of the nonlinear reaction terms. The considerations are taken within the framework of weak solutions.

Monotone techniques for first order boundary value problem

In this paper, we discuss the application of monotone iterative techniques and upper-lower solution schemes to the nonlinear first-order periodic boundary value problem in a real Banach space. Monotone sequences are constructed which converge to extremal solutions of *.

Existence of coupled quasi-solutions of systems of nonlinear reaction-diffusion equations

Systems of nonlinear parabolic initial boundary value problems arise in many applications, such as epidemies, ecology, biochemistry, biology, chemical and nuclear engineering. Constructive methods of proving existence results for such problems, which can also provide numerical procedures for the computation of solutions, are of greater value than theoretical existence results. The method of upper and lower solutions coupled with monotone iterative technique has been employed successfully to prove existence of multiple solutions of nonlinear reaction-diffusion equations, in special cases, by various authors [335, 10, 11, 15, 181. Recently, in [6, 171 weakly coupled systems of reaction-diffusion equations, when the nonlinear terms are independent of gradient terms, are discussed and some special type of results are obtained. We, in this paper, investigate general systems of nonlinear reaction-diffusion problems when the nonlinear terms possess a mixed quasi-monotone property. We discuss a very general situation and obtain coupled extremal quasi-solutions, which in special cases reduce to minimal and maximal solutions. We shall also indicate how one-step cyclic monotone iterative schemes can be generated which yield accelerated rate of convergence of iterates. This work is in the spirit of our recent paper [12] for elliptic systems.

Monotone method and periodic solution of non linear parabolic boundary value problem for systems

A system of parabolic equations is considered: Lui = uit − uix = fi (x, t, u, uix) on Q, Biui(j, t) = ωij(t), t ∈ (−∞, ∞), j = 0, 1, i = 1, 2, …, n, where Bi is one of the boundary operators Biui = ui or Biui = ∂ui/∂v + βi(x, t)ui, x = 0, 1, Ω = (0, 1), Q = Ω × R, u(=(u1, …, un)): Q → Rn, v(x) is the outward normal to the boundary ∂Ω, f, u, ω0, ω1 are n-valued functions and f, ω0, ω1 are periodic in t with period T and Bi is a positive function.The paper is classified into two parts. The first part deals with the existence and uniqueness of periodic solutions of the above system of parabolic equations. The second part deals with a monotone iterative method which develops a monotone iterative scheme for the solution of the above system of equations. In this paper we establish the existence of coupled quasi-solutions of the above equation. Also we give a monotone iterative scheme for the construction of such a solution.

On application of the monotone iteration scheme to noncoercive elliptic and hyperbolic problems

On montre que le schema d'iteration monotone peut s'appliquer avec succes aux equations autres que coercives elliptiques et hyperboliques