Initial-Value Technique for Singularly-Perturbed Turning-Point Problems Exhibiting Twin Boundary Layers

The initial-value technique that was originally developed for solving singularly-perturbed nonturning-point problems (Ref. 1) is used here to solve singularly-perturbed turning-point problems exhibiting twin boundary layers. In this method, the required approximate solution is obtained by combining solutions of the reduced problem, an initial-value problem, and a terminal-value problem. Error estimates for approximate solutions are obtained. The implementation of the method on parallel architectures is discussed. Numerical examples are presented to illustrate the present technique.     

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.     

New initial-value method for singularly perturbed boundary-value problems

An initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint. The idea is to replace the original two-point boundary value problem by two suitable initial-value problems. The method is very easy to use and to implement. Nontrivial text problems are used to show the feasibility of the given method, its versatility, and its performance in solving linear and nonlinear singularly perturbed problems.This work was supported in part by the Consiglio Nazionale delle Ricerche, Contract No. 86.02108.01, and in part by the Ministero della Pubblica Istruzione.     

Helly’s Principle and Its Application to an Infinite-Horizon Optimal Control Problem

In this paper, a numerical method is presented to solve singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a discontinuous source term. First, an asymptotic expansion approximation of the solution of the boundary-value problem is constructed using the basic ideas of the well-known WKB perturbation method. Then, some initial-value problems and terminal-value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial-value problems and terminal-value problems are singularly-perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples are provided to illustrate the method.     

A proof of the Pontryagin maximum principle for initial-value problems

Many optimization problems in economic analysis, when cast as optimal control problems, are initial-value problems, not two-point boundary-value problems. While the proof of Pontryagin (Ref. 1) is valid also for initial-value problems, it is desirable to present the potential practitioner with a simple proof specially constructed for initial-value problems. This paper proves the Pontryagin maximum principle for an initial-value problem with bounded controls, using a construction in which all comparison controls remain feasible. The continuity of the Hamiltonian is an immediate corollary. The same construction is also shown to produce the maximum principle for the problem of Bolza.     

Initial-value methods for second-order singularly perturbed boundary-value problems

In this paper, we present a numerical method for solving linear and nonlinear second-order singularly perturbed boundary-value-problems. For linear problems, the method comes from the well-known WKB method. The required approximate solution is obtained by solving the reduced problem and one or two suitable initial-value problems, directly deduced from the given problem. For nonlinear problems, the quasilinearization method is applied. Numerical results are given showing the accuracy and feasibility of the proposed method.This work was supported in part by the Consiglio Nazionale delle Ricerche (Contract No. 86.02108.01 and Progetto Finalizzatto Sistemi Informatia e Calcolo Paralello, Sottoprogetto 1), and in part by the Ministero della Pubblica Istruzione, Rome, Italy.     

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

Symmetry analysis of initial-value problems

Symmetry analysis is a powerful tool that enables the user to construct exact solutions of a given differential equation in a fairly systematic way. For this reason, the Lie point symmetry groups of most well-known differential equations have been catalogued. It is widely believed that the set of symmetries of an initial-value problem (or boundary-value problem) is a subset of the set of symmetries of the differential equation. The current paper demonstrates that this is untrue; indeed, an initial-value problem may have no symmetries in common with the underlying differential equation. The paper also introduces a constructive method for obtaining symmetries of a particular class of initial-value problems.     

Initial-Value Technique for Singularly Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations Arising in Chemical Reactor Theory

An initial-value technique is presented for solving singularly perturbed two-point boundary-value problems for linear and semilinear second-order ordinary differential equations arising in chemical reactor theory. In this technique, the required approximate solution is obtained by combining solutions of two terminal-value problems and one initial-value problem which are obtained from the original boundary-value problem through asymptotic expansion procedures. Error estimates for approximate solutions are obtained. Numerical examples are presented to illustrate the present technique.     

Some relationships between Green's functions and invariant imbedding

The method of invariant imbedding has been used to resolve the solution of linear two-point boundary-value problems into contributions associated with the homogeneous equation with homogeneous boundary conditions, with inhomogeneous boundary conditions, and with an inhomogeneous source term in the equation. The relationship between the Green's function and the invariant imbedding equations is described, and it is shown that the Green's function can be determined from an initial-value problem. Several numerical examples are given which illustrate the efficacy of the initial-value algorithm.This work was supported by the US Atomic Energy Commission.     

On the complexity of parabolic initial-value problems with variable drift

We study the intrinsic difficulty of solving linear parabolic initial-value problems numerically at a single point. We present a worst-case analysis for deterministic as well as for randomized (or Monte Carlo) algorithms, assuming that the drift coefficients and the potential vary in given function spaces. We use fundamental solutions (parametrix method) for equations with unbounded coefficients to relate the initial-value problem to multivariate integration and weighted approximation problems. Hereby we derive lower and upper bounds for the minimal errors. The upper bounds are achieved by algorithms that use Smolyak formulas and, in the randomized case, variance reduction. We apply our general results to equations with coefficients from Hölder classes, and here, in many cases, the upper and lower bounds almost coincide and our algorithms are almost optimal.     

Invariant imbedding and optimal control theory

A method for directly converting an optimal control problem to a Cauchy problem is presented. No use is made of the Euler equations, Pontryagin's maximum principle, or dynamic programming in the derivation. The initial-value problem, in addition to being desirable from the computational point of view, possesses stable characteristics. The results are directly applicable in the study of guidance and control and are particularly useful for obtaining numerical solutions to control problems.     

Solving Volterra Integral Equations with ODE Codes

Some nonlinear Volterra integral equations are equivalent toan initial-value problem for a system of ordinary differentialequations (ODEs). Because effective ODE codes are widely available,some authors have sought to exploit this connection for thenumerical solution of the integral equations. There are twomajor difficulties: One would like to solve large systems ofODEs. The initial-value problem for the ODEs may be stiff. Theinitial-value problems are shown to have a remarkable structurewhich can be exploited to overcome these two difficulties.     

微分方程初值问题泰勒级数法的实现

提出了用微分变换来求解常微分方程初值问题的一个方法,该方法能通过迭代获得问题解析解的高阶Taylor级数的展开式,从而实现了高阶泰勒级数方法.     

Numerical solution of second-order nonlinear singularly perturbed boundary-value problems by initial-value methods

Nonlinear singularly perturbed boundary-value problems are considered, with one or two boundary layers but no turning points. The theory of differential inequalities is used to obtain a numerical procedure for quasilinear and semilinear problems. The required solution is approximated by combining the solutions of suitable auxiliary initial-value problems easily deduced from the given problem. From the numerical results, the method seems accurate and solutions to problems with extremely thin layers can be obtained at reasonable cost.This work was supported by CNR, Rome, Italy (Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo, Sottoprogetto 1).     

A numerical ODE solver that preserves the fixed points and their stability

In this paper, we provide a one-step predictor-corrector method for numerically solving first-order differential initial-value problems with two fixed points. The method preserves the stability behaviour of the fixed points, which results in an efficient integrator for this kind of problem. Some numerical examples are provided to show the good performance of the method.     

Initial-value technique for a class of nonlinear singular perturbation problems

An initial-value technique, which is simple to use and easy to implement, is presented for a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. It is distinguished by the following fact: The original second-order problem is replaced by an asymptotically equivalent first-order problem and is solved as an initial-value problem. Numerical experience with several examples is described.     

The b and h functions for integral equations with displacement kernels: A computational method and an application to radiative transfer

Many problems in the theory of radiative transfer reduce to the solution of Fredholm integral equations with displacement kernels. Frequently, we are interested in the solutions of the Fredholm integral equations as well as certain functionals on the solution (reflection and transmission coefficients, etc.). Earlier it was shown that these functionals can be expressed algebraically in terms of the basic functions b and h. Normally, these functions are computed as solutions of an initial-value problem. Since they represent internal interactions due to isotropic illuminations, they are also solutions of a linear two-point boundary-value problem, which, unfortunately, s unstable. The purpose of this paper is to show that this unstable problem can be solved using a Gram-Schmidt orthogonalization scheme. This is demonstrated by making comparisons against earlier calculations using the initial-value method. In addition, the process is ideally suited to take advantage of multitasking on parallel processors.     

Resonance elliptic variational inequalities with discontinuous nonlinearities of linear growth

We consider resonance elliptic variational inequalities with second-order differential operators and discontinuous nonlinearities of linear growth. The theorem on existence of a strong solution is proved. The initial-value problem is reduced to the problem of existence of a fixed point for a compact multivalued mapping and then the existence of this point is established by the Leray–Schauder method.     

An inexact Newton method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is an application of Newton's method to a nonlinear differential operator equation. Since the linear boundary-value problem to be solved at each iteration must be discretized, it is natural to consider quasilinearization in the framework of an inexact Newton method. More importantly, each linear problem is only a local model of the nonlinear problem, and so it is inefficient to try to solve the linear problems to full accuracy. Conditions on size of the relative residual of the linear differential equation can then be specified to guarantee rapid local convergence to the solution of the nonlinear continuous problem. If initial-value techniques are used to solve the linear boundary-value problems, then an integration step selection scheme is proposed so that the residual criteria are satisfied by the approximate solutions. Numerical results are presented that demonstrate substantial computational savings by this type of economizing on the intermediate problems.This work was supported in part by DOE Contract DE-AS05-82-ER13016 and NSF Grant RII-89-17691 and was part of the author's doctoral thesis at Rice University. It is a pleasure to thank the author's thesis advisors, Professor R. A. Tapia and Professor J. E. Dennis, Jr.