Browse Prior Art Database

Approximate Solutions and Error Bounds for Quasilinear Hyperbolic Initial-boundary Value Problems

IP.com Disclosure Number: IPCOM000128505D
Original Publication Date: 1972-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 15 page(s) / 33K

Publishing Venue

Software Patent Institute

Related People

To-yat Cheung: AUTHOR [+3]

Abstract

Constrained minimization problems are formilated from a quasilinear hyperbolic initial-boundary value problem, making use of a convexity assumption on its nonlinear part and a minimum property of the linear hyperbolic equation. Approximate solutions and two error bounds can be obtained by solving these minimization problems by linear:programming and discretization techniques. Numerical results are presented.

This text was extracted from a PDF file.
This is the abbreviated version, containing approximately 19% of the total text.

Page 1 of 15

THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Approximate Solutions and Error Bounds for Quasilinear Hyperbolic Initial-boundary Value Problems*

To-yat Cheung

Technical Report 72-1 January 1972

*This research was supported in part by the National Science Foundation under Grant GJ 0362.

ABSTRACT

Constrained minimization problems are formilated from a quasilinear hyperbolic initial-boundary value problem, making use of a convexity assumption on its nonlinear part and a minimum property of the linear hyperbolic equation. Approximate solutions and two error bounds can be obtained by solving these minimization problems by linear:programming and discretization techniques. Numerical results are presented.

1. INTRODUCTION

This paper is concerned with the numerical determination of approximate S.Olutious,and error bounds by linear programming for the following quasilinear hyperbolic initial-boundary value problem*.

(Equation Omitted)

is a uniformly hyperbolic

(Equation Omitted)

operator and

(Equation Omitted)

Let R 2 be an open domain in E 2 , bounded by the segments FA on the positive

(Equation Omitted)

on the positive y-axis and the characteristic curve 'E (defined by

(Equation Omitted)

Then the above regions R and R are defined as:

(Equation Omitted)

Let R 3 {A) and R Ro U R 2 V RV It is assumed that

University of Minnesota Page 1 Dec 31, 1972

Page 2 of 15

Approximate Solutions and Error Bounds for Quasilinear Hyperbolic Initial-boundary Value Problems

(Equation Omitted)

and

(Equation Omitted)

In Section 2, some preliminary material is presented. In Section 3, two error bounds for a given approximate solution to Q(g) are derived, making use of. a minimum principle of the linear case of Q(g) and some convexity assumption on g(x,y,u). The error bounds depend on the absolute defects of the operator equations. In Section 4, by mems 6f these error bounds, theoretical constrained minimization problems are,formulated by which we can determine approximate solutions and error bounds. In Section 5, computational schemes making use of linear programming are suggested to solve these problems. Section 6 presents some numerical results. By similar approach, Rosen [19701 approximated the quasilinear first elliptic boundary value problem, and Cheung (1971 a,b] approximated the quasilinear first and third elliptic boundary value problems and parabolic boundary value problems with both linear and quasilinear boundary conditions.

2. A MINIMUM PRINCIPLE

We first consider the linear case of

(Equation Omitted)

In terms of the coefficients of L , we define an operator 1.1 2 ax and two expressions

(Equation Omitted)

Let

(Equation Omitted)

The following minimum principle is given by D. Sather in his Theorem V (1966]:

Lemma.1 For the,problem

(Equation Omitted)

Let u e V(R) satisfy

(Equation Omitted)

Then, if the minimum of u in R is nonpositive, it can only be attained in

This lemma immediately implies the following inverse-positive property o...