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Interactive Graphical Solution of Boundary Value Problem Using Linear Programing

IP.com Disclosure Number: IPCOM000128502D
Original Publication Date: 1971-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 8 page(s) / 26K

Publishing Venue

Software Patent Institute

Related People

J. B. Rosen: AUTHOR [+3]

Abstract

The solution of generalized approximation problems by the use of mathematical programming has been the subject of a number of papers recently (for an early discussion see 141). In order to permit a user to formulate and solve a problem rapidly, an interactive graphical system has been developed for solving such problem. This system can be used to approximate discrete data or functions in one or two variables and to approximate the solution to linear ordinary or partial differential equation boundary value problems [5,9,61. Auxiliary conditions on the approximation may also be imposed. For two-dimensional boundary value problems an approximate solution is obtained in terms of a convenient spline basis using linear programming to minimize a weighted sum of the interior and boundary errors in the maximum norm.. A specific problem is formulated using an interactive graphics terminal, by means of function keys and a light pen, and taking advantage of symbolic notation where possible. The solution to the original problem together with certain error information is then obtained and may be graphically dis played in various forms allowing the user to modify his formulation on line. The formulation and method are discussed in the next Section, and error estimates are obtained in Section 3. The system and its use are briefly described and Illustrated by an example in Sections 4 & 5.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Interactive Graphical Solution of Boundary Value Problem Using Linear Programing*

by J. B. Rosen Technical Report 71-2 December 1971

*To be published in Proceedings of 4 th IFIP Colloquium on Optimization Techniques, (A. V. Balakrishnan, Ed.), Academic Press.,1972.

1. Introduction

The solution of generalized approximation problems by the use of mathematical programming has been the subject of a number of papers recently (for an early discussion see 141). In order to permit a user to formulate and solve a problem rapidly, an interactive graphical system has been developed for solving such problem. This system can be used to approximate discrete data or functions in one or two variables and to approximate the solution to linear ordinary or partial differential equation boundary value problems [5,9,61. Auxiliary conditions on the approximation may also be imposed. For two-dimensional boundary value problems an approximate solution is obtained in terms of a convenient spline basis using linear programming to minimize a weighted sum of the interior and boundary errors in the maximum norm.. A specific problem is formulated using an interactive graphics terminal, by means of function keys and a light pen, and taking advantage of symbolic notation where possible. The solution to the original problem together with certain error information is then obtained and may be graphically dis played in various forms allowing the user to modify his formulation on line.

The formulation and method are discussed in the next Section, and error estimates are obtained in Section 3. The system and its use are briefly described and Illustrated by an example in Sections 4 & 5.

2. Formulation and Approximation Method

We consider a general class of linear boundary value problems with sufficiently smooth solutions. Specifically, let A denote a bounded domain in Rn with boundary 4, and let Ltul denote the linear-differential operator of order 2m, defined by

*This research was supported in part by the National Science Foundation under Grant GJ Q362.

-2-

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where the coefficients a 0 (x) are bounded on the closure Similarly, let B [u] denote a linear boundary differential operator of order

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University of Minnesota Page 1 Dec 31, 1971

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Interactive Graphical Solution of Boundary Value Problem Using Linear Programing

, is ... a where the coefficients of B are bounded on an (see [33). The boundary problem considered may be stated in the form

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where f and the g i are defined on 9 and an, respectively. We assume that f and the g i are sufficiently smooth (together with other required assumptions on the domain a and the operators B and L) to insure the existence of a unique

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In the Sobelev space

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. For details of notation see, for example, Chap. 3 of (11]. PC We generate a finite-dimensional subspAce of W do 0)...