http://orcid.org/0000-0002-2329-2881

A method for the numerical resolution of differential analytical models and the program PEEI that computerizes it.

Giacomo Lorenzoni

Summary

Are exposed the mathematical bases of a computer program to numerically solve differential analytical models, i.e. systems of partial differential equations. Are described the analytical properties of two well known models to approximate functions of a variable (the interpolating polynomial and the cubic spline) and new upper bounds for the errors of the second are obtained. Are presented essential aspects of a curve in the multidimensional Euclidean space, in order to define the directional derivative of a function and to obtain an upper bound for the absolute maximum value of a derivative definite on a curve. Is shown, for a point where intersect more curves, the expression of a partial derivative as a linear combination of directional derivatives, and of it is deduced the optimum approximation when the values of these are approximate. Is formulated the expression of a generic differential analytical model, highlighting in detail its arguments, identifying the principal impediment to the knowledge of an its exact solution in not knowing its partial derivatives, circumstantiating the context of information contingently available, and showing how it can be calculated an its numerical solution solving an inherent system of not differential equations. Is exposed in detail the algorithm that expresses a derivative of this system as a linear combination of his variables. Finally it is introduced, describing his use characteristics and essential actions, the program PEEI that was realized, on the basis of the previous treatment, to calculate numerical solutions of differential analytical models.

Keywords: differential analytical models, numerical solution, systems of partial differential equations.

INDEX

1  INTRODUCTION

2  PRELIMINARY COGNITIONS.

2.1  Two models to approximate functions of a variable: the interpolating polynomial and the cubic spline.

2.1.1  The interpolating polynomial.

2.1.2  The cubic spline.

2.1.2.1  Other upper bounds for the errors of the cubic spline.

2.2  A curve in the multidimensional Euclidean space. The directional derivative. The absolute maximum value of a derivative definite

on a curve.

2.3  The approximation of a linear combination of directional derivatives that expresses a partial derivative in a point where intersect

more curves.

3  THE FORMULATION OF A DIFFERENTIAL ANALYTICAL MODEL, AND HIS NUMERICAL SOLUTION AS THE VARIABLES

OF A TOTAL SYSTEM.

4  THE APPROXIMATION OF A DERIVATIVE OF THE TOTAL SYSTEM WITH A LINEAR COMBINATION OF LOCAL VALUES

OF THE FUNCTION.

4.1  The set of rectilinear segments.

4.2  The expression, for means of a tree graph, of the linear combination that approximates a derivative of total system.

5  PEEI: A COMPUTER PROGRAM FOR THE NUMERIC RESOLUTION OF DIFFERENTIAL ANALYTICAL MODELS.

CONCLUSIONS

BIBLIOGRAPHY

Date of release:  Tuesday 27 May 2008

Language: italian

Number of downloads: 2545

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Related link: 

 A method to numerically solve every differential analytical model (English paper)

 PEEI: a computer program for the numerical solution of systems of partial differential equations.