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

A method to numerically solve every differential analytical model

Giacomo Lorenzoni

Summary

A differential analytical model is a system of so many PDEs in the same (or lesser) number of unknown functions where each PDE is of any order and can be nonlinear. Usually the operative interest of such a model is to determine an its solution when is subject to additional conditions like those boundary or initial.
This work exposes the mathematical basis of a program (freeware in http:// www.giacomo.lorenzoni.name/peei/ ) to numerically solve every differential analytical model with every set of additional conditions. In particular is exposed what follows.
Are described the analytical properties of two well known models to approximate a function: the interpolating polynomial and the cubic spline. The values of a natural cubic spline and of its derivatives, in the interpolation nodes, are expressed as linear combinations of the known values of the function to be interpolated and whose coefficients depend only on the nodes. Are obtained new bounds for the errors of a cubic spline. Are presented essential aspects of a curve in a multidimensional Euclidean space, in order to obtain an upper bound for the absolute maximum value of a derivative defined on a curve. Is shown the expression of a partial derivative as a linear combination of directional derivatives and is deduced its optimal approximation. Is formulated the expression of the generic differential analytical model, is identified the main impediment to knowledge of an its exact solution in not knowing its partial derivatives, is circumstantiated the context of information contingently available and is showed how, solving an inherent system of nonlinear equations, can be calculated an its numerical solution. Is exposed an original algorithm that, in this system of nonlinear equations, expresses a derivative as a linear combination of unknowns.

Keywords: numerical solution of PDEs systems, differential analytical models, numerical differentiation, splines, graph algorithms, interpolation.

INDEX

1  INTRODUCTION

2  TWO MODELS FOR APPROXIMATE A FUNCTION: THE INTERPOLATING POLYNOMIAL AND THE CUBIC SPLINE.

2.1  The interpolating polynomial

2.2  The cubic spline

2.3  New bounds for the errors of a cubic spline

3  THE MAXIMUM ABSOLUTE VALUE OF A DERIVATIVE DEFINED ON A CURVE OF THE MULTIDIMENSIONAL EUCLIDEAN SPACE

4  THE APPROXIMATION OF A LINEAR COMBINATION OF DIRECTIONAL DERIVATIVES THAT EXPRESSES A PARTIAL DERIVATIVE AT A INTERSECTION OF SOME CURVES

5  THE FORMULATION OF A DIFFERENTIAL ANALYTICAL MODEL AND ITS NUMERICAL SOLUTION AS THE UNKNOWNS OF A TOTAL SYSTEM

6  THE APPROXIMATION OF A DERIVATIVE OF THE TOTAL SYSTEM WITH A LINEAR COMBINATION OF LOCAL VALUES OF THE FUNCTION TO BE DERIVED

6.1  The set of rectilinear segments

6.2  An original algorithm that expresses, by means of a tree graph, the linear combination that approximates a derivative of the total system.

CONCLUSION

REFERENCES

Date of release:  Wednesday 11 February 2015

Language: English

Number of downloads: 146

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

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