NUMERICAL METHOD TO SELECT AN ANALYTICAL POLYNOMIAL SOLUTION FOR LINEAR OR NONLINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS OF SECOND ORDER

July 15, 2021

The aim of this book is to consistently solve a basic type of equation - linear or nonlinear - presenting in detail the fundamental relations. It is desired that, using the "single cell" method, any graduate of a mathematics or engineering faculty be able to programme the solving of these basic PDEs. The types of PDEs analyzed are deliberately limited. They have not been diversified because this would have made difficult to follow the solving procedure, which would be drastically augmented.

The first chapters set out the mathematical procedure used to obtain an analytical solution of an elliptic PDE and criteria for evaluating the accuracy of the results. To complete the procedure it is useful to establish a methodology, based on which the user can decide whether the solution can be accepted or has to be rejected. To solve this problem the author was inspired by the concept of the "allowable stress" used in the strength of the materials. When had to decide if a machine part executed from a known material can be safely used or not, the engineer had to answer to two challenges: a mathematical part to calculate the maximum stress (sometimes found by solving an elliptic PDEs), which is usually a "time consuming" activity and a decision part to verify if the maximum stress is "less than or at most equal" to a value selected based on the experience of a large number of engineers, designated as "the allowable stress". This decision, based on a single condition (therefore not "time consuming") is not always easy to be taken, due to the fact that the "allowable stress", which can be found in specialty books or building codes, is given with different value ranges.