Numerical verification of equations in applied mechanics: comments on the inexpensive alternative to computer algebra

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2018
• Θέματα: Numerical verification; Verification of equations; Hypotheses and conclusions; Computer algebra; Gröbner bases; Applied mechanics; Elasticity; Compatibility equations; Movement of a particle; Slider-crank mechanism; Parallel numerical method; Maple; Αριθμητική επαλήθευση; Επαλήθευση εξισώσεων; Υποθέσεις και συμπεράσματα; Υπολογιστική άλγεβρα; Βάσεις Gröbner; Εφαρμοσμένη μηχανική; Ελαστικότητα; Εξισώσεις συμβιβαστού; Κίνηση υλικού σημείου; Μηχανισμός διωστήρα–στροφάλου; Παράλληλη αριθμητική μέθοδος

Computer algebra methods play a continually increasing rôle in the proof of equations and theorems. Gröbner bases and characteristic sets have been extensively used in this task. Here we attempt a critical view of this approach, which is frequently extremely computer-memory- and time-consuming. In fact, we suggest the direct verification of our conclusions on the basis of the existing `hypotheses' numerically and not algebraically. This approach can be incorporated into `tomorrow's semi-rigorous mathematical culture' commented by Zeilberger although a strictly rigorous related approach can also be used on the basis of the parallel numerical method. Several examples from applied mechanics (including compatibility equations in plane elasticity, the classical Newton–Kepler and related movement problems for a particle and problems during movements in mechanisms) illustrate this extremely elementary approach and its advantages over computer algebra methods in the applied mechanics environment. The case of differential polynomials constitutes a standard part of the present method.