2nd order PDEs: geometric and functional considerations

TL;DR

This paper explores second-order linear PDEs with real characteristics, emphasizing geometric and invariant-based approaches, and reformulates classical theory using differential operators and coordinate transformations.

Contribution

It introduces a formalized, invariant-based framework for analyzing second-order PDEs, connecting geometric invariants with operator properties and coordinate transformations.

Findings

Invariant properties of differential operators analyzed.

Coordinate transformations expressed via invariants.

Examples illustrating different solution methods provided.

Abstract

INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the applied physicist, but with a weakness for formalization: look inside the black box of the formulas, try to compact them (for example, proceeding from an inverse transformation of coordinates) and make them smart (in the context, reformulating the theory by means of differential operators and related invariants), applying them with awareness and then connecting them to geometry or to spatial categories, which are in mathematics what is closest to the sensible reality. Finally, proposing examples that are exercise and corroborating for theory. TOPICS The geometric meaning of invariant to a differential operator. Operator Principal Part and its factorization: commutativity and product with and without residues(first order terms). Related conditions by operators and invariants derivatives. Coordinate transformation by invariants and expression of the hyperbolic and parabolic operators in the new coordinates. Properties of the Jacobian Matrix and relations between invariants derivatives and inverse coordinates transformation or the initial variables derivatives. Commutativity conditions and product without residues in terms of inverse coordinate transformations that allow to build commutative differential operators or whose product is without residues (or both). Diffeomorphisms and plane transformations: new operators and invariants in the new coordinate space which lead to the chain rule in compact form. Conclusive considerations and examples who compares different methods of solution.