Generalizing Bagarello's operator approach to solve a class of partial differential equations

TL;DR

This paper extends Bagarello's non-commutative operator method from ordinary differential equations to solve certain classes of partial differential equations, including evolution and Navier-Stokes equations, using operator theory and series solutions.

Contribution

It introduces a novel extension of Bagarello's operator approach to PDEs, providing a new analytical framework for solving complex evolution equations.

Findings

Successfully applied to evolution equations and Navier-Stokes equations

Demonstrated effectiveness through numerous examples

Provided explicit series solutions using operator methods

Abstract

The non-commutative strategy developed by Bagarello (see Int. Jour. of Theoretical Physics, 43, issue 12 (2004), p. 2371 - 2394) for the analysis of systems of ordinary differential equations (ODEs) is extended to a class of partial differential equations (PDEs), namely evolution equations and Navier-Stokes equations. Systems of PDEs are solved using an unbounded self-adjoint, densely defined, Hamiltonian operator and a recursion relation which provides a multiple commutator and a power series solution. Numerous examples are given in this work.