A ghost perturbation scheme to solve ordinary differential equations

We propose an algebraic method to systematically approach the solution of an ordinary differential equation (ODE) with any boundary conditions. We define an extended ODE (eODE) composed of a linear generic differential operator that depends on free parameters plus an ε formal perturbation formed by the original ODE minus the same linear term. After an eODE’s formal ε expansion, we can solve order by order a hierarchy of linear ODEs. We get a sequence of functions that converge exponentially fast to the solution/s when ε = 1 and after determining the free set of parameters by minimising a distance-to-the solution function. Therefore, we get a formal expansion of the solution that we call Ghost Expansion that can be used as a multiscaling decomposition of the ODE’s solution. The method permits the detection of several solutions to Boundary Value Problems just by looking at the number of minima of the distance function. We present the method by its application to several cases where we discuss its properties, benefits and shortcomings and some practical algorithmic improvements on it.

Ponente: Pedro Garrido. Universidad de Granada.

Fecha y hora: miércoles, 1 de junio de 2022 a las 11:00. 

Lugar: Seminario de Física Computacional, planta baja del edificio de Física (junto a las pantallas). Facultad de Ciencias.