Symbolic analysis of second-order ordinary differential equations with polynomial coefficients

Abstract

The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. It is shown that the $\theta$-operator method can be used as a symbolic computational approach to obtain the indicial equation and the recurrence relation. Consequently, the singularity structure leads to the transformations that yield a solution in terms of a special function, if the equation is suitable. Hypergeometric and Heun-type equations are mostly employed in physical applications. Thus only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.

16 pages, 2 figures. Matches the published version. The codes, a sample worksheet, and the user manual can be obtained from https://github.com/tbirkandan/symODE2

References in corpus (8)

• Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces
• Dirac Equation in the Background of the Nutku Helicoid Metric
• Tensor calculus with open-source software: the SageManifolds project
• Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
• Slowly Rotating Homogeneous Stars and the Heun Equation
• An Analysis of the Wave Equation for the $U(1{)}^2$ Gauged Supergravity Black Hole
• Addentum to: Conformal Invariance and Near-extreme Rotating AdS Black Holes
• Computations of general Heun functions from their integral series representations

Cited by in corpus (2)

• Solutions of a charged scalar field in five-dimensional helicoid solution with electromagnetic field
• Analysis of scalar fields with series convolution