On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

Algorithms

AndrewR. Cameron¹  KyleR. Bryenton²  Patrick Strongman²  Nikita Volodin²  KeeganL. A. Kirk²  Nasser Saad² 

[1] Department of Physics, University of Prince Edward Island, Charlottetown, PE C1A 4P3, Canada;School of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, PE C1A 4P3, Canada;

关键词: symbolic computation;    algorithm;    polynomial solutions;    Scheffé criteria;    Heun equation;    Dirac equation;   

DOI  :  10.3390/a13110286

来源: DOAJ

【 摘 要 】

The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n, $n-1$, and $n-2$ respectively. We show that for $n\ge 3$ the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of $n=4$. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations.

【 授权许可】

Unknown