Exact solutions of Mathieu’s equation

• Published in: Progress of theoretical and experimental physics Vol. 2020; no. 4
• Main Author: Daniel, Derek J.
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01.04.2020
• Subjects: Charged particles; Eigenvalues; Electric fields; Helmholtz equations; Ordinary differential equations; A10; A02; A13; A61; A64
• ISSN: 2050-3911; 2050-3911
• DOI: 10.1093/ptep/ptaa024

Mathieu’s equation originally emerged while studying vibrations on an elliptical drumhead, so naturally, being a linear second-order ordinary differential equation with a Cosine periodic potential, it has many useful applications in theoretical and experimental physics. Unfortunately, there exists no closed-form analytic solution of Mathieu’s equation, so that future studies and applications of this equation, as evidenced in the literature, are inevitably fraught by numerical approximation schemes and nonlinear analysis of so-called stability charts. The present research work, therefore, avoids such analyses by making exceptional use of Laurent series expansions and four-term recurrence relations. Unexpectedly, this approach has uncovered two linearly independent solutions to Mathie’s equation, each of which is in closed form. An exact and general analytic solution to Mathieu’s equation, then, follows in the usual way of an appropriate linear combination of the two linearly independent solutions.