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Lecture

Bessel Equation: Introduction

In course

PHYS-216: Mathematical methods (for SPH)

Ce cours est un complément aux cours d'analyse et d'algèbre linéaire qui apporte des méthodes et des techniques mathématiques supplémentaires requises pour les cours de physique de 3e année, notamment l'électrodynamique et la mécanique quantique. ...

Description

This lecture introduces the Bessel equation, which is crucial in solving physical problems like wave equations in circular systems. It covers the systematic procedure for solving scalar linear 2nd order ODEs, including the Frobenius method and variation of parameters. The Fuchs Theorem and its application around regular singular points are discussed, along with the significance of Bessel functions in eigenvalue problems. Examples of eigenmodes of vibration in circular systems are presented, illustrating the practical application of the series substitution method.

Instructors (3)

Jonathan Graves

Prof. Jonathan P. Graves is a Senior Scientist at EPFL and Honorary Visiting Professor at the University of York, UK. He achieved first class joint honours in Electronic Engineering and Mathematics from the University of Nottingham, UK in 1996. He completed his Ph.D. in Theoretical Mechanics from the University of Nottingham, UK, three years later in 1999. During his Ph.D. he was based in the Culham theory group of the United Kingdom Atomic Energy Authority, developing kinetic descriptions of the internal kink instability, and participating in deuterium-tritium experimental analysis in the Joint European Torus. After a short time in industry, and a postdoc at Nottingham University, he took a position at the Swiss Plasma Center at EPFL, becoming a Senior Scientist in 2014, and became an Honorary Visiting Professor at the University of York, UK, in 2020. In 2015 he became a member of the EUROfusion Scientific and Technical Advisory Committee (STAC) and a member of the EUROfusion DEMO Technical Advisory Group. He is on the editorial board for the journal Plasma Physics and Controlled Fusion, and in 2020 became Scientific Secretary of the Varenna-Lausanne International Workshop in Theory of Fusion Plasmas.

Stephan Brunner

Sylvain Bréchet

Sylvain Bréchet was born on October 13th, 1981 in Moudon (legal origin Epesses, VD, Switzerland). He obtained a Master of Science in physics at EPFL in 2005. He went on to Cambridge for his PhD studies in theoretical cosmology from 2005 to 2009 under the supervision of Prof. Lasenby (FRS) and Prof. Michael Hobson at the Cavendish Laboratory of the University of Cambridge. He went back to EPFL where he is since 2010 university lecturer and research scientist in the Institute of Condensed Matter Physics. He taught classical mechanics, special relativity and thermodynamics to mechanical, electrical engineering students and physics students. He is currently writing a textbook in thermodynamics.

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Related concepts (22)

Wave equation

The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields - as they occur in classical physics - such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation, which is much easier to solve and also valid for inhomogeneous media.

Bessel function

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number , which represents the order of the Bessel function. Although and produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of . The most important cases are when is an integer or half-integer.

One-way wave equation

A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resulting from superposition of two waves in opposite directions. In the one-dimensional case, the one-way wave equation allows wave propagation to be calculated without the mathematical complication of solving a 2nd order differential equation.

Variation of parameters

In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations.

Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE).

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