Lecture

Frobenius Method: General Case of Regular Singular Point

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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 presents a constructive proof of Fuchs' Theorem, focusing on applying the series substitution method to the general case of an ordinary or regular singular point xo. It explores scenarios at xo, considering a 2nd order ODE with power series expressions for p(x) and q(x). The lecture delves into the nature of roots S1 and S2, the indicial equation, and deriving the second solution in different cases. It illustrates the method through examples and emphasizes the importance of ordinary points. The Frobenius series solutions and their properties are thoroughly discussed, providing insights into linearly independent solutions and the Wronskian. The lecture concludes by showcasing the application of the method to the harmonic oscillator ODE.

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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

Teaching & PhD PhD Students Hugo Jean Elie Chkroun, Nicole Vadot, Oraz Anuaruly, Haomin Sun, Isabel Muzio, Fabien Nicolas Jeanquartier Past EPFL PhD Students Ajay Chandrarajan Jayalekshmi, Arnas Volcokas Past EPFL PhD Students as codirector Xavier Lapillonne, Thibaut Vernay, Julien Stanislas Pierre Dominski, Gabriele Merlo, Emmanuel Lanti, Noé Ohana, Moahan Murugappan Courses Plasma instabilities PHYS-736 To complete the theoretical knowledge acquired before the graduate studies.

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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Mathematics

Algebra: Linear algebra

Related concepts (29)

Linear independence

In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple since it makes all three equations valid. The word "system" indicates that the equations should be considered collectively, rather than individually.

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Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.

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Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a .

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