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Lecture

Sturm-Liouville Problem: 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 Sturm-Liouville problem, focusing on linear 2nd order ODEs with boundary value problems. It covers the general solution to ODEs, the existence and uniqueness of solutions, types of boundary conditions, and the Sturm-Liouville eigenvalue problem. The lecture also discusses the weight function, eigenvalues, and eigenfunctions associated with the problem.

Instructors (3)

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.

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.

Official source

Ontological neighbourhood

Mathematics

Discrete mathematics: Discrete exterior calculus

Related concepts (22)

Boundary value problem

In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems.

Cauchy boundary condition

In mathematics, a Cauchy (koʃi) boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition. It is named after the prolific 19th-century French mathematical analyst Augustin-Louis Cauchy.

Neumann boundary condition

In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative applied at the boundary of the domain. It is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution itself (as opposed to its derivative) on the boundary, whereas the Cauchy boundary condition, mixed boundary condition and Robin boundary condition are all different types of combinations of the Neumann and Dirichlet boundary conditions.

Dirichlet boundary condition

In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.

Robin boundary condition

In mathematics, the Robin boundary condition (ˈrɒbɪn; properly ʁɔbɛ̃), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition.

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