Code
Φ-212
Level
Undergraduate
Category
A
Teacher
E. Kyritsis
ECTS
7
Hours
6
Semester
Spring
Display
Yes
Offered
Yes
Teacher Webpage
Goal of the course
The course is intended for second-year students and is the continuation and extension of the course Differential Equations I. Differential equations with partial derivatives are considered as they mainly appear in classical boundary value problems of physics. The concepts of the eigenvalue problem, Fourier series, Fourier transformation, special Legendre and Bessel functions, Green’s functions are introduced and developed.
Program
Monday 11:00-13:00, Room 3
Wednesday 9:00-11:00, Room 3
Thursday 9:00-11:00, Room 3
Friday, 14:00-16:00, Amphitheater X/P
Wednesday 9:00-11:00, Room 3
Thursday 9:00-11:00, Room 3
Friday, 14:00-16:00, Amphitheater X/P
Syllabus
1. Partial differential equations of separable form. Partial differential equations of separable form. The wave equation, Laplace’s equation, the heat equation. Initial and boundary conditions leading to unique solution. The example of vibrating string.(2 weeks)
2. Sturm-Liouville theory, Fourier series. Boundary value problems for ordinary differential equations: Sturm-Liouville theory. The basic theorems of the eigenvalue problem. Consideration through operators. Singular eigenvalue problems. Eigenfunction expansions. Fourier series.(2 weeks)
3. The wave equation, Laplace’s equation, the heat equation on bounded regions. One-dimensional heat equation (cooling of a metal plate in a bath of zero temperature, thermal equilibrium of a thermally insulated wall). The two-dimensional Laplace’s equation in Cartesian, polar coordinates and the three-dimensional Laplace’s equation in spherical coordinates (electric field inside a square, cylindrical capacitor, spherical capacitor). The two-dimensional wave equation in Cartesian, polar coordinates (vibration of a rectangular, circular membrane). Problems with non-homogeneous boundary conditions.(3 weeks)
4. The wave equation, Laplace’s equation, the heat equation on unbounded regions. Extension of the basic theory to eigenvalue problems with continuous spectrum. Fourier transformation. The heat equation on the infinite or semi-infinite interval. The evolution function of the temperature field. The wave equation on the infinite plane.(3 weeks)
5. Inhomogeneous differential equations: Green’s functions. Boundary value problems for inhomogeneous differential equations. Definition and construction of the Green’s function for ordinary differential equations. The method of Green’s function for partial differential equations. Green’s functions for bounded regions.
(2 weeks)
6. Special functions of Mathematical Physics. The notion of generating function and recursion relations. Legendre polynomials and Bessel functions: basic properties and computational techniques. (1 week)
2. Sturm-Liouville theory, Fourier series. Boundary value problems for ordinary differential equations: Sturm-Liouville theory. The basic theorems of the eigenvalue problem. Consideration through operators. Singular eigenvalue problems. Eigenfunction expansions. Fourier series.(2 weeks)
3. The wave equation, Laplace’s equation, the heat equation on bounded regions. One-dimensional heat equation (cooling of a metal plate in a bath of zero temperature, thermal equilibrium of a thermally insulated wall). The two-dimensional Laplace’s equation in Cartesian, polar coordinates and the three-dimensional Laplace’s equation in spherical coordinates (electric field inside a square, cylindrical capacitor, spherical capacitor). The two-dimensional wave equation in Cartesian, polar coordinates (vibration of a rectangular, circular membrane). Problems with non-homogeneous boundary conditions.(3 weeks)
4. The wave equation, Laplace’s equation, the heat equation on unbounded regions. Extension of the basic theory to eigenvalue problems with continuous spectrum. Fourier transformation. The heat equation on the infinite or semi-infinite interval. The evolution function of the temperature field. The wave equation on the infinite plane.(3 weeks)
5. Inhomogeneous differential equations: Green’s functions. Boundary value problems for inhomogeneous differential equations. Definition and construction of the Green’s function for ordinary differential equations. The method of Green’s function for partial differential equations. Green’s functions for bounded regions.
(2 weeks)
6. Special functions of Mathematical Physics. The notion of generating function and recursion relations. Legendre polynomials and Bessel functions: basic properties and computational techniques. (1 week)
Bibliography
1. Trahanas, S., Partial Differential Equations, Crete University Press (2015).
2. Brown, J., Churchill, R., Fourier Series and Boundary Value Problems (5th ed.), McGraw-Hill (1993).
3. Vergados, I., Mathematical Methods of Physics, ΟΕΔΒ (1986).
2. Brown, J., Churchill, R., Fourier Series and Boundary Value Problems (5th ed.), McGraw-Hill (1993).
3. Vergados, I., Mathematical Methods of Physics, ΟΕΔΒ (1986).
