Course Outline
Methods
METHODS
Self-adjoint ODEs
- Periodic functions. Fourier series: definition and simple properties; Parseval’s theorem. Equations of second order. Self-adjoint differential operators. The Sturm-Liouville equation; eigenfunctions and eigenvalues; reality of eigenvalues and orthogonality of eigenfunctions; eigenfunction expansions (Fourier series as prototype), approximation in mean square, statement of completeness. [5]
PDEs on bounded domains: separation of variables
- Physical basis of Laplace’s equation, the wave equation and the diffusion equation. General method of separation of variables in Cartesian, cylindrical and spherical coordinates. Legendre’s equation: derivation, solutions including explicit forms of P0, P1 and P2, orthogonality. Bessel’s equation of integer order as an example of a self-adjoint eigenvalue problem with non-trivial weight. Examples including potentials on rectangular and circular domains and on a spherical domain (axisymmetric case only), waves on a finite string and heat flow down a semi-infinite rod. [5]
Inhomogeneous ODEs: Green’s functions
- Properties of the Dirac delta function. Initial value problems and forced problems with two fixed end points; solution using Green’s functions. Eigenfunction expansions of the delta function and Green’s functions. [4]
Fourier transforms
- Fourier transforms: definition and simple properties; inversion and convolution theorems. The discrete Fourier transform. Examples of application to linear systems. Relationship of transfer function to Green’s function for initial value problems. [4]
PDEs on unbounded domains
- Classification of PDEs in two independent variables. Well posedness. Solution by the method of characteristics. Green’s functions for PDEs in 1, 2 and 3 independent variables; fundamental solutions of the wave equation, Laplace’s equation and the diffusion equation. The method of images. Application to the forced wave equation, Poisson’s equation and forced diffusion equation. Transient solutions of diffusion problems: the error function. [6]
Appropriate books
- G.B. Arfken, H.J. Weber & F.E. Harris Mathematical Methods for Physicists. Elsevier 2013
- M.L. Boas Mathematical Methods in the Physical Sciences. Wiley 2005
- J. Mathews and R.L. Walker Mathematical Methods of Physics. Benjamin/Cummings 1970
- K. F. Riley, M. P. Hobson, and S.J. Bence Mathematical Methods for Physics and Engineering: a comprehensive guide. Cambridge University Press 2002
- Erwin Kreyszig Advanced Engineering Mathematics. Wiley
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