Department of Physics, Unit Catalogue 2007/08 |
PH30025 PDEs & complex analysis |
| Credits: 6 |
| Level: Honours |
| Semester: 2 |
| Assessment: EX100 |
| Requisites: |
| Before taking this unit you must take PH20019 and take PH20020 |
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Aims: The aim of this unit is to continue the development of students' mathematical knowledge and skills by introducing concepts and methods used in a mathematical description of the physical world.
Learning Outcomes: After taking this unit the student should be able to: * recognise and solve some of the key equations which arise in the natural sciences; * apply the separation of variables method to linear partial differential equations, and solve the resulting ordinary differential equations by series solution; * use superposition methods for inhomogeneous equations; * derive theorems of analytic functions and use them to evaluate integrals. Skills: Numeracy T/F A, Problem Solving T/F A. Content: Linear equations of science (15 hours): Derivation of the diffusion equation as an example of how PDEs arise in nature. Introduction to Laplace's, Poisson, wave and Schrödinger's equations. Linearity and superposition. Boundary conditions. Solution by separation of variables in Cartesian, cylindrical and spherical coordinate systems. Series solution of ODEs, including Legendre polynomials and Bessel functions. Sturm-Liouville theory. Orthogonality of functions. Green's functions. Solution of inhomogeneous ODEs. Functions of a complex variable (7 hours): Functions of z, multivalued functions, branch points and branch cuts. Differentiation, analytic functions, Cauchy-Riemann equations. Complex integration; Cauchy's theorem and integral. Taylor and Laurent expansions. Residue theorem, evaluation of real integrals. |