 Description:
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Aims & Learning Objectives:
Aims:
To introduce students to the applications of advanced analysis to the solution of PDEs, including examples drawn from topics in advanced physics.
Objectives:
Students should be able to obtain solutions to certain important PDEs using a variety of techniques e.g. Green's functions, separation of variables, and in cases interpret these in physical terms. They should also be familiar with important analytic properties of the solution.
Content:
Topics will be chosen from the following: Elliptic equations in two independent variables: Harmonic functions. Mean value property. Maximum principle (several proofs). Dirichlet and Neumann problems. Representation of solutions in terms of Green's functions. Continuous dependence of data for Dirichlet problem. Uniqueness.
Parabolic equations in two independent variables: Representation theorems. Green's functions.
Self-adjoint second-order operators: Eigenvalue problems (mainly by example). Separation of variables for inhomogeneous systems.
Green's function methods in general: Method of images. Use if integral transforms. Conformal mapping.
Calculus of variations: Maxima and minima. Lagrange multipliers. Extrema for integral functions. Euler's equation and its special first integrals. Integral and non-integral constraints.
Applications to physical problems: Translate advanced physical problems into mathematical form; obtain and interpret mathematical solutions.
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