Aims & Learning Objectives: Aims: To furnish the student with a range of analytic techniques for the solution of ODEs and PDEs.
Objectives:
Students should be able to obtain the solution of certain ODEs and PDEs. They should also be aware of certain analytic properties associated with the solution e.g. uniqueness.
Content: SturmLiouville theory: Reality of eigenvalues. Orthogonality of eigenfunctions. Expansion in eigenfunctions. Approximation in mean square. Statement of completeness.
Fourier Transform: As a limit of Fourier series. Properties and applications to solution of differential equations. Frequency response of linear systems. Characteristic functions.
Linear and quasilinear firstorder PDEs in two and three independent variables: Characteristics. Integral surfaces. Uniqueness (without proof).
Linear and quasilinear secondorder PDEs in two independent variables: CauchyKovalevskaya theorem (without proof). Characteristic data. Lack of continuous dependence on initial data for Cauchy problem. Classification as elliptic, parabolic, and hyperbolic. Different standard forms. Constant and nonconstant coefficients.
Onedimensional wave equation: d'Alembert's solution. Uniqueness theorem for corresponding Cauchy problem (with data on a spacelike curve).
