I will be taking up a PDEs course next semester and would like to find some good references. The topics covered in the syllabus is given below.
Partial differential equations: Conservation laws, classifications, elementary analytical methods, initial/ boundary value problems. Diffusion equation: Fundamental solution, similarity solution, qualitative behavior of diffusion initial value problems, Cauchy problem with infinite domain, Initial boundary value problems in the semi- infinite domain, Green’s function, homogeneous boundary value problem with inhomogeneous boundary condition. Hyperbolic equations: Characteristic methods, initial value problems with non- continuous initial data, introduction to weak solutions. Basic option theory: Call option, put option, Asian option, Black – Sholes model and its derivatives. Numerical methods: Discretization of derivatives, boundary conditions, grids, finite difference methods for initial/ boundary value problems, consistency, stability, convergence, applications of finite difference methods in financial derivatives.
I hope someone could suggest a some reference books or maybe even a single book that may cover the above topics. Thanks and looking forward for some assistance. Cheers