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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

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  • $\begingroup$ Suggest you get any engineering/physics book and do exercises in explicit solutions for the heat equation using Fourier series. Not sure where to send you for one-dimensional wave equation. Anyway, all your stuff is one space dimension, no transforms, no elliptic, but it would appear they will never be showing you anything that has a closed-form solution, so your comfort level will improve if you practice such. Just remembered, vibrating string from physics, good example for this. $\endgroup$ – Will Jagy Jul 18 '14 at 17:17
  • $\begingroup$ Weinberger's A First Course in Partial Differential Equations: with Complex Variables and Transform Methods is a good reference to have that is mathematically solid (i.e. uniform convergence is explicitly dealt with) without being beyond the level of the typical undergraduate physics/engineering separation-of-variables PDE text. The Dover reprint is also quite cheap. I took a beginning graduate course that used this back in Fall 1989, and I think the hardback version cost over $60 back then. $\endgroup$ – Dave L. Renfro Jul 18 '14 at 18:28
  • $\begingroup$ Partial Differential Equations: Second Edition Graduate Studies in Mathematics. Excellent book with many examples and exercises. $\endgroup$ – Mathsource Jul 21 '14 at 15:43
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For something that has a little bit of everything, check out Partial Differential Equations by Walter A Strauss

It is a great intro to all of these topics.

For more in depth references, I reccommend these to anyone studying this field:

Partial Differential Equations- Lawrence C Evans

Numerical Solution of Partial Differential Equations: An Introduction- Morton, K. W.

Numerical Methods to Conservation Laws- Randall J. Leveque

Green's Functions and Boundary Value Problems - Ivar Stakgold

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A nice book focused on examples and treating much of the classical families of PDE step by step is the Partial Differential Equations of L. C. Evans. It seems to cover near all your syllabus, maybe without the discretization part.

Whatever, it is typically a good first course companion.

Hoping it will be of some help.

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