Partial differential equations for scientists and engineers

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1. photochemsynAug 2022
   The only reason for scientists and engineers (i.e. non-pure-mathematicians) to learn ordinary differential equations is so they can move on to partial differential equations. This is because:

   > "Most of the natural laws of physics, such as Maxwell's equations, Newton's law of cooling, the Navier-Stokes equations, Newton's equations of motion, and Schrodinger's equation of quantum mechanics, are stated (or can be) in terms of PDEs, that is these laws describe physical phenomena by relating space and time derivatives. Derivatives occur in these equations because the derivatives represent natural things (like velocity, acceleration, force, friction, flux, current). Hence, we have equations relating partial derivatives of some unknown quantity that we would like to find."

   For the vast majority of people working with DEs, they'll have two major goals: (1) Take a physical problem they wish to model and construct (formulate) the PDE (the actual mathematical model), and (2) Solve the PDE, taking into account the initial value and boundary conditions.

   Going through a introductory course in ODEs is a valuable process because a common method of solving a PDE is reducing it to an ODE. However, with real-world problems, the actual approach will almost always be computational:

   > Numerical methods: These methods change a PDE to a system of difference equations that can be solved by means of iterative techniques on a computer; in many cases this is the only technique that will work. In addition to methods that replace PDEs by difference equations, there are other methods that attempt to approximate solutions by polynomial surfaces - spline approximations.

   A related common approach are perturbation methods, which transform a nonlinear problem into a series of linear ones that approximate the original nonlinear equation, and to which in turn numerical methods can be applied. Wikipedia has a good writeup:

   https://en.wikipedia.org/wiki/Perturbation_theory

   Quotes from a very good source: "Partial Differential Equations for Scientists and Engineers, Stanley J. Farnow, 1993, Dover"

   Pure mathematicians tend to oppose the introduction of approximate and computational methods early on, but there's no reason they can't be taught side-by-side with the traditional 'exact solutions' material, and it would probably greatly interest and benefit students.

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2. btrettelMay 2022
   Stanley Farlow's "Partial Differential Equations for Scientists and Engineers" is a great introduction to partial differential equations. That book was at exactly the right level for me when I was 19 to be introduced to a wide variety of things in differential equations in general.
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3. gattrSep 2018
   On a related note, I can recommend Stanley J. Farlow's "Partial Differential Equations for Scientists and Engineers", which I bought around 2004 to better understand a fluid simulation paper I was trying to implement. (My Maths course ended with linear algebra and ODEs). Very good read with nice physics examples.
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4. btrettelOct 2017
   Stanley Farlow's "Partial Differential Equations for Scientists and Engineers" would be my suggestion for an introduction. It's a Dover book, so it's cheap. Look at the book as a set of simple introductions to a large number of PDE related topics, many of which have applications beyond PDEs, e.g., the Laplace transform, conformal mapping, perturbation methods, etc. The book would disappoint anyone looking for rigor. You can refer to a more advanced book for that.

   The book has relatively few exercises, but the ones present are definitely worth your time. I recall working out a large fraction of them before I took a PDE class as an undergraduate. Dover will also send you a scan of the solutions manual if you contact them. As I recall a few exercises had mistakes in their solutions, but I had no trouble spotting the mistakes as a novice.

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