PDE for Scientists and Engineers Farlow

[DrPapper]

Hello Fellow Physics People,

I am just now taking a math methods course for Physicists and we're using Mary Boas book. I wanted to supplement it for better understanding as saw Partial Differential Equations for Scientists and Engineers by Stanley J. Farlow.

Reading reviews for this book on Amazon made me think, perfect for me. I've done well in Calculus and ODEs. So I pick it up and get to the problems at the end of lesson 1 and lesson 2 and think to myself "How the l;kj would I do these?" I look at the solutions and have no idea how they got to them for some of these.

So is this book really that good? Am I missing something? Should it be able to be used for self-study alone or in conjunction with another text?

In isolation of the info an average student would recall from calculus and ODEs and info in this book prior to other lessons how for example would you solve problem 2 in lesson 2:

Suppose the rod has a constant internal heat source, so the basic equation describing the heat flow within the rod is: U_t=a²U_xx+1

o<x<1

Suppose we fix the boundaries' temperatures by u(0,t)=0 and u(1,t)=0. What is the steady-state temperature of the rod? In other words, does the temperature u(x,t) converge to a constant temperature U(x) independent of time?

Hint: Set u_t=0. It would be useful to graph this temperature. Also start with an initial temperature of zero and draw some temperature profiles. <-----How do I even draw this?Let me note that Farlow uses U_xx to mean ∇²u and U_t to be the differential of U with respect to time. At least as far as I can see...

 

[SredniVashtar]

I read Farlow's book a lot of years ago and I sort of liked it (to the extent I bought it when I saw it on Amazon). It's not a monumental masterwork, nor a revered reference book on PDE: it's just a quick hands-on collection of (almost unabridged) essays on the most common methods of solutions for PDEs. I would not use this book as the only book on PDEs, but I would use it to get my feet wet before starting my journey into the PDE Realm.
The best feature of Farlow's book is that most of the chapters (or strings of two-three chapters) can be read in whichever order you like. It gives you an idea of what are and how to solve the most common PDEs without requiring higher level math. In that, it is exceptionally good.

You will have to fill in the gaps, though.

As for your example: follow the hint.
Steady state means u does not change with temperature - hence the hint to set U_t=0. So, what happens to the PDE when you set U_t=0?
It becomes an Ordinary Differential Equation in x. Solve that and you will find the shape of your solution (might have to guess a couple of constants...)
Can you plot these solutions?

P.S.
Sorry for being nit-picking but U_xx is not Farlow's convention for nabla square of U. It's just that the problem is one-dimensional in space and the Laplacian (in rectangular coordinates) reduces to U_xx

 

[DrPapper]

I read Farlow's book a lot of years ago and I sort of liked it (to the extent I bought it when I saw it on Amazon). It's not a monumental masterwork, nor a revered reference book on PDE: it's just a quick hands-on collection of (almost unabridged) essays on the most common methods of solutions for PDEs. I would not use this book as the only book on PDEs, but I would use it to get my feet wet before starting my journey into the PDE Realm.
The best feature of Farlow's book is that most of the chapters (or strings of two-three chapters) can be read in whichever order you like. It gives you an idea of what are and how to solve the most common PDEs without requiring higher level math. In that, it is exceptionally good.

You will have to fill in the gaps, though.

As for your example: follow the hint.
Steady state means u does not change with temperature - hence the hint to set U_t=0. So, what happens to the PDE when you set U_t=0?
It becomes an Ordinary Differential Equation in x. Solve that and you will find the shape of your solution (might have to guess a couple of constants...)
Can you plot these solutions?

P.S.
Sorry for being nit-picking but U_xx is not Farlow's convention for nabla square of U. It's just that the problem is one-dimensional in space and the Laplacian (in rectangular coordinates) reduces to U_xx

Thank you so much for your help. The last bit wasn't nit picky at all, it's good to know. :D

I guess I need to take a breather, I'm stressing on an upcoming test. Ha ha, that ODE conversion should have been appearant to me. LOL

THANKS AGAIN!