Solving partial differential equations

[BlueStar123]

Homework Statement

There are two separate problems:

1) Consider the cube region defined between 0 and L in each of the three dimensions. A scalar field inside this cube satisfies $$\Delta \Psi = -c\Psi$$. c = 40/L². The boundary conditions are specified. $$\Psi = 0$$ on the planes y =0/L, x=0/L and z=L. On the plane z=o, the field is defined by a simple function in terms of x, y and L.
We need an analytical solution for the field.

2) A system of partial diff eqns are given:
$$\frac{\partial f}{\partial x}= \frac{\partial g}{\partial y} + cf^{2}$$
$$\frac{\partial f}{\partial y}= \frac{\partial g}{\partial x}$$
Once again, various boundary conditions like for problem (1) are given and we are asked for analytical solutions.

2. The attempt at a solution


I do not need help on the solution to these problems themselves. I would like to solve them all by myself. I just need help on knowing what math topics I need to read before I can tackle these two problems. I am not a math major and my math knowledge is equal to that of an engineering graduate who took the required calculus course in the undergraduate program. But I am very passionate about math and I believe I can pick up the required concepts quickly. So, can the good folks on this forum please answer the following questions for me:

1) What concepts and in what order should I read? You may say: Single variable calculus => multi-variable calculus => single variable differential eqns => ... etc

2) What textbooks/online articles/lecture notes/online free textbooks can I get that will help me learn the topics you outlined for the above question.

3) What search terms can I use to get more info on these questions. E.g. the first equation seems to be called Poisson equation or something. I ask for these search terms since I hope to get to some prof's lecture notes by searchin for these terms.

4) Finally, can you also tell me how difficult the solutions to these problems are for someone who already knows these topics.

 

[HallsofIvy]

I am a bit confused by this. You want to solve first order partial differential equations and you are asking about single variable calculus? Certainly you should know all of calculus as well as ordinary differential equations.

You might want to google "first order partial differential equations". I did and immediately got
http://www.scholarpedia.org/article...on/First-Order_Partial_Differential_Equations
which looks helpful.

 

[r.a.c.]

If you are up to level with engineering math course then you should be able to tackle this. In the end engineers do have to take partial differential equations. If you know all basic calculus and are comfortable with the complex plane (Not as in need to know full complex analysis, just a mild understanding) then I suggest the book Fourier analysis from the princeton lectures in analysis. This book actually takes you in depth into such problems with thorough understanding and proofs. A more applicational book is one also called Fourier analysis but by Gerald B. Folland.