PHYSICS 312
2001 Midterm review problem set

Problem 1:
Find the steady state solution of the problem

$$\frac{\partial}{\partial x}\left((b+fx)\frac{\partial u}{\partial x}\right)=\frac{1}{k} \frac{\partial u}{\partial t}$$

$$0<x<a,\;0<t$$

$$u(0,t)=T_1;\;u(a,t)=T_1,0<t$$

b,f,k are constants.


Problem 2:
Solve the boundary value problem

$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$

inside a square of side $2\pi$:

$$u(x,-\pi)=u(x,\pi)=\sin x;\;u(\pi,y)=u(\pi,y)=\sin y$$

Problem 3:
In which of the following cases can solutions be found by the method of separation of variables?

$$\frac{\partial^2u}{\partial x^2}+x\frac{\partial^2u}{\partial t^2}=0$$ (1)

$$\frac{\partial^2u}{\partial x^2}+t^2\frac{\partial^2u}{\partial t^2}=0$$ (2)

$$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial t^2}=e^{-x}$$ (3)

$$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=e^{-x}+ e^{-y}$$ (4)

In cases where the variables don't separate directly, transform the equations so that product solutions can be found. You don't need to solve the resulting ordinary differential equations.

Problem 4:
Show that the function

$$\phi(x,y)=\sin(\pi x)\sin(2\pi y)-\sin(2\pi x)\sin(\pi y)$$

is an eigenfunction of the triangle T bounded by the lines y=0,y=x,x=1 associated with the differential equation

$$\nabla^2\phi=-\lambda^2\phi$$

and the boundary condition

$$\phi=0$$

on the boundary of T. What is the eigenvalue $\lambda^2$ asociated with $\phi$?


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