PHYSICS 312
Problem set 4:

Problem 1:
Give a physical interpretation for this problem and thus explain why $u(x,t)$ should increase steadily as $t$ increases. (Assume that $S$ is a positive constant).

$$\frac{\partial ^2u}{\partial x^2}=\frac{1}{k}\frac{\partial u}{\partial t},\; 0<x<a,\;0<t$$

$$\frac{\partial u(x,t)}{\partial x}\vert _{x=0}=0,\;\frac{\partial u(x,t)}{\partial x}\vert _{x=a}=S,\;0<t$$

$$u(x,0)=0,\;0<x<a$$

Problem 2:
Show that

$$\nu(x,t)=\frac{S}{2a}(x^2+2kt)$$

satisfies the heat equations and boundary conditions in the problem above. Find $w(x,t)$ defined by

$$u(x,t)=\nu(x,t)+w(x,t)$$

Problem 3:
Show that the four functions

$$u_0=1,\;u_1=x,\;u_2=x^2+2kt,\;u_3=x^3+6kxt$$

are solutions to the heat equation. (These are sometimes called heat polynomials.) Find a linear combination of them that satisfies the boundary conditions $u(0,t)=0,\;u(a,t)=t$.

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