[Partial differential equation Tutorial Exercise Reference Solution]1.3 Fixed solution problem

1. Consider the Neumann boundary value problem of the Poisson equation $$\bex \sedd{\ba{ll} \lap u=f (x, Y, z),& (x, Y, z) \in\omega,\\ \frac{\p u}{\p n}|_{\vga}=0,& amp; (x, Y, z) \in\vga. \ea} \eex$$

(1). Is the solution of the above boundary value problem unique?

(2). The necessary condition for proving the solution of the boundary value problem by the divergence theorem is $$\bex \iiint_\omega f (x, Y, z) \rd x\rd y\rd z=0. \eex$$

Prove:

(1). Not unique. The solution can be a difference between any constant.

(2). $$\bex \iiint_\omega f (x, Y, z) \rd x\rd y\rd z =\iiint_\omega \lap u\rd x\rd y\rd z =\iint_{\p\omega} \frac{\p u}{\p n} \rd s=0. \eex$$

2. The absolute temperature of the object surface is $u $, when it radiates heat to the outside according to the Stefan-bolzmann law is proportional to the $u ^4$, namely $$\bex \rd q=\sigma u^4\rd s\rd T. \eex$$ This assumes that there is only heat between the object and the surrounding medium Radiation without heat conduction, the temperature of the surrounding medium is $f (x,y,z,t) $, the boundary condition of the thermal radiation problem is given.

Answer: $$\bex-k\frac{\p u}{\p n}=-\sigma u^4+\sigma f^4. \eex$$

[Partial differential equation Tutorial Exercise Reference Solution]1.3 Fixed solution problem