# кь ж ж п п × ж º я п • • ψ(x,0)

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```St&aelig;r&eth;fr&aelig;&eth;ileg e&eth;lisfr&aelig;&eth;i
7. d&aelig;mabla&eth;
&THORN;J/vor 2012
D&aelig;mi
• Find a solution ψ(x, t) of the heat equation on the half line x &gt; 0 whih satises the initial ondition ψ(x, 0) = g(x) and the boundary ondition ψ(0, t) = 0
for all t. Hint: Method of images.
Can you solve the same problem if the boundary ondition is ∂x ψ(0, t) = 0 ?
• Find a solution of the heat equation on the half line x &gt; 0 whih satises the
boundary ondition ψ(0, t) = f (t) where f is a periodi funtion of period T .
Hint: Fourier series and separation of variables.
Use the result to estimate how deep into the ground annual temperature
variations reah. For typial soil σ = 10−7 m2 /sek
• Consider the one dimensional heat equation on a nite strip [0, L] &times; R with
Neumann boundary ondition at 0 and Dirihlet boundary ondition at L.
Find the retarded Green funtion and use it to solve the Cauhy problem
with initial ondition
ψ(x, 0) = Ax2 (L − x)
and the given boundary onditions. Make the solution as expliit as you an.
Vi&eth; r&aelig;&eth;um &thorn;essi d&aelig;mi f&ouml;studaginn 20. apr&iacute;l. Leysi&eth; eins m&ouml;rg og &thorn;i&eth; geti&eth; og skili&eth;
&thorn;eim &iacute; t&iacute;manum.
```