Stærðfræðileg eðlisfræði 7. dæmablað ÞJ/vor 2012 Dæmi • Find a solution ψ(x, t) of the heat equation on the half line x > 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 > 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] × 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. Við ræðum þessi dæmi föstudaginn 20. apríl. Leysið eins mörg og þið getið og skilið þeim í tímanum.