Considerdiffusioninsideanenclosedcirculartube.Letitslength
Chapter 4, Problem 4.2.4(choose chapter or problem)
Considerdiffusioninsideanenclosedcirculartube.Letitslength(circumference) be 2l. Let x denote the arc length parameter wherel x l. Then the concentration of the diffusing substance satises ut =kuxx for l x l u(l,t)=u(l,t) and ux(l,t)=ux(l,t). These are called periodic boundary conditions. (a) Show that the eigenvalues are =(n/l)2 forn =0,1,2,3,.... (b) Show that the concentration isu(x,t)=1 2A0 + n=1An cos nx l + Bn sinnx l en22kt/l2.
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