Answer Happy

(1 point) In this problem we explore using Fourier series to solve nonhomogeneous boundary value problems. For un type un, for derivatives use the prime notation u, uh, .... Solve the heat equation a2u + te მე-2 au = ,0<x<7,1 > 0 at u(0,0) 0, (7,0) = 0,1 > 0 u(1,0) = 2,0<<<<7. To find a series solution u(x, t) we first must write the function cell as a Fourier series 2e-= пп (t) sin(974). n= Therefore Fr(e) = [ xe^(-6t)sin(((npi) 7x dar -1"(n+1)*(14e-6t))/(npi) Now we try to find a solution u of the form ulv,t) = Şu, (t) sin(97**). ) = n=1 Using this series in the PDE we get ди. au X 0 ot 8.rº n=1 6! au azu Since we want = Te their Fourier coefficients must be equal: at მე-2 0 which gives us an ODE in un (0) which we solve using constant cn, to get un(t) = 0 = F(t),

which gives us an ODE in un(t) which we solve using constant cn, to get 0 Un(t) п Now that we have a general form for u, we can find the constants Cn by using the initial condition u(x,0) = x. Plugging the formula we just derived for un(t) into the series for u we get ~ u(x,0) = [(-1^(n+1)8686+cn)/(npi(n^2pi^2-294))+cn]sin((npi)/7)x sin (***) = X. х 7 n=1 Recognizing that this is a Fourier series for X, we can solve for on: 7 2 Cn = } xsin(((npi)/7)x) da- [(-1^(n+1)686)/(npi(n^2pi^2-294))] = - ((-1^(n+1)*14)/(npi))(n^2pi^2-343)/(n^2pi^2-294))