Question

1. (P1, page 68, 10pts) Solve the following initial-boundary value problem: ut = kuzi+t, (2,0) = 2(L-1), (0, 1) = 0, u(Lit) = 0, for 0<x<L, t > 0, for 0 <r<L, for t>0, for t>

Answer 1

This ut = kxx tt for OTX KL ; tro Ulx,0) = f(x) ģ ulogt) = 0 = UCL gt). is non-homogeneous Heat Equation of the form Ut-k Uxx = Ox, t) where dogt). first we solve it for homogeneous part so

Cars lent // XX uco, t) 0 1 UCL gt) = 0 Uxx,0) = f(x). Consider wx t = X(X) TH) by method of separation of Nào, Bo G = XT' =kx| = ( 1 + m Lau if mxo = m. 27 lt if (t) I 2 T't di KT 20 TH)= A edert ① and X 12 X"+d²x zo e X = X 60 = Blesd at Smda AS ucot) 20 & XOH) 20 = XCO) co as TH to " XCO) = 0 7 Bes(0) + (Simo) co B=0 and B=0 = XbU=c8mda and as UCL) 20 X (4) TH=0 = X(L) o as TH) to X(4)=0 & CSndl=0 = Side Sinu as eto 90 Xnx) = An Sm (NT) ; Tn Lt): An e unkit[ xow] [Trict) ] = [ G48m (128) ] [Amellery By pubesposition principle o wwt - en moux) et U)! ; An Anch AB Ux,0) = f(x + f(x) = E n sm (Max). eo o f w sa Dn Sin (AEY) - ® This is fourier Sime series do Pn is given by

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