Question

Write a solution of the initial boundary value problem. Use whichever software to plot profiles for times = 0.2, 0.5, and 1.1 u = 3uxx for 0 < x <L,1 > 0, u(0,1) = u(L,t) = 0, u(x,0) = L(1 - Cos(21x/L))

Answer 1

o o t>0 Heat Conduchin Equation: 34 cod Son za 04 Ends are maintained at op u (ogt) = UCL,t)=0. UlX0 = f(x) ġ OSXEL Consider ucxt = XGJ TW by separation of variable = XT XA XIT to P ak Clonstant if kroz K= > I = -? Take LIAR E T'+49 TEO TH = 6 ed* ĐÓ. x=3 = x'4d9 x 20 = x(x) = Bestat comida >@ A8 uont) 20 - Xo)74) 20 > X(O)=0 as TW40 X(0)=0 = 60°(0)+cºm(0)=0 | = 8-0 9) X) - CSoda ) ubat) = 0 = XCL) TH) = 0 = X(L)=0 as 14t0 X(6=0 = C&md(L) = 0 0 0 - CsmLd20 - Smd=0 as GEO » Smled) Smmt 5 dan 0 = Thay (t): An é o ut 3 Xn60- Co Sm (N1Ix) DET By superposition poinciples yurt) = Xacowane) - ( ca am cerpe) fome Cu na unita may on (97) e * (*2. no

UNUL As u(x0) - 1 (1 - Cos (@Tx/L) ) = fwo. Menuft- (2x)} = 5 Dn Sin Mix). .' > 47–69 (8E) 1 = E º D Sim (13) This is sine fourier series. I Dn261- les carx)). 8. (TEX). t Jo - gluces (m1)-17 Inomhus 80 By putting values of Dno at pu les art=1) 1822-3 L no (124) J e becomes a Bas given? ucat) 84 (Cescott-I) m uxt.-3(*2)+ er mas no Cn2-4) (2) Answer, becomes o
considering L=10, the graph of u(X,t) at t=0.2,0.5,1.1 is given below

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