Question

1. Consider the following inhomogeneous wave equation on (0,7) : utt - 4uxx = (1 - x) cost Uz(0,t) = cost-1, uz(7,t) = cost u(3,0) = 2(7,0) = cos 3x (a) Convert the PDE to an equation with homogeneous boundary conditions by making an appropriate substitution u(x, t) = u(x, t) - p(x, t), implying u(,t) = v(x, t) + p(2,t) for an appropriate function p(x, t). (b) Finish solving the PDE using the Method of Eigenfunction expansion.

Answer 1

NUNLPN let-quan = (1-x) cost ocrca, to Ux(0,t) - cost-I 4xca,t) - cest , eso 2160,0) = 22 osaar le (7,0) = cos3x our ex we look solution u of of the foom, ulxit) = vexit) + wexit) - ince we dealing with Neumann Berundary vecte problem 2 -- we take the function wenit)= x(cost-1) at 22 [cost- (cost-1)] = x(cost-1) + 5 Since we = -xcist, Wxx = and wo(a,0) = ( 0) 0 0 Now the function & must be solution of the problen NEE-48xx = cost + A - Olack, too Vyloft) = x (x,x)=0 to, o v (7,0)=0 xx ex VE(X,O) = cos3x, ocasa The formal solution vixit) 5 Tn (t) X₂ (x) Where Xm (t) are Elgen function of the starn-houbille problen Xll(x) +dX(x)=0 oux cx X'(o)= x (x)=0 Ergenvalue is dn = n², Xom (x) = cosna , noo --=== So vext = 3 Tolt) cosna m2, o

a Riff we with respect to t and x and substitute into Equation we get VE - 9 x x = { ["(t) HANT HI] carne a cost + 1 - 2 720 This genesus Mo 04 I "+4n²T (t) = n71 is t=0 then o r (x,o) - Z Ty (o) casna nzo showing that Tnlos zo #n7,0 Nome we get cos3x = Vq (3,0) = E Tn'(o) cosna' n20 showing that I lo] = { ! +3,070 The General solution of O is Talt) = Ancospint) + Bn sment Smee Inco), An=0 so that Talt) = Bysmat and Thlt) = 2n By cosent Then Tolos=0 for 07, Sheory that Theo for all nal and nt3 ef n=3, then til 10) = 6 B3 = 1 =) B3 = -2 Hence Tight) = Ismet finaly of no then Tort) - 710) tk Test 14.3ds = | lese * 4. Jedes constige

and To(t) = To los + 1 To (s) ds = f ( sms + 2 ds = [-cess + et = (1-cost) + 2th Summing up we get was Vuxit) = (1-cost) +21² + 7 sinse cose anel x2 celnit) = v(act) twent) fucrit) = (1-Cost) at 2t2 + I sonst cossrt 2