Question

a) Use the d'Alembert solution to solve au au - <r< ,t> 0, at2 48,2 ux,0) = cos 3x, u(,0) = 21 b) Consider the heat equation диди 0<x<1, t > 0, at ax? with boundary conditions uz (0,t) = 0, uz(1,t) = 0, > 0, and initial conditions u(x,0) = { 0, 2.0, 0<r < 0.5, 0.5 <<1. Use the method of separation of variables to solve the equation.

Answer 1

a) Utt = (ar Uxz c = 1/2 U(7,0) = CODBX foo, Uf(x,0)=e = e = gai Now by D'Alembert's we have UCX,+) = tech + Soluhon [fGtet) +fG-ct) ] + ste Socce guids 3 [ f(x-bit) + f(x+<z>] + § 2 ()S Ś [ Cos{314-38+${3(2+$}}] + atbt gisids x-let x+ 4 Szat ezt els - 5 [ cos{i (21-47} +cos {{{2x++4] + Śle 247 Brate 12-6 10+) = [ cos{ {(92-43% cost (94+kif] + (2x) €=6] CS Scanned with CamScanner

# -) let ocall; tso, then Up = 'Uson In Sepevahon at uariable method: take U1947)= X(x). TCH) X?.T - Varan X.T'=ut X.T'= X".T u sepevahun Constant X"+gX=0 s T'+at=0 then 1 X (D1, D2 ) Boundary Concihons : Ulo Cost) = 0 X'10).TCH)=0 = x1(0)=6 Lolit) so 7 X'(1)-TCH) = 0 =) X'(lje o Cal TCH) 70 ;&lso). (*) Now we solve SLP! x" tax=0 x 10)=0-X(1) Rekelt! X\\+Xx=0, xcovEx'(6)=0 → An= (7) ; Xo = cool meyve) ; 920,53;. Hence soluhon of (*) () An= (map) ? 1:0,1,2,- (Ergen-coace) cs scénned CºlCnny) snegl,.-- CEigen-veckor) CamScanner muutmanannnnnnnnnnnnilinedeleledelilleliling >

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