Question

3. In the problems below, you may use the formal solution of the appropriate partial differential equation and boundary conditions from course notes and the text. You do not have to derive the formal solution. (a) (15 points) Find the solution of the initial-boundary value problem du du ət – Ər2 t> 0, 0 < x <7, u(0,t) = , t>0, u( ,t) = 0, t>0, u(x,0) = sin 2x, 0<x< 7. (b) (10 points) Solve the initial-boundary value problem Utt = 16Uzr, t>0, 0<x<T, u(0,t) = u(17,t) = 0, t> 0, u(x,0) = 2 sin(x), OSxST, ut(x,0) = –2 sin(3x), 0<x<.

Answer 1

Date : 3. (al du at du an² , t70, o caca z = uloit) =t tzo I 770 ulat) = 0 ulno) = sinar o custo Compare the given partial differential equation with standard heat conduction problem given by ut= kuna 0<x<l, t70 uloit) = lo tro. ull,t=0 elno) = f(x) bcaal whose solution is kt ulat) Z au e din ( ηπα nal l. 7 whene au 2 (f(xl - Uo) sin (nox) dx solution af PDE o is given 1. The by

Pagai Dato: au c²uar o cual, to ulho ulna) f(n) { ounce ly (a,0) g(x) ulo,t = 0, ull,t=0 , 120 which has the solution ulnut) = I fintct) + fla-ct) utct 2 g (2) dz a-ct ď Alembert's for mula . x-ct by Now the solue the given PDE 2 we have flw= 2 sina g(x) = -2 binßa) et cu o company i. The solution of PDE (2 is . atht ulut) = 2 bin (atht) 4 2 sin(n-ut 1-2rin / 3Tdi xat

Posge: D019 ulnit) 2 sin (x+4t) + sin(x- ut to a 3 + 2 ( cas 37 patut a 4t 2 2. sin(x+4+* **ut) cas (35+ 44 – 60+) + Love [ casB[x+4+) = cas 3 La=4137 12 L a sin (x) cas (4+) tte Icas (3x+12+) | -- cas (32 = pat)? E a finx cou[4) + +, -asis (3x+12+ +37–122| 12 L & sin 3x+127- (30-12t) 2 sina casht - I sin (3x) & sin(127) ulat) & Zwinx casut - I sin (3x) sin (127) le is the solution of ginen PDE.