Question

Problem 1: Solve the initial value / Dirichlet problem on the half-line and find the value u(1, 2): (8 points) uu(t, x) – uzz(t, x) = x +t, (t, x) € Rx [0, +00), u(0, 2) = cos(V), U(0,x) = e, u(t,0) = 1+t.
pls solve

Answer 1

Please refer below images for the solution.

Given ean is: To find the value of u (1,2) " H (t, x) - Uax (t, x) = xtt, (1,2) ER:[0,) u 10, x) = cos (5x), u660, x) = ex, ult,0) = 1 tt the general from of the wave eqn (2-dimensional) ty of the form ax? day ay flats } U(1,0) = f(a) u (2,0) = g(x), ult,0) = n(t) solution is of the form u (t) = 1/2 [ f(x + 4) + (x-ct)] + 1 e 1 g(s) ds the x+ct. a_ct atc (t-o) + antes Jr (ses de de a-c6-0) Now comparing with the given eau ③ we get f (x) = coste, g(n)= ez, h (4)= Itt &e=1 f(x, t) = x+t so the solution of giver en is att 8(s) ds u CE, n) =3 [ +(2+4)** 60-43) +45 x+(t-o) t + 2 S J 415, 6) di de att-o)

att I ulte) = 1/2 ( cas de ser los utt) + 2 ed. at ot + j (sto) Ids.de 2 St atto att Now seeds= les part ent) = (ente e (Ste) ds de tatt-o & s att-o do + so atto S { (art-0² _ (A-t to? to (att-o) 2 o - 0 (x-t rådo} - 15 s [ @ Chat - 420) +2 +0 -20° -20 } do $(ort -320 tato -30% de East (at-20440 -6%) de &fat () I cooby +4.609 41L00169 32 {xt – et² 2 te + 2 2

2 { x + + (28-26) { )) کی 3 S xt 2 z t 3 the a-t xt ² + / +3. so from we we have the solution of giver equation ① is re(t, x) = 1 / 2 (Los datt & cos dart] ť + lat Now - .: u(1,2) I (osos + cos + Casu) + 1 3 - 1 + 1 + 1 -) (1,2)= 1 (los d] + cos1) + 1 ele²-1) + 7 - se se ett ? + 6 2 २ 2