Question

1. Solve the Cauchy problem (2.1)-(2.2) for the following initial condition a) $(x) = 1 if |2<1 and $(x) = 0 if |z| > 1. b) p(x) = e-x, x > 0; $(x) = 0, x < 0.

Answer 1

ANSWER:- solve the cachy problem (a-1)-(2-2) for the following initial conditions. those are. (@ ex)=if Ixke and $(x)=0 if IX/>/ (b) $(x) =é X, X>0 , 004=0, 220. The begin with the heat, or diffusion, Eslation on the real time. Here the land 22 Equations are Uz= kura; XER,t>- (211) U (10)=0(X), XER —(-a) ca 0(x)= if Ixk) and 011)=0 if 1x1x1 DaSi if XX</ 2o if Ixl>l The fundamental solution are given by -(x-4) . ülxit) = { oly). Fent ue 4KÉ -dy . . ranke

-(1-4) 4Kt uixit)= e-(x-43/4kt 417kt dy + 10 dy actual song and strong dengan cepat they are to 9/16, abuela e ta Jow a few artist og han Stream or the first -(1-47/4kt 1 ) uenit) - St (98703 +5 inte troligt eyya Ke (4 4 sat . 12 1 Sunkt ulut) = / ulost) = 0 + 5' obra de utrit)= '_ J'e-(8-94akt dy 40 .-01-9° 4KE ta ca 297 e de Mente) Tone E Se 12-9/4kt dy artikE' kt

(acoubsector de ses entry (b) 0(X)=é ?, 230; $(x) =0, 22.0 00x)= ser x>0 no 120 Then fundamental Solution are given by U(1/4)= $ry). I 642-432/4KŁ N4IKE dy. =499). 2(x+43/444 - dy + loy). I e(x-99/kt I dyt } 8ly) Verkt dy = 0+ (8) a entender - book net 240 95/4 kali pucano - Piante -C4 04-2Plate) -X-4944kt dy. V 4Tikt