Question

au du atua = 90, дх with the initial conditions at t = 0: u=0 if u=-1-1 u=1 - << -1, if -1 <I<0, if 0 < I< 0. (Define u(r, t), x,t and the constant qo appropriately.) (b) Use the method of characteristics along suitable curves r(t) to obtain the implicit equation satisfied by the general solution ur,t) of the PDE given in the first problem (do not have to use the initial conditions at this stage, so there would be an undetermined function). Confirm that if the initial condition is given as u(1,0) = ax + b with constants a and b, we have the explicit form of the solution uI, t) = ar + qot + a2 + b 1+ at and the characteristic curve +2 + bt + (1+ at).10, where To is a constant. (Hint: du is not zero along the characteristic curve. You may be able to check that u(I(t), t) computed from the above explicit expressions satisfies the evolution equation of u along the characteristic curves you have got. ]

Answer 1

tų 249 at it dx . U (21,0)=0 2176 NC5o = s. ECO =0 @u ( solast! dx dz = 4 at 1 dz. 44 yt 2+G 561.20 Ezz = u = 29C2 asthan az l - do = 2q + C 4= 2% + as tb] X = 229 + C22+ C b = 2290 ps +62 +6 is - G X = z29 (9s+b) 2 AS.

x = 290 + Slaz+1 + b 2 js = (x-bZ - 249.). azt I u = 29. tax-ab2-a z ²9 th azt 9 +208 -2abz-az ² + 2b az tab

uma a 229 +2ax taq. 26 2 (92+1) - Now put z=t un,t = a + 9 + 2ax +299 +.26 24t+11 X(Z) = 9 z² slaz+1) + b2 x peel z= t ED S = No. SXCZ) 9612 Nolat+1) + bt