Question

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Problem 2. Given the PDE ut + uy = u? u(3,0) = g(x) for IER, >0, for ER. (a) Sketch the characteristic curve that passes through P(2,3) in the xt-plane. Find u(2,3) without using the exact solution ulit, t). (b) Use the method of characteristics to find the solution u(, t).

Answer 1

+ 4y = ut, for eit tra ult, o) = S(r) nklin Xo's $ Y = 0 4o - giss @ - (микас+ems ft 4 444+25 dx - @ dy dy du - ut - chaructivisti 3x = ean in xt. Plane by 0 1.0 = x= d t it is essing throug Point P (2,3; tienic 223ti 5 : sharacteristic cah 1 x=t C, dxal 3 X = c.l.3 Y + 4 dt t = 4 + 4 * tool * =0 dit :!7. du . ut = du 1 = 4 3 - 1 1 = y + 3 - 2 = 6) = -1 Lio Tie yu) -- Y Z by ② und by x=y+$- 7 x=t+ s = ult, x = $Y-- s=x-* -f(x +! #g12-)-1