Consider the following quasi-linear PDE, ???? du t + (1+2u) where the initial condition is...

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Consider the following quasi-linear PDE, ди ????х du Ət + (1+2u) where the initial condition is f (x) = -{₁ = 1, 2-x, =-u; |x| > 1 x≤1 u (x,0) = f (x) 1, 2+x, 2x, 1, x < -1 −1≤ x ≤0 0 < x≤ 1 x > 1 = (a) [8 points] Find the parametric solution, using r as your parameter along a charac- teristic and s to label the characteristic (i.e. the initial value of x). First write down the relevant ODEs for at/ar, ax/ar, du/ar. Take the initial conditions t 0 and x s at r = 0. Using the initial condition in (8), write down the IC for u at r = 0, in terms of s. Solve for t, u and x (in that order!) as functions of r, s. When integrating for x, be careful: u depends on r! Consider the following quasi-linear PDE, ди ????х du Ət + (1+2u) where the initial condition is f (x) = -{₁ = 1, 2-x, =-u; |x| > 1 x≤1 u (x,0) = f (x) 1, 2+x, 2x, 1, x < -1 −1≤ x ≤0 0 < x≤ 1 x > 1 = (a) [8 points] Find the parametric solution, using r as your parameter along a charac- teristic and s to label the characteristic (i.e. the initial value of x). First write down the relevant ODEs for at/ar, ax/ar, du/ar. Take the initial conditions t 0 and x s at r = 0. Using the initial condition in (8), write down the IC for u at r = 0, in terms of s. Solve for t, u and x (in that order!) as functions of r, s. When integrating for x, be careful: u depends on r!

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