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Partial Differential Equations of Applied Mathematics

Zauderer, 3rd

2.2.6. Show that the initial value problem n(x, t) + v2(x, t) = 0, v(x, t) = x on a2 +t2 1, has no solution. However, if the initial data are given only over the semicircle that lies in the half-plane r +t 0, the solution exists but is not differentiable along the characteristic base curves that issue from the two endpoints of the semicircle

Answer 1

SOLUTION

Nu_t (x, t) + nu_x (x, t) = 0, nu (x, t) = x on x^2 + t^2 = 1 Lagrange equation for (1) is dt/1 = dx/1 = dv/0 therefore dv = 0 implies nu = c_1 (constant c_1) now dt = dx implies integral dt = integral dx implies t = x + c_2 now we nu (x, t) = x on x^2 + t^2 = 1 let x = p t^2 = 1 - p^2 therefore t = squareroot 1 - p^2 and nu = p therefore c_1 = p and t = x + c_2 implies squareroot 1 - p^2 = p + c_2 \r\n therefore squareroot 1 - c_1^2 = c_1 + c_2 therefore 1 - nu^2 = nu + (t - x) ---(1) now if nu^2 > 1 1 - v^2 < 0 squareroot 1- v^2 is not a real number therefore solution does not exist (therefore v + (t - x) is real (1) can't hold.