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A set of problems related to the solution of partial differential equations in fluid dynamics. The problems involve finding implicit solutions, using the method of characteristics, and deriving euler-lagrange equations. Some problems also include finding the conservation of total momentum and integrability of the hamiltonian system.
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is ∂u ∂t
Initially, at t = 0, the flow is given by u(x, 0) = e−x^2.
(i) Show that an implicit form of the solution for u(x, t) is
x = ut ± [log(1/u)]^1 /^2.
(ii) At some time ts, a shock forms in the solution to this problem at a point x = xs. Find ts and xs and the corresponding value of u at this point.
e−x^ = e−t^ + e−ζ^ − 1
where ζ labels the fluid particle which is at x = ζ at t = 0. Find the speed, at time t = 1, of the fluid particle which is at x = 1 when t = 0.
(ii) Solve the following partial differential equation by the method of characteristics:
x
∂u ∂t
∂u ∂x = 0, with u = e−x 2 when t = 0.
(iii) The kinematic wave equation for u(x, t) is ∂u ∂t
where k is a positive constant. Suppose initial conditions for this partial differential equation are
u(x, 0) =
1 , x ≤ 0 , 1 + x, 0 ≤ x ≤ 1 , 2 , x ≥ 1.
Sketch the characteristics for this equation in the (x, t)-plane.
x˙^2 + ˙y^2
(i) Derive the Euler-Lagrange equations. (ii) Show that the Euler-Lagrange equations remain invariant if we replace A(x, y), B(x, y) with A^ ˆ = A + ∂f ∂x
∂f ∂y
where f = f (x, y) is an arbitrary function. (iii) Derive the Hamiltonian function. (iv) Let Φ(x, y) = sin(x) + sin(y), A(x, y) = cos(x) sin(y), B(x, y) = sin(x) cos(y). Show that in this case the Hamiltonian system is integrable.
