Consider the equation uux + uy = 0 with the initial condition u(x, 0) = h(x)...

Consider the equation uux + uy = 0 with the initial condition

u(x, 0) = h(x) = ⇢ 0 for x > 0

u_o for x < 0, with u_o< 0.

Show that there is a second weak solution with a shock along the line x = u_o y / 2

The solution in both mathematical and graphical presentation before and after the shock.

Expert Solution

Colby Messinger answered 2 years ago

Consider the equation: ?̇ +2? = ?(?) with initial condition x(0) = 2 (a) If u(t)...

Consider the equation: ?̇ +2? = ?(?) with initial condition x(0) = 2 (a) If u(t) = 0, find the solution ?(?). What is ?(?) as t -> ∞? (b) If u(t) = 4+t, find the solution ?(?). What is ?(?) as t -> ∞? (c) If u(t) = ?3?, find the solution ?(?). What is ?(?) as t -> ∞? (d) If u(t) = δ(t), find the solution ?(?). What is ?(?) as t -> ∞?

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0...

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0 ≤ x ≤ 2 and t ≥ 0. ∂ 2u ∂t2 = 0.01 ∂ 2u ∂x2 (0 ≤ x ≤ 2, t ≥ 0) (1) ∂u ∂x(t, 0) = 0 and ∂u ∂x(t, 2) = 0 (2) u(0, x) = f(x) = x if 0 ≤ x ≤ 1 1 if 1 ≤ x ≤ 2. (a) Compute the coefficients a0, a1, a2, ....

x2uxx --2xyuxy -- 3y2uyy = 0 u(x,1) = x, uy(x,1) = 1

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0)...

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0) = 0 ut(x, 0) = 0.1x(π − x) u(0,t) = u(π,t) = 0 (a) Solve the problem using the separation of variables. (b) Solve the problem using D’Alembert’s solution. Hint: I would suggest doing an odd expansion of ut(x,0) first; the final solution should be exactly like the one in (a).

Consider the following hypothesis test: H 0: u = 16 H a: u ≠ 16 A...

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Solve the equation y'=5-8y with initial condition y'(0)=0

? ′′ − 8? ′ − 20? = xe^(x) initial condition is ?(0) = 0 y'(0)=0....

? ′′ − 8? ′ − 20? = xe^(x) initial condition is ?(0) = 0 y'(0)=0. laplace/inverse/partialfract. show all work please!

Consider the equation utt = uxx x ∈ (0, pi) ux(0,t) = u(pi,t) = 0 Write...

Consider the equation utt = uxx x ∈ (0, pi) ux(0,t) = u(pi,t) = 0 Write the series expansion for a solution u(x,t)

Consider the differential equation y′(t)+9y(t)=−4cos(5t)u(t), with initial condition y(0)=4, A)Find the Laplace transform of the solution...

Consider the differential equation y′(t)+9y(t)=−4cos(5t)u(t), with initial condition y(0)=4, A)Find the Laplace transform of the solution Y(s).Y(s). Write the solution as a single fraction in s. Y(s)= ______________ B) Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the form (c/(s-p)), where c is a constant and the root p is a constant. Both c and p may be complex. Y(s)= ____ + ______ +______ C)...

Solve x(y^2+U)Ux -y(x^2+U)Uy =(x^2-y^2)U, U(x,-x)=1

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