Consider a Laplace solution with no-flux boundary conditions on three sides of a

Question

Consider a Laplace solution with no-flux boundary conditions on three sides of a rectangle and a non-homogeneous
flux condition on the top side uy(x;H) = f(x).
(a) Explain what condition on f(x) must be true for this equation to have a solution. Hint: evoke the
zero net flux property.
(b) Solve for u(x; y) if f(x) = sin(2x=L). Does this condition satisfy what you discovered from part
(a)?
(c) a 0 is not determined by the boundary condition and therefore is a free parameter. If g(x; y) is an
initial condition to the heat equation, and u(x; y) the resulting equilibrium solution, specify how
you would find a0?

Explanation / Answer

One of the important things to note here is that unlike the heat equation we will not have any initial conditions here. Both variables are spatial variables and each variable occurs in a 2nd order derivative and so we’ll need two boundary conditions for each variable. Next, let’s notice that while the partial differential equation is both linear and homogeneous the boundary conditions are only linear and are not homogeneous. This creates a problem because separation of variables requires homogeneous boundary conditions.To completely solve Laplace’s equation we’re in fact going to have to solve it four times. Each time we solve it only one of the four boundary conditions can be nonhomogeneous while the remaining three will be homogeneous.

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