Service Type- Research and Writing

Topic: partial differential equation

1. Elliptic PDE: Solve Laplace's equation u_xx + u_yy, = 0 in the rectangle 0 < x < a, 0 < y < b with the boundary conditions: At x = 0, ∂u/∂x = 0; At x = a, ∂u/∂x = 0; At y = 0, u = 0; At y = b, u = f (x),dx

where f is a given arbitrary function. The problem is shown schematically below. The desired answer is therefore the specification of u(x, y) for all points within the given rectangle.

2. Parabolic PDE: A long rod (which can therefore be considered one-dimensional) is subjected to an initial temperature distribution along its axis. One end of the rod is maintained at a constant temperature, while the other end is kept insulated, i.e., there is no heat transfer from the rod into the surroundings at that end point.

The heat equation describes this situation: u_t = ku_xx, for 0 < x < L, 0 < t < ∞ , where u(x,t) is the temperature at a point x in the rod at time t. The relevant initial and boundary conditions are:

At t = 0, u(x, 0) = f (x), where f(x) is a given but arbitrary function;

At x = 0, u(0, t) = 0;

At x = L, ∂u/∂x (L, t) = 0. As we've discussed in class, remember that this means that the derivative is taken first, and the limit second.

Solve for the temperature u(x,t), which gives the temperature at any point x at any time t.

3. Hyperbolic PDE: Consider the wave equation y_tt = a²y_xx for 0 < x < L, 0 < t < ∞. This equation describes the elastic oscillations of a bar of length L, where y(x,t) represents the displacement of the point x in the bar as a function of time t. The relevant initial and boundary conditions are:

At t = 0, y(x, 0) = y_t(x, 0) = 0;

At x = 0, y/(0,t) = 0;

At x = L, y_x(L, t) = F/E'

where a, F and E are positive constants. Solve for the displacement of the end of the bar, y(L,t), as a function of time. [Note that y(L,t) is different from y(x,t)!]