Partial Differential Equations (Dirichlet Boundary Condition; Seperation of Variables)

2. Consider a thin rod of heat-conducting material with length L. Suppose that the rod is initially heated to a temperature of T uniformly throughout the tod, and is dropped into a bucket of ice water at t = 0. Suppose that the rod is everywhere insulated, except for its left end (x = 0), which is expoosed to the ice water. (a) Write down the system of equations that describes this scenario. (You can use Dirichlet boundary condition for the left end of the rod.) (b) Find the temperature of the rod using seperation of variables. (c) At any given time, which part of the rod will be the warmest? Estimate when the maximum temperature of the rod will be less than 1% of T.

Partial Differential Equation (PDE)

3. Consider the PDE problem: {see attachment} Suppose v(x,y) represents the temperature of some heat-conducting material. What physical scenario could be described by this PDE problem? What does each equation mean physically? Solve for v(x,y). Your final answer should indicate how all constants are obtained from g(x).

Gibb's Phenomenon (Spurious Oscillations; Truncated Fourier Series; Overshoot; Undershoot)

4. In this problem, you will devise a computer experiment to investigate Gibb's phenomenon, which is the presence of spurious oscillations in the graph of a truncated Fourier series near the places where the full Fourier series is discontinous. Choose any function you like that demonstrates Gibb's phenomenon. Your goal is to answer these two questions: (a) You should find that the amount of overshoot only depends on the height of the discontinuity of your function. Expressed as a ratio to the height of the discontinuity, what is the approximate amount of overshoot/undershoot? (b) What happens to the amount of overshoot/undershoot as you increadse the number of terms in your truncated Fo... click for more

The heat equation on a ring

Let u(x,t) describe the temperature of a thin metal ring with circumference... solve each PDE and write down the final solution (see attachment for complete question)

Separation of variables

Solve for u(x,t) using separation of variables (see attachment for full question)