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. ( ...continues
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 ...continues
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)
Solve for u(x,t) using separation of variables (see attachment for full question)
Partial Differential Equations
Suppose that a string of length L is held fixed at one end and is being moved up and down, say with a displacement of f(t), at the other end. Assume that the string is initially at reast and has a zero displacement, and similarly, that the forcing of the string, f(t), also has a zero initial velocity and displacement. a) Writ ...continues
In this problem, you will find the electrostatic potential inside an infinitely long, grounded, metal cylinder of unit radius whose axis coincides with the z-axis (See figure below). In cylindrical coordinates, the potential, V(r, theta, z), satisfies Laplace's equation... Please see attached... Let us assume that the po ...continues
Separation of variables - turkey
You are trying to cook a turkey for Thanksgiving, but the instructions on the turkey are for a 15-pound turkey and you have a 20-pound turkey. In this problem, you will use separation of variables to estimate how long it should take to cook your 2-pound turkey.
ODEs: Sturm-Liouville Form; Weight Function; Inner Product
Write each of these ODEs in Sturm-Liouville form, and identify the weight function that woudl be used in defining an appropriate inner product {see attachment for ODEs}
num solutions to PDE that approach zero as x -> (+/-)infinity
Does y' = y - e^(-(x^2)) have solutions that approach zero as x goes to infinity and negative infinity? how many?
