Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MATH 471 Project

Quantum Rotors & Curvature Flows

I) Quantum rotor. Consider the eigenvalue problem,

a) Write down a set of difference equations for this problem and reduce it to a N × N matrix equation Be sure to show how you do the differencing at the end points.

b) Write a code to find the lowest eigenvalue and tabulate it for N = 32, 64, 128, 256, 512.

II) Curvature flow. Consider a closed curve that is evolving in time whose coordinates x are parameterized by α, that is, The governing equation for the curvature flow is given by,

where

Implement the forward Euler method in MATLAB to solve (1). Take the initial shape to be:

The parameter and the time Let N denote the number of time-steps and M, the number of spatial discretization points.

Using your code, perform the following steps:

a) Using N = 100 and M = 32, simulate the evolution of the curve until t = 4. Plot the curve shape at t = 0, 1, 2, 3, 4.

b) The length of the curve at time t is given by

Compute L(4) using trapezoidal rule with M = 64 and N = 1600. Using this as the reference value, tabulate the errors in computing L(4) with N = 100, 200, 400, 800. Deduce the order of accuracy of your numerical algorithm.

c) Repeat the simulation in step 1 with M = 128. What do you observe? Explain the output.

Submit your MATLAB script along with the above. Make sure that the code can be executed as is without any external dependencies.