Question 1:
1. Code and test Euler Solver for 1st Order IVP. Be sure to compare results against exact answers where known, to include problems that have no answer over the domain in question, and to try problems where you do not know the answer. Convince us that all is working well!
2. Repeat with Runge-Kutta (R-K) algorithm.
3. Apply Euler's and/or R-K for systems of 1st-order equations.
4. Use a 2nd order solver based on Euler's for 2nd differences. Compare results to those obtained in Step 3.
5. Apply solvers from Steps 3 and/or 4 to generate phase diagrams. Discuss critical points, compare theory to numerical results, seek fascinating plots. This involves solving many IVP for many different IV.
Example:
y' = z(z2 - 1)
z' = -y
6. Apply solvers from Steps 3 and/or 4 to generate bifurcation diagrams. This involves using the shooting method, where one IV is a parameter that can be varied in order to satisfy a BV.
Example One: For s ∈ (0, 9Π2), solve
y" + sy + y3 = 0 for x ∈ (0,1)
y = 0 for x ∈ {0,1}.
Consider the associated eigenvalue problem -y" = sy, y(0) = 0 = y(1).
Example Two: For s ∈ (0, 4Π2), solve
y" + sy + 4(x -.5)2y3 = 0 for x ∈ (0,1)
y = 0 for x ∈ {0, 1}.
Classify the observed symmetries of solutions and note the symmetry-breaking bifurcations.