Homework Problems in Mathematics: Equilibrium Points, Phase Portraits, and PDE Solutions, Assignments of Mathematics

Download Homework Problems in Mathematics: Equilibrium Points, Phase Portraits, and PDE Solutions and more Assignments Mathematics in PDF only on Docsity! Homework Problems 1) Consider the model: dx1 dt = ( 1 − x1 − 1 2 x2 )x1 dx2 dt = ( 1 − 2 3 x1 − x2 )x2 (a) Find the equilibrium points in the quadrant x1, x2 ≥ 0, and classify their types. (b) Draw the phase portrait, with the nullclines. What is the predicted qualitative behavior of solutions in this model as time t → +∞? Try to do part (b) by hand. Then go to the course webpage: http://www.math.umass.edu/∼kevrekid/math697 and download the code pplane7.m. Opening a Matlab window and executing pplane7 in the command window, use the program to produce the phase portrait that you will hand in (do make sure that it agrees with your predictions of the equilibrium points and of their types !). 2) Consider the standard diffusion equation ut = kuxx in the interval 0 ≤ x ≤ l and for times t ≥ 0, with u(0, t) = u(l, t) = 0. Solve the PDE using separation of variables u(x, t) = X(x)T (t). Now, solve the corresponding lattice problem un+1j − u n j = k∆t ∆x2 (uj+1 + uj−1 − 2uj) with un0 = u n J = 0 also by separation of variables. Make sure that you get the same answer in the limit of ∆t → 0 and ∆x → 0 (and J → ∞, so that J∆x → l. Practice Problems 1) Consider the initial value problem dx dt = − x log x , x(0) = x0 > 0 ,