Project 5 MAT 362 Due Sunday November 15th, Midnight. The goal of this project is to investigate the heat and wave partial differential equations (PDE), as well as a nonlinear elliptic ordinary BVP. (1) Code and test explicit, implicit, Crank-Nichols, and method of lines (MOL) solvers for the heat equation ut = cuxx, t > 0, x ∈ (0, 1) with 0-D BC and initial temperature distribution u(0, x) = f(x). Test with initial temperatures given by f(x) = sin(pix), sin(pix) + 12 sin(3pix), and something more interesting like a bump function. Compare approximate solutions to actual solutions. Try different values of c > 0, dx, and dt. Note the conditional or unconditional stabilities of the algorithms. Try 0- Neumann BC at one or both ends. (2) Code and test a MOL solver for the wave equation utt = c 2uxx, t > 0, x ∈ (0, 1) with 0-D BC and initial displacement and velocity given by u(0, x) = f(x) and ut(0, x) = g(x), respectively. Test with initial displacements given by f(x) = sin(pix) and the piecewise defined f(x) = (x− 14 )( 34 −x), x ∈ [ 14 , 34 ], f(x) = 0 otherwiswe, assuming g = 0. Compare approximate solutions to actual solutions obtained by the general solution or D’Alembert’s formula. Try different values of c, dx, and dt. Run for a long time an observe sta- bility, phase drift, or amplitude dissipation. Try 0-Neumann BC at one or both ends. (3) Solve y′′ + sy + y3 with 0-D BC using Newton’s method. Vary s and draw some bifurcation branches in the ||y||∞ vs s plane. Final Project ideas: (1) Repeat problems 1), 2), or 3) on the square (0, 1)2. (2) Write a general region code to produce the D2 matrix for a subregion of (0, 1)2. Apply to solve an elliptic, hyperbolic, or parabolic problem on that region. (3) Experiment with 1), 2), or 3) adding other terms like convection or damp- ing, and/or for other boundary conditions.
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