Parabolic PDEs 1. Implement in MATLAB/Octave or Python a finite difference method to solve a parabolic PDE problem in 1D, which is defined as:

ut(x, t) = c 2 uxx (x, t) in [a, b] [0, T] with u(a, t) = 0, u(b, t) = 0 and u(x, 0) = f (x).

You are required to write MATLAB/Octave function FDparabolic.m with the following input-output format:

function [x,t,u] = FDparabolic( a, b, n, T, m, c, f ) or for Python def FDparabolic( a, b, n, T, m, c, f ): ...

where the inputs a and b are the extrema, n is the number of subintervals, T is the final time, m is the number of time steps, c is the coefficient c and f is the initial value. The outputs x contains the nodes of partition, t contains the nodes in time and u contains the values of the computed solution at the nodes

.

2. Use the implemented code FDparabolic.m to compute the solution of the problem:

ut(x, t) = uxx (x, t), u(x, 0) = sin(x) in [0, ] [0, 1] with u(0, t) = u(, t) along the boundary. Use 20 nodes in space and 100 in time.

3. Produce a visualisation of the solution using gnuplot or MATLAB/Octave command surf( t, x, u )

3.3.2 Hyperbolic PDEs 1. Implement in MATLAB/Octave or Python a finite difference method to solve a hyperbolic PDE problem in 1D, which is defined as: utt(x, t) = c 2 uxx (x, t)

in [a, b] [0, T] with u(a, t) = 0, u(b, t) = 0 and u(x, 0) = f (x), ut(x, 0) = g(x).

You are required to write MATLAB/Octave function FDparabolic.m with the following input-output format: function [x,t,u] = FDhyperbolic( a, b, n, T, m, c, f, g )

Or in Python def FDhyperbolic( a, b, n, T, m, c, f, g ):.

.. where the inputs a and b are the extrema, n is the number of subintervals, T is the final time, m is the number of time steps, c is the coefficient c, f is the initial value for u and g is the initial value for ut . The outputs x contains the nodes of partition, t contains the nodes in time and u contains the values of the computed solution at the nodes.

2. Use the implemented code FDhyperbolic.m to compute the solution of the problem: ut(x, t) = 4uxx (x, t), u(x, 0) = sin(x), ut(x, 0) = 0

in [0, ] [0, 1] with u(0, t) = u(, t) along the boundary. Use 20 nodes in space and 40 in time. 3. Produce a visualisation of the solution using gnuplot or MATLAB/Octave command surf( t, x, u )

3.3.3 Numerical Instability Investigate the effects on the solutions of various combinations of number of subintervals n and time steps m by using the functions developed (FEparabolic.m and hyperbolic.m) to compute the solutions of u(0.5, 0.5) with time steps of 10, 100, 200 and 400; and subintervals of 4, 16 and 32, thus complete a table similar to that shown in Table 3 for each case.

m n u(/2, 0.5)

10 4

16

32

100 4

16

32

200 4

16

32

400 4

16

32