heatEq1D.m

function FPS_heat_equation
% 1D-Heat Equation: Fourier Pseudospectral
% RBF Course: HW#2
% Demo code of ode45/Fourier-PS solution of 
% PDE u_t = u_xx
% IC u(x,0) = 0.05/(1.05-cos(pi*x))
% for time t=0 to t=0.2

clear all
close all
clc


t0 = 0;
tf = 0.2;

n = 32;             % set space resolution

x = linspace(-1,1,n+1); x[end] = []; %layout datapoints; skip last.
u0 = (0.05./(1.05-cos(pi*x)) )';        % give Init. cond.

v = (-pi^2*[0:n/2, n/2-1:-1:1].^2)';    % vector to multiply in fourier space
                                        % to get second derivatives.
                                        
[t,u] = ode45(@dudt, t0:0.01:tf, u0, [], v);    % use ode45 to advance in time

u(:,n+1) = u(:,1);                  % Fill in right-edge in graphics.
mesh([x,1],t,u); colormap([0,0,0])  %Plot result


function uxx = dudt(t,u,v)
'''Calculate uxx by Fourier Pseudospectral Method'''
uf = fft(u);
uf = uf.*v;
uxx = real(ifft(uf));