1-Dim wave equation

Consider the following 1D heat problem where the high function depends on space and time

with initial condition

and boundary conditions .

Solve this problem for the parameter values

if we define

clear all;

%-------------------------------------------

% Initialize variables

%-------------------------------------------

dt = 0.01; %time step

dx = 1/100; %space step

c = 0.05; %propagation coeff.

r = c/dx^2; % dt*c/dx^2; the dt term is added in the solver step

x = 0:dx:1;

m = length(x);

u = zeros(1,m);

v = zeros(1,m);

%--------------------

% Time evolution

%--------------------

numIter = 750;

t = 0; %time

for step=1:numIter

t = t+dt;

%-----------------------------------------------------

% compute acceleration values (forces) that modify velocity

%-----------------------------------------------------

for j=1:m

jleft = j-1;

if (j==1), jleft=1; end

jright = min(j+1,m);

if (j==m), jright=m; end

force(j) = r*(u(jleft)-2*u(j)+u(jright));

end

%------------------------------------

%Solver: Semi-Implicit Euler Method

%------------------------------------

v = v + dt*force; %update velocities

u = u + dt*v; %update positions with the updated velocities

%------------------------------------

u(1) = 3*sin(5*t); %impose boundary conditions;

u(end) = 0;

% -----------------------------------

plot(x, u);

axis([-0.1,1.1,-10,10]);

text(0,8,['time = ' num2str(t)]);

drawnow;

end

initial condition and the following initial conditions to create an initial wave:

and

boundary conditions .