Finite difference - wave equation - boundary conditions and setting things up - matlab

Question

I am working on a project that has to do with solving the wave equation in 2D (x, y, t) numericaly using the central difference approximation in MATLAB with the following boundary conditions:
The general assembly formula is:
I understand some of the boundary conditions (BC), like
du/dy=0 at j=m,
,
but I am not sure how to implement these boundary conditions in MATLAB.
A friend has given me these equations:
Here is my try with the MATLAB code,
but I am not able to progress any further:
% The wave function
% Explicit
% Universal boundary conditions for all 3 cases:
% u=0 at t=0
% du/dt=0 at t=0
% Case 1 boundary conditions
% At x=0, u=2sin(2*pi*t/5);
% At y=0, du/dy=0;
% At y=2, du/dy=0;
% At x=5, du/dx=0;
% u=0 and du/dt=0 at t=0;
%-------------------------------------------------------------------------%
% Setting up
clc; clear all; close all;
% length, time, height
L = 5; % [m]
h = 2; % [m]
T = 10; % [s]
% Constants
c_x = 1; % arbitrary
c_y = 1; % arbitrary
dx = 0.1; % <x> increment
dy = 0.1; % <y> increment
dt = 0.1; % time increment
nx = L/dx + 1; % number of <x> samples
ny = h/dy + 1; % number of <y> samples
nt = T/dt + 1; % number of time samples
t(:,1) = linspace(0, T, nt);
theta_x = c_x*(dt^2)/(dx^2);
theta_y = c_y*(dt^2)/(dy^2);
% theta_x = theta_y
theta = theta_x;
%-------------------------------------------------------------------------%
% The matrix
U = zeros(nt, nx, ny);
% Setting up the <U> matrix with the boundary conditions - case 1
U(1, :, :) = 0; % U=0 at t=0
for tt=1:nt % U=2sin(2pi/5*t) at x=0
for jj=1:ny
U(tt, 1, jj)=2*sin(2*pi/5.*t(tt));
end
end
for it=2:t
for ix=2:nx-1
for iy=2:ny-1
% Boundary conditions
% General case (internal):
U1 = -U(it-1, ix, iy);
U2 = 2*(1-2*theta)*u(it, ix, iy);
U3 = theta*U(it, ix-1, iy);
U4 = theta*U(it, ix+1, iy);
U5 = theta*U(it, ix, iy-1);
U6 = theta*U(it, ix, iy+1);
end
end
end

Answer 1

The general assembly formula you have kind of applies to the boundaries as well.
The complication is that when you apply the formula when j = 1 and j = m, you have j = 0 and j = m+1 term that are off of your grid.
To ameliorate this problem, boundary conditions give you a relationship between the points off the grid and on the grid.
As you have indicated, the dudy = 0 condition has given you the relation that u(i,m-1) == u(u,m+1) on the boundary. So you use the general assembly formula and replace all of the m+1 terms with m-1 on the boundary. You'll have a similar relation for the lower boundary as well.