////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
//solution to computational problem of hw#6 using backward differencing in 
//space and forward euler in time
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

xinitial=input("Enter x initial>");
xfinal=input("Enter x final>");
tinitial=input("Enter t initial>");
tfinal=input("Enter t final>");

J=input("Enter number of space steps>");
dt=input("Enter time step>");

dx=(xfinal-xinitial)/J;
N=(tfinal-tinitial)/dt;

lambda=dt/dx;

U=zeros(J+1,N+1);

for j=1:J+1
	U(j,1)=init((j-1)*dx);
end

for n=2:N+1

	U(1,n)=(1-lambda)*U(1,n-1)+lambda*U(J,n-1);

	for j=2:J+1
		U(j,n)=(1-lambda)*U(j,n-1) + lambda*U(j-1,n-1);
	end
end

x=linspace(xinitial,xfinal,J+1);
plot2d(x', [U(:,1),U(:,N+1)], leg="exact solution@numerical solution");