sandbox/easystab/boundary_conditions.m

    How to impose boundary conditions

    This code is built upon diffmat.m where the differentiation matrices are built. Hee we solve a linear non-homogeneous differential equation in 1D with non-homogeneous boundary conditions

    clear all; clf

% parameters
L=2*pi; % domain length
N=25; % number of points

% the grid
x=linspace(0,L,N)';
h=x(2)-x(1); % the grid size

% first derivative
D=zeros(N,N);
D(1,1:3)=[-3/2, 2, -1/2]/h;
for ind=2:N-1
    D(ind,ind-1:ind+1)=[-1/2, 0, 1/2]/h;
end
D(end,end-2:end)=[1/2, -2, 3/2]/h;

% second derivative
DD=zeros(N,N);
DD(1,1:3)=[1, -2, 1]/h^2;
for ind=2:N-1
    DD(ind,ind-1:ind+1)=[1, -2, 1]/h^2;
end
DD(end,end-2:end)=[1, -2, 1]/h^2;

    The differential equation

    We solve this \displaystyle f_{xx}=1 with the boundary conditions \displaystyle f(1)=0, f(L)=1 

    % I build the linear differential equation
A=D;

% second order derivative
A=DD;

% boundary conditions
I=eye(N);
A([1,N],:)=I([1,N],:);
b=1+zeros(N,1); b([1,N])=[0,0];

% solve the system
f=A\b;

% plotting
plot(x,f,'b.-',x,x.^2/2-pi*x,'r-');
xlabel('x');ylabel('f');
legend('numerical','theory')
xlim([0,L]); grid on

set(gcf,'paperpositionmode','auto')
print('-dpng','-r75','boundary_conditions.png')

    Exercices/Contributions

    • Please put the two boundary conditions on the same side of the domain (for instance, impose the value of f and its derivative at x=0)
    • Please solve f_{xx}=\sin(x)
    • Please check the convergence with the number of gridpoints
    • Please use Chebychev differentiation matrix chebdif.m and compare with the finite differences
    • Please use a non-constant coefficient i your differential equation a(x)f_{xx}=0 (first find the way to code that using differentiation matrices) %}