I am unable to input the dirichlet condition into this non homogenous ode. Please point out my mistake.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% -u" + u = – 5x + 4 0 < x <= 1 %%%%
%%%% u(0) = 0, u(1) = 2 %%%%
%%%% Exact solution u = (exp(x)*(3*exp(1) + 4))/(exp(2) – 1) – 5*x – (exp(-x)*(3*exp(1) + 4*exp(2)))/(exp(2) – 1) + 4 %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
format long;
N = 20;
x0 = 0;
xN = 1;
h = 1/N;
for j = 1:N+1
X(j) = x0 + (j-1)*h;
end
f = @(x)(- 5*x +4);
A = zeros(N+1,N+1);
F = zeros(N+1,1);
%%%% Local stiffness matrix %%%%
a = [1/h -1/h ; -1/h 1/h] ;
for i=1:N
phi1 = @(x)(X(i+1)-x)/h; %%%% linear basis function %%%%
phi2 = @(x)(x-X(i))/h; %%%% linear basis function %%%%
f1 = @(x)f(x)*phi1(x); %%%% integrand for load vector %%%%
f2 = @(x)f(x)*phi2(x); %%%% integrand for load vector %%%%
v(1,i) = gauss(f1,X(i),X(i+1),1); %%%% element wise values of
v(2,i) = gauss(f2,X(i),X(i+1),1); %%%% load vector
end
%%%% Assembling %%%%
for i=1:N
A([i i+1],[i i+1]) = A([i i+1],[i i+1]) + a;
F([i i+1],1) = F([i i+1],1) + v([1 2],i);
end
%%%% Dirichlet Boundary condition %%%%
% F(N+1,1) = F(N+1,1)+2;
fullnodes = [1:N+1];
%%%%% Dirichlet boundary condition %%%%%
freenodes=setdiff(fullnodes,[1]);
Uh = zeros(N+1,1);
Uh(N+1,1) = 2;
%%%% Approximate solution %%%%
Uh(freenodes)=A(freenodes,freenodes)F(freenodes,1);
%%%% Exact solution %%%%
U = zeros(N+1,1);
for i =1:N+1
U(i) = (exp(X(i))*(3*exp(1) + 4))/(exp(2) – 1) – 5*X(i) – (exp(-X(i))*(3*exp(1) + 4*exp(2)))/(exp(2) – 1) + 4;
end
error = max(abs(U-Uh));
[U Uh]
plot(X,U,X,Uh,’o’)I am unable to input the dirichlet condition into this non homogenous ode. Please point out my mistake.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% -u" + u = – 5x + 4 0 < x <= 1 %%%%
%%%% u(0) = 0, u(1) = 2 %%%%
%%%% Exact solution u = (exp(x)*(3*exp(1) + 4))/(exp(2) – 1) – 5*x – (exp(-x)*(3*exp(1) + 4*exp(2)))/(exp(2) – 1) + 4 %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
format long;
N = 20;
x0 = 0;
xN = 1;
h = 1/N;
for j = 1:N+1
X(j) = x0 + (j-1)*h;
end
f = @(x)(- 5*x +4);
A = zeros(N+1,N+1);
F = zeros(N+1,1);
%%%% Local stiffness matrix %%%%
a = [1/h -1/h ; -1/h 1/h] ;
for i=1:N
phi1 = @(x)(X(i+1)-x)/h; %%%% linear basis function %%%%
phi2 = @(x)(x-X(i))/h; %%%% linear basis function %%%%
f1 = @(x)f(x)*phi1(x); %%%% integrand for load vector %%%%
f2 = @(x)f(x)*phi2(x); %%%% integrand for load vector %%%%
v(1,i) = gauss(f1,X(i),X(i+1),1); %%%% element wise values of
v(2,i) = gauss(f2,X(i),X(i+1),1); %%%% load vector
end
%%%% Assembling %%%%
for i=1:N
A([i i+1],[i i+1]) = A([i i+1],[i i+1]) + a;
F([i i+1],1) = F([i i+1],1) + v([1 2],i);
end
%%%% Dirichlet Boundary condition %%%%
% F(N+1,1) = F(N+1,1)+2;
fullnodes = [1:N+1];
%%%%% Dirichlet boundary condition %%%%%
freenodes=setdiff(fullnodes,[1]);
Uh = zeros(N+1,1);
Uh(N+1,1) = 2;
%%%% Approximate solution %%%%
Uh(freenodes)=A(freenodes,freenodes)F(freenodes,1);
%%%% Exact solution %%%%
U = zeros(N+1,1);
for i =1:N+1
U(i) = (exp(X(i))*(3*exp(1) + 4))/(exp(2) – 1) – 5*X(i) – (exp(-X(i))*(3*exp(1) + 4*exp(2)))/(exp(2) – 1) + 4;
end
error = max(abs(U-Uh));
[U Uh]
plot(X,U,X,Uh,’o’) I am unable to input the dirichlet condition into this non homogenous ode. Please point out my mistake.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% -u" + u = – 5x + 4 0 < x <= 1 %%%%
%%%% u(0) = 0, u(1) = 2 %%%%
%%%% Exact solution u = (exp(x)*(3*exp(1) + 4))/(exp(2) – 1) – 5*x – (exp(-x)*(3*exp(1) + 4*exp(2)))/(exp(2) – 1) + 4 %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
format long;
N = 20;
x0 = 0;
xN = 1;
h = 1/N;
for j = 1:N+1
X(j) = x0 + (j-1)*h;
end
f = @(x)(- 5*x +4);
A = zeros(N+1,N+1);
F = zeros(N+1,1);
%%%% Local stiffness matrix %%%%
a = [1/h -1/h ; -1/h 1/h] ;
for i=1:N
phi1 = @(x)(X(i+1)-x)/h; %%%% linear basis function %%%%
phi2 = @(x)(x-X(i))/h; %%%% linear basis function %%%%
f1 = @(x)f(x)*phi1(x); %%%% integrand for load vector %%%%
f2 = @(x)f(x)*phi2(x); %%%% integrand for load vector %%%%
v(1,i) = gauss(f1,X(i),X(i+1),1); %%%% element wise values of
v(2,i) = gauss(f2,X(i),X(i+1),1); %%%% load vector
end
%%%% Assembling %%%%
for i=1:N
A([i i+1],[i i+1]) = A([i i+1],[i i+1]) + a;
F([i i+1],1) = F([i i+1],1) + v([1 2],i);
end
%%%% Dirichlet Boundary condition %%%%
% F(N+1,1) = F(N+1,1)+2;
fullnodes = [1:N+1];
%%%%% Dirichlet boundary condition %%%%%
freenodes=setdiff(fullnodes,[1]);
Uh = zeros(N+1,1);
Uh(N+1,1) = 2;
%%%% Approximate solution %%%%
Uh(freenodes)=A(freenodes,freenodes)F(freenodes,1);
%%%% Exact solution %%%%
U = zeros(N+1,1);
for i =1:N+1
U(i) = (exp(X(i))*(3*exp(1) + 4))/(exp(2) – 1) – 5*X(i) – (exp(-X(i))*(3*exp(1) + 4*exp(2)))/(exp(2) – 1) + 4;
end
error = max(abs(U-Uh));
[U Uh]
plot(X,U,X,Uh,’o’) finite element method MATLAB Answers — New Questions

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