Finite Difference Method for second order ODE in MATLAB
$begingroup$
I have an ordinary differential equation,
$$frac{d^2Y}{dX^2} - frac{1}{R} frac{dY}{dX} = 0,$$
where $R$ is a constant. I have to numerically integrate this ODE for a range from $0$ to $1$ using the central difference method and the finite difference method. I have a couple of questions.
Our amount of steps are $30$ and I've rearranged the differential equation, substituting the difference equations for the differential equations in the formula. After I've done this, I create the central difference matrix using the three coefficients for $X_{i+1}$, $X_i$, and $X_{i-1}$ and multiply the matrix by a matrix with the $30$ steps. The output matrix in my code, $B$, doesn't seem to be correct.
Can anyone please tell me what is wrong with my algorithm? I'm having a hard time trying to find the problem. Thank you so much.
% initial conditions and constants
yn = 1;
y0 = 0;
N = 30; #steps
R = .01;
h = (yn-y0)/(N-1);
alph = 2*R - h; %alpha coefficient
bet = -4*R; %Beta Coefficient
gam = 2*R + h; %Gamma Coefficient
% initialize
cent_diff = zeros(N,N); b = zeros(N,1); b(1) = 0;
y = zeros(N,1); y(1) = 0;
% creates the y vector
for i = 2:N
y(i) = y(i-1,1) + h;
%b(i) = cent_diff(1,:)*y;
end
% creates the cent_diff matrix
for i =1:N %rows
for j = 1:N %columns
if i == j
cent_diff(i,j) = bet;
elseif i == j+1
cent_diff(i,j) = alph;
elseif i == j-1
cent_diff(i,j) = gam;
end
end
end
for k = 2:N-1
c = 0;
% loop to add products of matrix and y vector
for i = 1:N
b_k = cent_diff(1,i)*y(i) + c;
c = b_k;
end
b(k) = c;
end % last value of vector b
b(N) = -gam;
numerical-methods finite-differences finite-difference-methods
edited Dec 3 '18 at 23:28
rafa11111
1,1262417
asked Dec 3 '18 at 22:08
Sebastian SanchezSebastian Sanchez
1
$endgroup$
add a comment |
$begingroup$
I have an ordinary differential equation,
$$frac{d^2Y}{dX^2} - frac{1}{R} frac{dY}{dX} = 0,$$
where $R$ is a constant. I have to numerically integrate this ODE for a range from $0$ to $1$ using the central difference method and the finite difference method. I have a couple of questions.
Our amount of steps are $30$ and I've rearranged the differential equation, substituting the difference equations for the differential equations in the formula. After I've done this, I create the central difference matrix using the three coefficients for $X_{i+1}$, $X_i$, and $X_{i-1}$ and multiply the matrix by a matrix with the $30$ steps. The output matrix in my code, $B$, doesn't seem to be correct.
Can anyone please tell me what is wrong with my algorithm? I'm having a hard time trying to find the problem. Thank you so much.
% initial conditions and constants
yn = 1;
y0 = 0;
N = 30; #steps
R = .01;
h = (yn-y0)/(N-1);
alph = 2*R - h; %alpha coefficient
bet = -4*R; %Beta Coefficient
gam = 2*R + h; %Gamma Coefficient
% initialize
cent_diff = zeros(N,N); b = zeros(N,1); b(1) = 0;
y = zeros(N,1); y(1) = 0;
% creates the y vector
for i = 2:N
y(i) = y(i-1,1) + h;
%b(i) = cent_diff(1,:)*y;
end
% creates the cent_diff matrix
for i =1:N %rows
for j = 1:N %columns
if i == j
cent_diff(i,j) = bet;
elseif i == j+1
cent_diff(i,j) = alph;
elseif i == j-1
cent_diff(i,j) = gam;
end
end
end
for k = 2:N-1
c = 0;
% loop to add products of matrix and y vector
for i = 1:N
b_k = cent_diff(1,i)*y(i) + c;
c = b_k;
end
b(k) = c;
end % last value of vector b
b(N) = -gam;
numerical-methods finite-differences finite-difference-methods
edited Dec 3 '18 at 23:28
rafa11111
1,1262417
asked Dec 3 '18 at 22:08
Sebastian SanchezSebastian Sanchez
1
$endgroup$
$begingroup$
Do you have initial or boundary conditions?
$endgroup$
– rafa11111
Dec 3 '18 at 22:21
add a comment |
$begingroup$
I have an ordinary differential equation,
$$frac{d^2Y}{dX^2} - frac{1}{R} frac{dY}{dX} = 0,$$
where $R$ is a constant. I have to numerically integrate this ODE for a range from $0$ to $1$ using the central difference method and the finite difference method. I have a couple of questions.
Our amount of steps are $30$ and I've rearranged the differential equation, substituting the difference equations for the differential equations in the formula. After I've done this, I create the central difference matrix using the three coefficients for $X_{i+1}$, $X_i$, and $X_{i-1}$ and multiply the matrix by a matrix with the $30$ steps. The output matrix in my code, $B$, doesn't seem to be correct.
Can anyone please tell me what is wrong with my algorithm? I'm having a hard time trying to find the problem. Thank you so much.
% initial conditions and constants
yn = 1;
y0 = 0;
N = 30; #steps
R = .01;
h = (yn-y0)/(N-1);
alph = 2*R - h; %alpha coefficient
bet = -4*R; %Beta Coefficient
gam = 2*R + h; %Gamma Coefficient
% initialize
cent_diff = zeros(N,N); b = zeros(N,1); b(1) = 0;
y = zeros(N,1); y(1) = 0;
% creates the y vector
for i = 2:N
y(i) = y(i-1,1) + h;
%b(i) = cent_diff(1,:)*y;
end
% creates the cent_diff matrix
for i =1:N %rows
for j = 1:N %columns
if i == j
cent_diff(i,j) = bet;
elseif i == j+1
cent_diff(i,j) = alph;
elseif i == j-1
cent_diff(i,j) = gam;
end
end
end
for k = 2:N-1
c = 0;
% loop to add products of matrix and y vector
for i = 1:N
b_k = cent_diff(1,i)*y(i) + c;
c = b_k;
end
b(k) = c;
end % last value of vector b
b(N) = -gam;
numerical-methods finite-differences finite-difference-methods
edited Dec 3 '18 at 23:28
rafa11111
1,1262417
asked Dec 3 '18 at 22:08
Sebastian SanchezSebastian Sanchez
1
$endgroup$
I have an ordinary differential equation,
$$frac{d^2Y}{dX^2} - frac{1}{R} frac{dY}{dX} = 0,$$
where $R$ is a constant. I have to numerically integrate this ODE for a range from $0$ to $1$ using the central difference method and the finite difference method. I have a couple of questions.
Our amount of steps are $30$ and I've rearranged the differential equation, substituting the difference equations for the differential equations in the formula. After I've done this, I create the central difference matrix using the three coefficients for $X_{i+1}$, $X_i$, and $X_{i-1}$ and multiply the matrix by a matrix with the $30$ steps. The output matrix in my code, $B$, doesn't seem to be correct.
Can anyone please tell me what is wrong with my algorithm? I'm having a hard time trying to find the problem. Thank you so much.
% initial conditions and constants
yn = 1;
y0 = 0;
N = 30; #steps
R = .01;
h = (yn-y0)/(N-1);
alph = 2*R - h; %alpha coefficient
bet = -4*R; %Beta Coefficient
gam = 2*R + h; %Gamma Coefficient
% initialize
cent_diff = zeros(N,N); b = zeros(N,1); b(1) = 0;
y = zeros(N,1); y(1) = 0;
% creates the y vector
for i = 2:N
y(i) = y(i-1,1) + h;
%b(i) = cent_diff(1,:)*y;
end
% creates the cent_diff matrix
for i =1:N %rows
for j = 1:N %columns
if i == j
cent_diff(i,j) = bet;
elseif i == j+1
cent_diff(i,j) = alph;
elseif i == j-1
cent_diff(i,j) = gam;
end
end
end
for k = 2:N-1
c = 0;
% loop to add products of matrix and y vector
for i = 1:N
b_k = cent_diff(1,i)*y(i) + c;
c = b_k;
end
b(k) = c;
end % last value of vector b
b(N) = -gam;
numerical-methods finite-differences finite-difference-methods
numerical-methods finite-differences finite-difference-methods
edited Dec 3 '18 at 23:28
rafa11111
1,1262417
asked Dec 3 '18 at 22:08
Sebastian SanchezSebastian Sanchez
1
edited Dec 3 '18 at 23:28
rafa11111
1,1262417
asked Dec 3 '18 at 22:08
Sebastian SanchezSebastian Sanchez
1
edited Dec 3 '18 at 23:28
rafa11111
1,1262417
edited Dec 3 '18 at 23:28
rafa11111
1,1262417
edited Dec 3 '18 at 23:28
rafa11111
1,1262417
1,1262417
asked Dec 3 '18 at 22:08
Sebastian SanchezSebastian Sanchez
1
asked Dec 3 '18 at 22:08
Sebastian SanchezSebastian Sanchez
1
asked Dec 3 '18 at 22:08
Sebastian SanchezSebastian Sanchez
1
1
$begingroup$
Do you have initial or boundary conditions?
$endgroup$
– rafa11111
Dec 3 '18 at 22:21
add a comment |
$begingroup$
Do you have initial or boundary conditions?
$endgroup$
– rafa11111
Dec 3 '18 at 22:21
$begingroup$
Do you have initial or boundary conditions?
$endgroup$
– rafa11111
Dec 3 '18 at 22:21
$begingroup$
Do you have initial or boundary conditions?
$endgroup$
– rafa11111
Dec 3 '18 at 22:21
add a comment |
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$begingroup$
Do you have initial or boundary conditions?
$endgroup$
– rafa11111
Dec 3 '18 at 22:21