Mixing
How to choose Eigenvaues for system extending in perpendicular direction?
$begingroup$
I have two linear third order ODEs with a separation constant (eigenvalue parameter) on a rectangular domain where $x in (0,1)$ and $y in (0,1)$as follows:
$$
lambda_h F''' - 2 lambda_h beta_h F'' + ( (lambda_h beta_h - 1) beta_h ) F' + beta_h^2 F = mu F',\
V lambda_c G''' - 2 V lambda_c beta_c G'' +( (lambda_c beta_c - 1) V beta_c ) G' + V beta_c^2 G = -mu G',
$$
$mu in mathbb{R}$.
The b.c for $F(x)$ are $F(0) = 0, frac{F''(0)}{F'(0)}=beta_h,frac{F''(1)}{F'(1)}=beta_h $
For $G(y)$ they are $G(0) = 0, frac{G''(0)}{G'(0)}=beta_c,frac{G''(1)}{G'(1)}=beta_c$
For these bc(s) and $lambda_h=0.02, beta_h = 10$, I calculated the Eigenvalues using chebfun in MATLAB which came out to be
F = -37.6413, -32.9463, -28.6002, -24.5873, -20.8885, -17.4846, -14.3643 -11.5367, -9.0383, -6.9287,......
G = 6.9287, 9.0383, 11.5367, 14.3643, 17.4846, 20.8885, 24.5873, 28.6002, 32.9463, 37.6413, .....
I now need to find the solutions to these two separated ODEs, to form my final solution.
What bothers me is that EVs of $F$ are in infinite numbers in the $-x$ direction, while of $G$ are in infinite numbers in $+y$ direction with the same magnitudes.
I cannot figure out what EVs should I choose and how many i should to build my solution ? I know that they are separated by the same constant. Basically, how should i proceed. haven't gotten any clue from texts.
Steps after finding the EVs
The general solution will be of the form
$$
F(x) = sum_k C_k e^{-delta_k(mu)x}
$$
where $delta_k(mu)$ is a root of the characteristic equation dependent on $mu$. There would be three roots of the char. eqn. So each EV when substituted in the characteristic equation should give three roots which would then be used to calculate $C_1,C_2,C_3$ constants using the b.c.
This is where i am facing problems in determining, how many and which EVs should be considered.
ordinary-differential-equations eigenvalues-eigenvectors problem-solving matlab boundary-value-problem
asked Jan 22 at 16:42
Indrasis MitraIndrasis Mitra
80111
$endgroup$
add a comment |
$begingroup$
I have two linear third order ODEs with a separation constant (eigenvalue parameter) on a rectangular domain where $x in (0,1)$ and $y in (0,1)$as follows:
$$
lambda_h F''' - 2 lambda_h beta_h F'' + ( (lambda_h beta_h - 1) beta_h ) F' + beta_h^2 F = mu F',\
V lambda_c G''' - 2 V lambda_c beta_c G'' +( (lambda_c beta_c - 1) V beta_c ) G' + V beta_c^2 G = -mu G',
$$
$mu in mathbb{R}$.
The b.c for $F(x)$ are $F(0) = 0, frac{F''(0)}{F'(0)}=beta_h,frac{F''(1)}{F'(1)}=beta_h $
For $G(y)$ they are $G(0) = 0, frac{G''(0)}{G'(0)}=beta_c,frac{G''(1)}{G'(1)}=beta_c$
For these bc(s) and $lambda_h=0.02, beta_h = 10$, I calculated the Eigenvalues using chebfun in MATLAB which came out to be
F = -37.6413, -32.9463, -28.6002, -24.5873, -20.8885, -17.4846, -14.3643 -11.5367, -9.0383, -6.9287,......
G = 6.9287, 9.0383, 11.5367, 14.3643, 17.4846, 20.8885, 24.5873, 28.6002, 32.9463, 37.6413, .....
I now need to find the solutions to these two separated ODEs, to form my final solution.
What bothers me is that EVs of $F$ are in infinite numbers in the $-x$ direction, while of $G$ are in infinite numbers in $+y$ direction with the same magnitudes.
I cannot figure out what EVs should I choose and how many i should to build my solution ? I know that they are separated by the same constant. Basically, how should i proceed. haven't gotten any clue from texts.
Steps after finding the EVs
The general solution will be of the form
$$
F(x) = sum_k C_k e^{-delta_k(mu)x}
$$
where $delta_k(mu)$ is a root of the characteristic equation dependent on $mu$. There would be three roots of the char. eqn. So each EV when substituted in the characteristic equation should give three roots which would then be used to calculate $C_1,C_2,C_3$ constants using the b.c.
This is where i am facing problems in determining, how many and which EVs should be considered.
ordinary-differential-equations eigenvalues-eigenvectors problem-solving matlab boundary-value-problem
asked Jan 22 at 16:42
Indrasis MitraIndrasis Mitra
80111
$endgroup$
$begingroup$
A remark : is $V$ (which is factorizable in the LHS of the second equation) a known or unknown quantity ? This question is linked to the fact that I think that replacing $mu$ by $mu/V$ would may be simplify this issue (with Neumann boundary conditions, more difficult to tackle than Cauchy initial conditions).
$endgroup$
– Jean Marie
Jan 22 at 22:01
$begingroup$
What are the values of $lambda_c$ and $beta_c$ ?
$endgroup$
– Jean Marie
Jan 22 at 22:16
$begingroup$
@JeanMarie $lambda_c=0.02$ $beta_c=10$ and $V=1$ can be consodered a case
$endgroup$
– Indrasis Mitra
Jan 22 at 23:52
$begingroup$
@JeanMarie I tired the problem with $V=1$, all the eigenvalues that i have reported for $G(y)$ are calculated for $V=1$. I am just at a loss in interpreting them. I have edited my original question to reflect how i am trying to approach the problem.
$endgroup$
– Indrasis Mitra
Jan 23 at 2:24
$begingroup$
Now posted to MO, mathoverflow.net/questions/321528/…
$endgroup$
– Gerry Myerson
Jan 23 at 11:59
add a comment |
$begingroup$
I have two linear third order ODEs with a separation constant (eigenvalue parameter) on a rectangular domain where $x in (0,1)$ and $y in (0,1)$as follows:
$$
lambda_h F''' - 2 lambda_h beta_h F'' + ( (lambda_h beta_h - 1) beta_h ) F' + beta_h^2 F = mu F',\
V lambda_c G''' - 2 V lambda_c beta_c G'' +( (lambda_c beta_c - 1) V beta_c ) G' + V beta_c^2 G = -mu G',
$$
$mu in mathbb{R}$.
The b.c for $F(x)$ are $F(0) = 0, frac{F''(0)}{F'(0)}=beta_h,frac{F''(1)}{F'(1)}=beta_h $
For $G(y)$ they are $G(0) = 0, frac{G''(0)}{G'(0)}=beta_c,frac{G''(1)}{G'(1)}=beta_c$
For these bc(s) and $lambda_h=0.02, beta_h = 10$, I calculated the Eigenvalues using chebfun in MATLAB which came out to be
F = -37.6413, -32.9463, -28.6002, -24.5873, -20.8885, -17.4846, -14.3643 -11.5367, -9.0383, -6.9287,......
G = 6.9287, 9.0383, 11.5367, 14.3643, 17.4846, 20.8885, 24.5873, 28.6002, 32.9463, 37.6413, .....
I now need to find the solutions to these two separated ODEs, to form my final solution.
What bothers me is that EVs of $F$ are in infinite numbers in the $-x$ direction, while of $G$ are in infinite numbers in $+y$ direction with the same magnitudes.
I cannot figure out what EVs should I choose and how many i should to build my solution ? I know that they are separated by the same constant. Basically, how should i proceed. haven't gotten any clue from texts.
Steps after finding the EVs
The general solution will be of the form
$$
F(x) = sum_k C_k e^{-delta_k(mu)x}
$$
where $delta_k(mu)$ is a root of the characteristic equation dependent on $mu$. There would be three roots of the char. eqn. So each EV when substituted in the characteristic equation should give three roots which would then be used to calculate $C_1,C_2,C_3$ constants using the b.c.
This is where i am facing problems in determining, how many and which EVs should be considered.
ordinary-differential-equations eigenvalues-eigenvectors problem-solving matlab boundary-value-problem
asked Jan 22 at 16:42
Indrasis MitraIndrasis Mitra
80111
$endgroup$
I have two linear third order ODEs with a separation constant (eigenvalue parameter) on a rectangular domain where $x in (0,1)$ and $y in (0,1)$as follows:
$$
lambda_h F''' - 2 lambda_h beta_h F'' + ( (lambda_h beta_h - 1) beta_h ) F' + beta_h^2 F = mu F',\
V lambda_c G''' - 2 V lambda_c beta_c G'' +( (lambda_c beta_c - 1) V beta_c ) G' + V beta_c^2 G = -mu G',
$$
$mu in mathbb{R}$.
The b.c for $F(x)$ are $F(0) = 0, frac{F''(0)}{F'(0)}=beta_h,frac{F''(1)}{F'(1)}=beta_h $
For $G(y)$ they are $G(0) = 0, frac{G''(0)}{G'(0)}=beta_c,frac{G''(1)}{G'(1)}=beta_c$
For these bc(s) and $lambda_h=0.02, beta_h = 10$, I calculated the Eigenvalues using chebfun in MATLAB which came out to be
F = -37.6413, -32.9463, -28.6002, -24.5873, -20.8885, -17.4846, -14.3643 -11.5367, -9.0383, -6.9287,......
G = 6.9287, 9.0383, 11.5367, 14.3643, 17.4846, 20.8885, 24.5873, 28.6002, 32.9463, 37.6413, .....
I now need to find the solutions to these two separated ODEs, to form my final solution.
What bothers me is that EVs of $F$ are in infinite numbers in the $-x$ direction, while of $G$ are in infinite numbers in $+y$ direction with the same magnitudes.
I cannot figure out what EVs should I choose and how many i should to build my solution ? I know that they are separated by the same constant. Basically, how should i proceed. haven't gotten any clue from texts.
Steps after finding the EVs
The general solution will be of the form
$$
F(x) = sum_k C_k e^{-delta_k(mu)x}
$$
where $delta_k(mu)$ is a root of the characteristic equation dependent on $mu$. There would be three roots of the char. eqn. So each EV when substituted in the characteristic equation should give three roots which would then be used to calculate $C_1,C_2,C_3$ constants using the b.c.
This is where i am facing problems in determining, how many and which EVs should be considered.
ordinary-differential-equations eigenvalues-eigenvectors problem-solving matlab boundary-value-problem
ordinary-differential-equations eigenvalues-eigenvectors problem-solving matlab boundary-value-problem
asked Jan 22 at 16:42
Indrasis MitraIndrasis Mitra
80111
asked Jan 22 at 16:42
Indrasis MitraIndrasis Mitra
80111
edited Jan 23 at 4:29
Indrasis Mitra
asked Jan 22 at 16:42
Indrasis MitraIndrasis Mitra
80111
asked Jan 22 at 16:42
Indrasis MitraIndrasis Mitra
80111
asked Jan 22 at 16:42
Indrasis MitraIndrasis Mitra
80111
80111
$begingroup$
A remark : is $V$ (which is factorizable in the LHS of the second equation) a known or unknown quantity ? This question is linked to the fact that I think that replacing $mu$ by $mu/V$ would may be simplify this issue (with Neumann boundary conditions, more difficult to tackle than Cauchy initial conditions).
$endgroup$
– Jean Marie
Jan 22 at 22:01
$begingroup$
What are the values of $lambda_c$ and $beta_c$ ?
$endgroup$
– Jean Marie
Jan 22 at 22:16
$begingroup$
@JeanMarie $lambda_c=0.02$ $beta_c=10$ and $V=1$ can be consodered a case
$endgroup$
– Indrasis Mitra
Jan 22 at 23:52
$begingroup$
@JeanMarie I tired the problem with $V=1$, all the eigenvalues that i have reported for $G(y)$ are calculated for $V=1$. I am just at a loss in interpreting them. I have edited my original question to reflect how i am trying to approach the problem.
$endgroup$
– Indrasis Mitra
Jan 23 at 2:24
$begingroup$
Now posted to MO, mathoverflow.net/questions/321528/…
$endgroup$
– Gerry Myerson
Jan 23 at 11:59
add a comment |
$begingroup$
A remark : is $V$ (which is factorizable in the LHS of the second equation) a known or unknown quantity ? This question is linked to the fact that I think that replacing $mu$ by $mu/V$ would may be simplify this issue (with Neumann boundary conditions, more difficult to tackle than Cauchy initial conditions).
$endgroup$
– Jean Marie
Jan 22 at 22:01
$begingroup$
What are the values of $lambda_c$ and $beta_c$ ?
$endgroup$
– Jean Marie
Jan 22 at 22:16
$begingroup$
@JeanMarie $lambda_c=0.02$ $beta_c=10$ and $V=1$ can be consodered a case
$endgroup$
– Indrasis Mitra
Jan 22 at 23:52
$begingroup$
@JeanMarie I tired the problem with $V=1$, all the eigenvalues that i have reported for $G(y)$ are calculated for $V=1$. I am just at a loss in interpreting them. I have edited my original question to reflect how i am trying to approach the problem.
$endgroup$
– Indrasis Mitra
Jan 23 at 2:24
$begingroup$
Now posted to MO, mathoverflow.net/questions/321528/…
$endgroup$
– Gerry Myerson
Jan 23 at 11:59
$begingroup$
A remark : is $V$ (which is factorizable in the LHS of the second equation) a known or unknown quantity ? This question is linked to the fact that I think that replacing $mu$ by $mu/V$ would may be simplify this issue (with Neumann boundary conditions, more difficult to tackle than Cauchy initial conditions).
$endgroup$
– Jean Marie
Jan 22 at 22:01
$begingroup$
A remark : is $V$ (which is factorizable in the LHS of the second equation) a known or unknown quantity ? This question is linked to the fact that I think that replacing $mu$ by $mu/V$ would may be simplify this issue (with Neumann boundary conditions, more difficult to tackle than Cauchy initial conditions).
$endgroup$
– Jean Marie
Jan 22 at 22:01
$begingroup$
What are the values of $lambda_c$ and $beta_c$ ?
$endgroup$
– Jean Marie
Jan 22 at 22:16
$begingroup$
What are the values of $lambda_c$ and $beta_c$ ?
$endgroup$
– Jean Marie
Jan 22 at 22:16
$begingroup$
@JeanMarie $lambda_c=0.02$ $beta_c=10$ and $V=1$ can be consodered a case
$endgroup$
– Indrasis Mitra
Jan 22 at 23:52
$begingroup$
@JeanMarie $lambda_c=0.02$ $beta_c=10$ and $V=1$ can be consodered a case
$endgroup$
– Indrasis Mitra
Jan 22 at 23:52
$begingroup$
@JeanMarie I tired the problem with $V=1$, all the eigenvalues that i have reported for $G(y)$ are calculated for $V=1$. I am just at a loss in interpreting them. I have edited my original question to reflect how i am trying to approach the problem.
$endgroup$
– Indrasis Mitra
Jan 23 at 2:24
$begingroup$
@JeanMarie I tired the problem with $V=1$, all the eigenvalues that i have reported for $G(y)$ are calculated for $V=1$. I am just at a loss in interpreting them. I have edited my original question to reflect how i am trying to approach the problem.
$endgroup$
– Indrasis Mitra
Jan 23 at 2:24
$begingroup$
Now posted to MO, mathoverflow.net/questions/321528/…
$endgroup$
– Gerry Myerson
Jan 23 at 11:59
$begingroup$
Now posted to MO, mathoverflow.net/questions/321528/…
$endgroup$
– Gerry Myerson
Jan 23 at 11:59
add a comment |
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$begingroup$
A remark : is $V$ (which is factorizable in the LHS of the second equation) a known or unknown quantity ? This question is linked to the fact that I think that replacing $mu$ by $mu/V$ would may be simplify this issue (with Neumann boundary conditions, more difficult to tackle than Cauchy initial conditions).
$endgroup$
– Jean Marie
Jan 22 at 22:01
$begingroup$
What are the values of $lambda_c$ and $beta_c$ ?
$endgroup$
– Jean Marie
Jan 22 at 22:16
$begingroup$
@JeanMarie $lambda_c=0.02$ $beta_c=10$ and $V=1$ can be consodered a case
$endgroup$
– Indrasis Mitra
Jan 22 at 23:52
$begingroup$
@JeanMarie I tired the problem with $V=1$, all the eigenvalues that i have reported for $G(y)$ are calculated for $V=1$. I am just at a loss in interpreting them. I have edited my original question to reflect how i am trying to approach the problem.
$endgroup$
– Indrasis Mitra
Jan 23 at 2:24
$begingroup$
Now posted to MO, mathoverflow.net/questions/321528/…
$endgroup$
– Gerry Myerson
Jan 23 at 11:59