Mixing
Can this problem be reduced to a Sturm-Liouville form?
$begingroup$
From a system of three coupled PDEs
begin{eqnarray}
frac{partial theta_h}{partial x} + beta_h (theta_h - theta_w) &=& 0,\
frac{partial theta_c}{partial y} + beta_c (theta_c - theta_w) &=& 0,\
lambda_h frac{partial^2 theta_w}{partial x^2} + lambda_c V frac{partial^2 theta_w}{partial y^2} - frac{partial theta_h}{partial x} - Vfrac{partial theta_c}{partial y} &=& 0
end{eqnarray}
with the bc(s) as
The boundary conditions for the problem are as follows:
The PDE(s) needs to be solved on a rectangular region where $x$ varies between $0$ to $1$ and $y$ varies between $0$ to $1$.
$$frac{partial theta_w(0,y)}{partial x}=frac{partial theta_w(1,y)}{partial x}=0 $$
$$frac{partial theta_w(x,0)}{partial y}=frac{partial theta_w(x,1)}{partial y}=0 $$
$$theta_h(0,y)=1 $$$$theta_c(x,0)=0$$
The first two were solved and substituted in the following form as
begin{eqnarray}
0 &=& e^{-beta_h x} left( lambda_h e^{beta_h x} frac{partial^2 theta_w}{partial x^2} - beta_h e^{beta_h x} theta_w + beta_h^2 int e^{beta_h x} theta_w , mathrm{d}x right) +\
&& + V e^{-beta_c y} left( lambda_c e^{beta_c y} frac{partial^2 theta_w}{partial y^2} - beta_c e^{beta_c y} theta_w + beta_c^2 int e^{beta_c y} theta_w , mathrm{d}y right).
end{eqnarray}
The following assumption was made then
$theta_w(x,y) = e^{-beta_h x} f(x) e^{-beta_c y} g(y)$ where $F(x) := int f(x) , mathrm{d}x$ and $G(y) := int g(y) , mathrm{d}y$ to reach the following two separated third order ODEs
begin{eqnarray}
lambda_h F''' - 2 lambda_h beta_h F'' + left( (lambda_h beta_h - 1) beta_h - mu right) F' + beta_h^2 F &=& 0,\
V lambda_c G''' - 2 V lambda_c beta_c G'' + left( (lambda_c beta_c - 1) V beta_c + mu right) G' + V beta_c^2 G &=& 0,
end{eqnarray}
Can these ODEs be expressed in Sturm-Liouville form of EigenValue problems (All texts i see show examples for second order problem ?)
My attempts at solving these two ODEs as eigen value problems were failing
1 Eigen values of a Third Order Linear Homogenous ODE
2 How to choose Eigenvaues for system extending in perpendicular direction?
3 Eigenvalues keep giving trivial solutions everytime.
I must mention here that user @Christoph ,@Cesareo and @LutzL have been extremely helpful in getting me this far.
So I have been trying other approaches and Sturm - Liouville problems was a form of problem i came across.
ATTEMPT
If i try the following form:
$$
lambda_h f'' - 2 lambda_h beta_h f' + ( (lambda_h beta_h - 1) beta_h) f' + beta_h^2 int f mathrm{d}x = mu f
$$
But this form has an integral operator on the LHS, Differentiating it w.r.t. $x$ makes it a third order problem.
ordinary-differential-equations multivariable-calculus pde eigenfunctions sturm-liouville
asked Jan 25 at 9:21
Indrasis MitraIndrasis Mitra
80111
$endgroup$
add a comment |
$begingroup$
From a system of three coupled PDEs
begin{eqnarray}
frac{partial theta_h}{partial x} + beta_h (theta_h - theta_w) &=& 0,\
frac{partial theta_c}{partial y} + beta_c (theta_c - theta_w) &=& 0,\
lambda_h frac{partial^2 theta_w}{partial x^2} + lambda_c V frac{partial^2 theta_w}{partial y^2} - frac{partial theta_h}{partial x} - Vfrac{partial theta_c}{partial y} &=& 0
end{eqnarray}
with the bc(s) as
The boundary conditions for the problem are as follows:
The PDE(s) needs to be solved on a rectangular region where $x$ varies between $0$ to $1$ and $y$ varies between $0$ to $1$.
$$frac{partial theta_w(0,y)}{partial x}=frac{partial theta_w(1,y)}{partial x}=0 $$
$$frac{partial theta_w(x,0)}{partial y}=frac{partial theta_w(x,1)}{partial y}=0 $$
$$theta_h(0,y)=1 $$$$theta_c(x,0)=0$$
The first two were solved and substituted in the following form as
begin{eqnarray}
0 &=& e^{-beta_h x} left( lambda_h e^{beta_h x} frac{partial^2 theta_w}{partial x^2} - beta_h e^{beta_h x} theta_w + beta_h^2 int e^{beta_h x} theta_w , mathrm{d}x right) +\
&& + V e^{-beta_c y} left( lambda_c e^{beta_c y} frac{partial^2 theta_w}{partial y^2} - beta_c e^{beta_c y} theta_w + beta_c^2 int e^{beta_c y} theta_w , mathrm{d}y right).
end{eqnarray}
The following assumption was made then
$theta_w(x,y) = e^{-beta_h x} f(x) e^{-beta_c y} g(y)$ where $F(x) := int f(x) , mathrm{d}x$ and $G(y) := int g(y) , mathrm{d}y$ to reach the following two separated third order ODEs
begin{eqnarray}
lambda_h F''' - 2 lambda_h beta_h F'' + left( (lambda_h beta_h - 1) beta_h - mu right) F' + beta_h^2 F &=& 0,\
V lambda_c G''' - 2 V lambda_c beta_c G'' + left( (lambda_c beta_c - 1) V beta_c + mu right) G' + V beta_c^2 G &=& 0,
end{eqnarray}
Can these ODEs be expressed in Sturm-Liouville form of EigenValue problems (All texts i see show examples for second order problem ?)
My attempts at solving these two ODEs as eigen value problems were failing
1 Eigen values of a Third Order Linear Homogenous ODE
2 How to choose Eigenvaues for system extending in perpendicular direction?
3 Eigenvalues keep giving trivial solutions everytime.
I must mention here that user @Christoph ,@Cesareo and @LutzL have been extremely helpful in getting me this far.
So I have been trying other approaches and Sturm - Liouville problems was a form of problem i came across.
ATTEMPT
If i try the following form:
$$
lambda_h f'' - 2 lambda_h beta_h f' + ( (lambda_h beta_h - 1) beta_h) f' + beta_h^2 int f mathrm{d}x = mu f
$$
But this form has an integral operator on the LHS, Differentiating it w.r.t. $x$ makes it a third order problem.
ordinary-differential-equations multivariable-calculus pde eigenfunctions sturm-liouville
asked Jan 25 at 9:21
Indrasis MitraIndrasis Mitra
80111
$endgroup$
$begingroup$
@Christoph Any guiding direction would be helpful.
$endgroup$
– Indrasis Mitra
Jan 25 at 15:59
add a comment |
$begingroup$
From a system of three coupled PDEs
begin{eqnarray}
frac{partial theta_h}{partial x} + beta_h (theta_h - theta_w) &=& 0,\
frac{partial theta_c}{partial y} + beta_c (theta_c - theta_w) &=& 0,\
lambda_h frac{partial^2 theta_w}{partial x^2} + lambda_c V frac{partial^2 theta_w}{partial y^2} - frac{partial theta_h}{partial x} - Vfrac{partial theta_c}{partial y} &=& 0
end{eqnarray}
with the bc(s) as
The boundary conditions for the problem are as follows:
The PDE(s) needs to be solved on a rectangular region where $x$ varies between $0$ to $1$ and $y$ varies between $0$ to $1$.
$$frac{partial theta_w(0,y)}{partial x}=frac{partial theta_w(1,y)}{partial x}=0 $$
$$frac{partial theta_w(x,0)}{partial y}=frac{partial theta_w(x,1)}{partial y}=0 $$
$$theta_h(0,y)=1 $$$$theta_c(x,0)=0$$
The first two were solved and substituted in the following form as
begin{eqnarray}
0 &=& e^{-beta_h x} left( lambda_h e^{beta_h x} frac{partial^2 theta_w}{partial x^2} - beta_h e^{beta_h x} theta_w + beta_h^2 int e^{beta_h x} theta_w , mathrm{d}x right) +\
&& + V e^{-beta_c y} left( lambda_c e^{beta_c y} frac{partial^2 theta_w}{partial y^2} - beta_c e^{beta_c y} theta_w + beta_c^2 int e^{beta_c y} theta_w , mathrm{d}y right).
end{eqnarray}
The following assumption was made then
$theta_w(x,y) = e^{-beta_h x} f(x) e^{-beta_c y} g(y)$ where $F(x) := int f(x) , mathrm{d}x$ and $G(y) := int g(y) , mathrm{d}y$ to reach the following two separated third order ODEs
begin{eqnarray}
lambda_h F''' - 2 lambda_h beta_h F'' + left( (lambda_h beta_h - 1) beta_h - mu right) F' + beta_h^2 F &=& 0,\
V lambda_c G''' - 2 V lambda_c beta_c G'' + left( (lambda_c beta_c - 1) V beta_c + mu right) G' + V beta_c^2 G &=& 0,
end{eqnarray}
Can these ODEs be expressed in Sturm-Liouville form of EigenValue problems (All texts i see show examples for second order problem ?)
My attempts at solving these two ODEs as eigen value problems were failing
1 Eigen values of a Third Order Linear Homogenous ODE
2 How to choose Eigenvaues for system extending in perpendicular direction?
3 Eigenvalues keep giving trivial solutions everytime.
I must mention here that user @Christoph ,@Cesareo and @LutzL have been extremely helpful in getting me this far.
So I have been trying other approaches and Sturm - Liouville problems was a form of problem i came across.
ATTEMPT
If i try the following form:
$$
lambda_h f'' - 2 lambda_h beta_h f' + ( (lambda_h beta_h - 1) beta_h) f' + beta_h^2 int f mathrm{d}x = mu f
$$
But this form has an integral operator on the LHS, Differentiating it w.r.t. $x$ makes it a third order problem.
ordinary-differential-equations multivariable-calculus pde eigenfunctions sturm-liouville
asked Jan 25 at 9:21
Indrasis MitraIndrasis Mitra
80111
$endgroup$
From a system of three coupled PDEs
begin{eqnarray}
frac{partial theta_h}{partial x} + beta_h (theta_h - theta_w) &=& 0,\
frac{partial theta_c}{partial y} + beta_c (theta_c - theta_w) &=& 0,\
lambda_h frac{partial^2 theta_w}{partial x^2} + lambda_c V frac{partial^2 theta_w}{partial y^2} - frac{partial theta_h}{partial x} - Vfrac{partial theta_c}{partial y} &=& 0
end{eqnarray}
with the bc(s) as
The boundary conditions for the problem are as follows:
The PDE(s) needs to be solved on a rectangular region where $x$ varies between $0$ to $1$ and $y$ varies between $0$ to $1$.
$$frac{partial theta_w(0,y)}{partial x}=frac{partial theta_w(1,y)}{partial x}=0 $$
$$frac{partial theta_w(x,0)}{partial y}=frac{partial theta_w(x,1)}{partial y}=0 $$
$$theta_h(0,y)=1 $$$$theta_c(x,0)=0$$
The first two were solved and substituted in the following form as
begin{eqnarray}
0 &=& e^{-beta_h x} left( lambda_h e^{beta_h x} frac{partial^2 theta_w}{partial x^2} - beta_h e^{beta_h x} theta_w + beta_h^2 int e^{beta_h x} theta_w , mathrm{d}x right) +\
&& + V e^{-beta_c y} left( lambda_c e^{beta_c y} frac{partial^2 theta_w}{partial y^2} - beta_c e^{beta_c y} theta_w + beta_c^2 int e^{beta_c y} theta_w , mathrm{d}y right).
end{eqnarray}
The following assumption was made then
$theta_w(x,y) = e^{-beta_h x} f(x) e^{-beta_c y} g(y)$ where $F(x) := int f(x) , mathrm{d}x$ and $G(y) := int g(y) , mathrm{d}y$ to reach the following two separated third order ODEs
begin{eqnarray}
lambda_h F''' - 2 lambda_h beta_h F'' + left( (lambda_h beta_h - 1) beta_h - mu right) F' + beta_h^2 F &=& 0,\
V lambda_c G''' - 2 V lambda_c beta_c G'' + left( (lambda_c beta_c - 1) V beta_c + mu right) G' + V beta_c^2 G &=& 0,
end{eqnarray}
Can these ODEs be expressed in Sturm-Liouville form of EigenValue problems (All texts i see show examples for second order problem ?)
My attempts at solving these two ODEs as eigen value problems were failing
1 Eigen values of a Third Order Linear Homogenous ODE
2 How to choose Eigenvaues for system extending in perpendicular direction?
3 Eigenvalues keep giving trivial solutions everytime.
I must mention here that user @Christoph ,@Cesareo and @LutzL have been extremely helpful in getting me this far.
So I have been trying other approaches and Sturm - Liouville problems was a form of problem i came across.
ATTEMPT
If i try the following form:
$$
lambda_h f'' - 2 lambda_h beta_h f' + ( (lambda_h beta_h - 1) beta_h) f' + beta_h^2 int f mathrm{d}x = mu f
$$
But this form has an integral operator on the LHS, Differentiating it w.r.t. $x$ makes it a third order problem.
ordinary-differential-equations multivariable-calculus pde eigenfunctions sturm-liouville
ordinary-differential-equations multivariable-calculus pde eigenfunctions sturm-liouville
asked Jan 25 at 9:21
Indrasis MitraIndrasis Mitra
80111
asked Jan 25 at 9:21
Indrasis MitraIndrasis Mitra
80111
edited Jan 25 at 9:43
Indrasis Mitra
asked Jan 25 at 9:21
Indrasis MitraIndrasis Mitra
80111
asked Jan 25 at 9:21
Indrasis MitraIndrasis Mitra
80111
asked Jan 25 at 9:21
Indrasis MitraIndrasis Mitra
80111
80111
$begingroup$
@Christoph Any guiding direction would be helpful.
$endgroup$
– Indrasis Mitra
Jan 25 at 15:59
add a comment |
$begingroup$
@Christoph Any guiding direction would be helpful.
$endgroup$
– Indrasis Mitra
Jan 25 at 15:59
$begingroup$
@Christoph Any guiding direction would be helpful.
$endgroup$
– Indrasis Mitra
Jan 25 at 15:59
$begingroup$
@Christoph Any guiding direction would be helpful.
$endgroup$
– Indrasis Mitra
Jan 25 at 15:59
add a comment |
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$begingroup$
@Christoph Any guiding direction would be helpful.
$endgroup$
– Indrasis Mitra
Jan 25 at 15:59