Three coupled differential equations to be solved analytically
$begingroup$
I have three coupled DEs, two first order and the third partial second order with laplacian operator
$lambda, beta$ and $V$ are constants. Any advice on how to approach the problem analytically would be a huge help.
ordinary-differential-equations
edited Jan 3 at 9:45
Kevin
5,736823
asked Jan 2 at 9:48
Indrasis MitraIndrasis Mitra
50111
$endgroup$
add a comment |
$begingroup$
I have three coupled DEs, two first order and the third partial second order with laplacian operator
$lambda, beta$ and $V$ are constants. Any advice on how to approach the problem analytically would be a huge help.
ordinary-differential-equations
edited Jan 3 at 9:45
Kevin
5,736823
asked Jan 2 at 9:48
Indrasis MitraIndrasis Mitra
50111
$endgroup$
1
$begingroup$
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 2 at 10:14
$begingroup$
You should also mention that these are partial differential equations, and that the second order derivative operator is of Laplacian or elliptical type. What do you imagine under "matrix solution"?
$endgroup$
– LutzL
Jan 2 at 10:46
$begingroup$
@LutzL made the edits as suggested
$endgroup$
– Indrasis Mitra
Jan 2 at 10:53
add a comment |
$begingroup$
I have three coupled DEs, two first order and the third partial second order with laplacian operator
$lambda, beta$ and $V$ are constants. Any advice on how to approach the problem analytically would be a huge help.
ordinary-differential-equations
edited Jan 3 at 9:45
Kevin
5,736823
asked Jan 2 at 9:48
Indrasis MitraIndrasis Mitra
50111
$endgroup$
I have three coupled DEs, two first order and the third partial second order with laplacian operator
$lambda, beta$ and $V$ are constants. Any advice on how to approach the problem analytically would be a huge help.
ordinary-differential-equations
ordinary-differential-equations
edited Jan 3 at 9:45
Kevin
5,736823
asked Jan 2 at 9:48
Indrasis MitraIndrasis Mitra
50111
edited Jan 3 at 9:45
Kevin
5,736823
asked Jan 2 at 9:48
Indrasis MitraIndrasis Mitra
50111
edited Jan 3 at 9:45
Kevin
5,736823
edited Jan 3 at 9:45
Kevin
5,736823
edited Jan 3 at 9:45
Kevin
5,736823
5,736823
asked Jan 2 at 9:48
Indrasis MitraIndrasis Mitra
50111
asked Jan 2 at 9:48
Indrasis MitraIndrasis Mitra
50111
asked Jan 2 at 9:48
Indrasis MitraIndrasis Mitra
50111
50111
1
$begingroup$
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 2 at 10:14
$begingroup$
You should also mention that these are partial differential equations, and that the second order derivative operator is of Laplacian or elliptical type. What do you imagine under "matrix solution"?
$endgroup$
– LutzL
Jan 2 at 10:46
$begingroup$
@LutzL made the edits as suggested
$endgroup$
– Indrasis Mitra
Jan 2 at 10:53
add a comment |
1
$begingroup$
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 2 at 10:14
$begingroup$
You should also mention that these are partial differential equations, and that the second order derivative operator is of Laplacian or elliptical type. What do you imagine under "matrix solution"?
$endgroup$
– LutzL
Jan 2 at 10:46
$begingroup$
@LutzL made the edits as suggested
$endgroup$
– Indrasis Mitra
Jan 2 at 10:53
1
1
$begingroup$
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 2 at 10:14
$begingroup$
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 2 at 10:14
$begingroup$
You should also mention that these are partial differential equations, and that the second order derivative operator is of Laplacian or elliptical type. What do you imagine under "matrix solution"?
$endgroup$
– LutzL
Jan 2 at 10:46
$begingroup$
You should also mention that these are partial differential equations, and that the second order derivative operator is of Laplacian or elliptical type. What do you imagine under "matrix solution"?
$endgroup$
– LutzL
Jan 2 at 10:46
$begingroup$
@LutzL made the edits as suggested
$endgroup$
– Indrasis Mitra
Jan 2 at 10:53
$begingroup$
@LutzL made the edits as suggested
$endgroup$
– Indrasis Mitra
Jan 2 at 10:53
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The three partial differential equations (PDEs) are
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} + beta_h (theta_h - theta_w) + V beta_c (theta_c - theta_w) &=& 0
end{eqnarray}
(MathJax allows you to use LaTeX expressions in your posts).
These are linear PDEs with constant coefficients. Using the first two PDEs you can write $theta_h$ and $theta_c$ using $theta_w$:
begin{equation}
theta_h(x,y) = beta_h e^{-beta_h x} int e^{beta_h x} theta_w(x,y) , mathrm{d}x, quad theta_c(x,y) = beta_c e^{-beta_c y} int e^{beta_c y} theta_w(x,y) , mathrm{d}y
end{equation}
(keyword integrating factor).
Replacing $theta_h, theta_c$ by these expressions in the third PDE yields a single linear partial integro-differential equation (PIDE) for $theta_w$, which can probably be solved using some Fourier or Laplace transform technique.
answered Jan 3 at 4:18
ChristophChristoph
60616
$endgroup$
$begingroup$
Thanks for the help . I will try this approach. Also, will put the effort to use MathJax from now.
$endgroup$
– Indrasis Mitra
Jan 3 at 5:37
add a comment |
$begingroup$
@Christoph After your suggestions the equation takes the following form
begin{eqnarray}
lambda_h frac{partial^2 theta_w}{partial x^2} + lambda_c V frac{partial^2 theta_w}{partial y^2} +( -beta_h - V beta_c )theta_w +beta_h^2 e^{-beta_h x} int e^{beta_h x} theta_w(x,y) mathrm{d}x + beta_c^2 e^{-beta_c y}int e^{beta_c y} theta_w(x,y)mathrm{d}y = 0
end{eqnarray}
Although you did refer to the use of Laplace and Fourier transfom to solve the resulting PIDE, can you point to any reference where I could find examples analogous to such equations ? I tried some textbooks ( Partial differential equations, Harumi & Hattori ; A Journey into partial Differential equations, Bray) on PDE but they normally won't cover PIDEs.
answered Jan 3 at 9:41
Indrasis MitraIndrasis Mitra
50111
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The three partial differential equations (PDEs) are
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} + beta_h (theta_h - theta_w) + V beta_c (theta_c - theta_w) &=& 0
end{eqnarray}
(MathJax allows you to use LaTeX expressions in your posts).
These are linear PDEs with constant coefficients. Using the first two PDEs you can write $theta_h$ and $theta_c$ using $theta_w$:
begin{equation}
theta_h(x,y) = beta_h e^{-beta_h x} int e^{beta_h x} theta_w(x,y) , mathrm{d}x, quad theta_c(x,y) = beta_c e^{-beta_c y} int e^{beta_c y} theta_w(x,y) , mathrm{d}y
end{equation}
(keyword integrating factor).
Replacing $theta_h, theta_c$ by these expressions in the third PDE yields a single linear partial integro-differential equation (PIDE) for $theta_w$, which can probably be solved using some Fourier or Laplace transform technique.
answered Jan 3 at 4:18
ChristophChristoph
60616
$endgroup$
$begingroup$
Thanks for the help . I will try this approach. Also, will put the effort to use MathJax from now.
$endgroup$
– Indrasis Mitra
Jan 3 at 5:37
add a comment |
$begingroup$
The three partial differential equations (PDEs) are
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} + beta_h (theta_h - theta_w) + V beta_c (theta_c - theta_w) &=& 0
end{eqnarray}
(MathJax allows you to use LaTeX expressions in your posts).
These are linear PDEs with constant coefficients. Using the first two PDEs you can write $theta_h$ and $theta_c$ using $theta_w$:
begin{equation}
theta_h(x,y) = beta_h e^{-beta_h x} int e^{beta_h x} theta_w(x,y) , mathrm{d}x, quad theta_c(x,y) = beta_c e^{-beta_c y} int e^{beta_c y} theta_w(x,y) , mathrm{d}y
end{equation}
(keyword integrating factor).
Replacing $theta_h, theta_c$ by these expressions in the third PDE yields a single linear partial integro-differential equation (PIDE) for $theta_w$, which can probably be solved using some Fourier or Laplace transform technique.
answered Jan 3 at 4:18
ChristophChristoph
60616
$endgroup$
$begingroup$
Thanks for the help . I will try this approach. Also, will put the effort to use MathJax from now.
$endgroup$
– Indrasis Mitra
Jan 3 at 5:37
add a comment |
$begingroup$
The three partial differential equations (PDEs) are
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} + beta_h (theta_h - theta_w) + V beta_c (theta_c - theta_w) &=& 0
end{eqnarray}
(MathJax allows you to use LaTeX expressions in your posts).
These are linear PDEs with constant coefficients. Using the first two PDEs you can write $theta_h$ and $theta_c$ using $theta_w$:
begin{equation}
theta_h(x,y) = beta_h e^{-beta_h x} int e^{beta_h x} theta_w(x,y) , mathrm{d}x, quad theta_c(x,y) = beta_c e^{-beta_c y} int e^{beta_c y} theta_w(x,y) , mathrm{d}y
end{equation}
(keyword integrating factor).
Replacing $theta_h, theta_c$ by these expressions in the third PDE yields a single linear partial integro-differential equation (PIDE) for $theta_w$, which can probably be solved using some Fourier or Laplace transform technique.
answered Jan 3 at 4:18
ChristophChristoph
60616
$endgroup$
The three partial differential equations (PDEs) are
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} + beta_h (theta_h - theta_w) + V beta_c (theta_c - theta_w) &=& 0
end{eqnarray}
(MathJax allows you to use LaTeX expressions in your posts).
These are linear PDEs with constant coefficients. Using the first two PDEs you can write $theta_h$ and $theta_c$ using $theta_w$:
begin{equation}
theta_h(x,y) = beta_h e^{-beta_h x} int e^{beta_h x} theta_w(x,y) , mathrm{d}x, quad theta_c(x,y) = beta_c e^{-beta_c y} int e^{beta_c y} theta_w(x,y) , mathrm{d}y
end{equation}
(keyword integrating factor).
Replacing $theta_h, theta_c$ by these expressions in the third PDE yields a single linear partial integro-differential equation (PIDE) for $theta_w$, which can probably be solved using some Fourier or Laplace transform technique.
answered Jan 3 at 4:18
ChristophChristoph
60616
answered Jan 3 at 4:18
ChristophChristoph
60616
answered Jan 3 at 4:18
ChristophChristoph
60616
answered Jan 3 at 4:18
ChristophChristoph
60616
60616
$begingroup$
Thanks for the help . I will try this approach. Also, will put the effort to use MathJax from now.
$endgroup$
– Indrasis Mitra
Jan 3 at 5:37
add a comment |
$begingroup$
Thanks for the help . I will try this approach. Also, will put the effort to use MathJax from now.
$endgroup$
– Indrasis Mitra
Jan 3 at 5:37
$begingroup$
Thanks for the help . I will try this approach. Also, will put the effort to use MathJax from now.
$endgroup$
– Indrasis Mitra
Jan 3 at 5:37
$begingroup$
Thanks for the help . I will try this approach. Also, will put the effort to use MathJax from now.
$endgroup$
– Indrasis Mitra
Jan 3 at 5:37
add a comment |
$begingroup$
@Christoph After your suggestions the equation takes the following form
begin{eqnarray}
lambda_h frac{partial^2 theta_w}{partial x^2} + lambda_c V frac{partial^2 theta_w}{partial y^2} +( -beta_h - V beta_c )theta_w +beta_h^2 e^{-beta_h x} int e^{beta_h x} theta_w(x,y) mathrm{d}x + beta_c^2 e^{-beta_c y}int e^{beta_c y} theta_w(x,y)mathrm{d}y = 0
end{eqnarray}
Although you did refer to the use of Laplace and Fourier transfom to solve the resulting PIDE, can you point to any reference where I could find examples analogous to such equations ? I tried some textbooks ( Partial differential equations, Harumi & Hattori ; A Journey into partial Differential equations, Bray) on PDE but they normally won't cover PIDEs.
answered Jan 3 at 9:41
Indrasis MitraIndrasis Mitra
50111
$endgroup$
add a comment |
$begingroup$
@Christoph After your suggestions the equation takes the following form
begin{eqnarray}
lambda_h frac{partial^2 theta_w}{partial x^2} + lambda_c V frac{partial^2 theta_w}{partial y^2} +( -beta_h - V beta_c )theta_w +beta_h^2 e^{-beta_h x} int e^{beta_h x} theta_w(x,y) mathrm{d}x + beta_c^2 e^{-beta_c y}int e^{beta_c y} theta_w(x,y)mathrm{d}y = 0
end{eqnarray}
Although you did refer to the use of Laplace and Fourier transfom to solve the resulting PIDE, can you point to any reference where I could find examples analogous to such equations ? I tried some textbooks ( Partial differential equations, Harumi & Hattori ; A Journey into partial Differential equations, Bray) on PDE but they normally won't cover PIDEs.
answered Jan 3 at 9:41
Indrasis MitraIndrasis Mitra
50111
$endgroup$
add a comment |
$begingroup$
@Christoph After your suggestions the equation takes the following form
begin{eqnarray}
lambda_h frac{partial^2 theta_w}{partial x^2} + lambda_c V frac{partial^2 theta_w}{partial y^2} +( -beta_h - V beta_c )theta_w +beta_h^2 e^{-beta_h x} int e^{beta_h x} theta_w(x,y) mathrm{d}x + beta_c^2 e^{-beta_c y}int e^{beta_c y} theta_w(x,y)mathrm{d}y = 0
end{eqnarray}
Although you did refer to the use of Laplace and Fourier transfom to solve the resulting PIDE, can you point to any reference where I could find examples analogous to such equations ? I tried some textbooks ( Partial differential equations, Harumi & Hattori ; A Journey into partial Differential equations, Bray) on PDE but they normally won't cover PIDEs.
answered Jan 3 at 9:41
Indrasis MitraIndrasis Mitra
50111
$endgroup$
@Christoph After your suggestions the equation takes the following form
begin{eqnarray}
lambda_h frac{partial^2 theta_w}{partial x^2} + lambda_c V frac{partial^2 theta_w}{partial y^2} +( -beta_h - V beta_c )theta_w +beta_h^2 e^{-beta_h x} int e^{beta_h x} theta_w(x,y) mathrm{d}x + beta_c^2 e^{-beta_c y}int e^{beta_c y} theta_w(x,y)mathrm{d}y = 0
end{eqnarray}
Although you did refer to the use of Laplace and Fourier transfom to solve the resulting PIDE, can you point to any reference where I could find examples analogous to such equations ? I tried some textbooks ( Partial differential equations, Harumi & Hattori ; A Journey into partial Differential equations, Bray) on PDE but they normally won't cover PIDEs.
answered Jan 3 at 9:41
Indrasis MitraIndrasis Mitra
50111
edited Jan 3 at 10:06
answered Jan 3 at 9:41
Indrasis MitraIndrasis Mitra
50111
answered Jan 3 at 9:41
Indrasis MitraIndrasis Mitra
50111
answered Jan 3 at 9:41
Indrasis MitraIndrasis Mitra
50111
50111
add a comment |
add a comment |
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1
$begingroup$
Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 2 at 10:14
$begingroup$
You should also mention that these are partial differential equations, and that the second order derivative operator is of Laplacian or elliptical type. What do you imagine under "matrix solution"?
$endgroup$
– LutzL
Jan 2 at 10:46
$begingroup$
@LutzL made the edits as suggested
$endgroup$
– Indrasis Mitra
Jan 2 at 10:53