Three coupled differential equations to be solved analytically












0












$begingroup$


I have three coupled DEs, two first order and the third partial second order with laplacian operator



enter image description here



$lambda, beta$ and $V$ are constants. Any advice on how to approach the problem analytically would be a huge help.






ordinary-differential-equations







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  • 1




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    Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
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    – 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
















0












$begingroup$


I have three coupled DEs, two first order and the third partial second order with laplacian operator



enter image description here



$lambda, beta$ and $V$ are constants. Any advice on how to approach the problem analytically would be a huge help.






ordinary-differential-equations







share|cite|improve this question











$endgroup$








  • 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














0












0








0





$begingroup$


I have three coupled DEs, two first order and the third partial second order with laplacian operator



enter image description here



$lambda, beta$ and $V$ are constants. Any advice on how to approach the problem analytically would be a huge help.






ordinary-differential-equations







share|cite|improve this question











$endgroup$




I have three coupled DEs, two first order and the third partial second order with laplacian operator



enter image description here



$lambda, beta$ and $V$ are constants. Any advice on how to approach the problem analytically would be a huge help.






ordinary-differential-equations




ordinary-differential-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 3 at 9:45









Kevin

5,736823




5,736823










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














  • 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










2 Answers
2






active

oldest

votes


















0












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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



















0












$begingroup$

@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.






share|cite|improve this answer











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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      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
















    0












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      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














    0












    0








    0





    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$



    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.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    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


















    • $begingroup$
      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











    0












    $begingroup$

    @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.






    share|cite|improve this answer











    $endgroup$


















      0












      $begingroup$

      @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.






      share|cite|improve this answer











      $endgroup$
















        0












        0








        0





        $begingroup$

        @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.






        share|cite|improve this answer











        $endgroup$



        @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.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 3 at 10:06

























        answered Jan 3 at 9:41









        Indrasis MitraIndrasis Mitra

        50111




        50111






























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