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.

$\endgroup$
  • 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
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.