# Three coupled differential equations to be solved analytically

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.

• Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures. – José Carlos Santos Jan 2 at 10:14
• 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"? – LutzL Jan 2 at 10:46
• @LutzL made the edits as suggested – Indrasis Mitra Jan 2 at 10:53

## 2 Answers

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.

• Thanks for the help . I will try this approach. Also, will put the effort to use MathJax from now. – Indrasis Mitra Jan 3 at 5:37

@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.