I have the following three 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} - V\frac{\partial \theta_c}{\partial y} &=& 0 \end{eqnarray} with boundary conditions as. The PDE needs to be solved on a rectangular region where $x$ varies between $0$ to $1$ and $y$ varies between $0$ to $1$.

$$\frac{\partial \theta_w(0,y)}{\partial x}=\frac{\partial \theta_w(1,y)}{\partial x}=0 $$

$$\frac{\partial \theta_w(x,0)}{\partial y}=\frac{\partial \theta_w(x,1)}{\partial y}=0 $$

$$\theta_h(0,y)=1 $$$$\theta_c(x,0)=0$$

I want to find a suitable ansatz which could eliminate $\theta_c$ and $\theta_h$ from the third equation, making it an equation only in $\theta_w$ and still keep it second order finally making it variable separated.

I already tried the following attempt (with help from a MSE user)


\begin{eqnarray} \theta_h(x,y) &=& \beta_h e^{-\beta_h x} \int e^{\beta_h x} \theta_w(x,y) \, \mathrm{d}x,\\ \theta_c(x,y) &=& \beta_c e^{-\beta_c y} \int e^{\beta_c y} \theta_w(x,y) \, \mathrm{d}y. \end{eqnarray}

Substituting in the third equation and using $\theta_w(x,y) = e^{-\beta_h x} f(x) e^{-\beta_c y} g(y)$, I obtain

\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} with some separation constant $\mu \in \mathbb{R}$ where $F(x) := \int f(x) \, \mathrm{d}x$ and $G(y) := \int g(y) \, \mathrm{d}y$

These are third order Eigen Value problems which need to be solved analytically.

I was curious if there could be some ansatz which could keep this a second - order problem . This would make it solving analytically much recognizable as then i could bring them under the domain of Sturm Liouville type.

I have seen examples where having a complex ansatz helps, but i cannot figure out how to move forward here.


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