I had a system of three PDEs $$\frac{\partial \theta_h}{\partial x}+\beta_h (\theta_h-\theta_w) = 0$$
$$\frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c-\theta_w) = 0$$
$$ \lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V\frac{\partial^2 \theta_w}{\partial y^2}-\frac{\partial \theta_h}{\partial x} - V\frac{\partial \theta_c}{\partial y} = 0 $$ On eliminating $\theta_h$ and $\theta_c$ from the third equation I reach $$ \lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 \theta_w}{\partial y^2} +( -\beta_h - V \beta_c )\theta_w +\beta_h^2 e^{-\beta_h x} \int e^{\beta_h x} \theta_w(x,y) \mathrm{d}x + \beta_c^2 e^{-\beta_c y}\int e^{\beta_c y} \theta_w(x,y)\mathrm{d}y = 0 $$ The bc(s) for the system are :
$$\frac{\partial \theta_w(0,y)}{\partial x}=\frac{\partial \theta_w(1,y)}{\partial x}=0 $$
$$\frac{\partial \theta_w(x,0)}{\partial y}=\frac{\partial \theta_w(x,1)}{\partial y}=0 $$
$$\theta_h(0,y)=1 $$$$\theta_c(x,0)=0$$
I must note here that i know an ansatz $\theta_w(x,y) = e^{-\beta_h x} f(x) e^{-\beta_c y} g(y)$ which can provide variable separation for the last PDE. But the Eigen value problems that would then come out of this are third order.
Is there any module in MATHEMATICA that can handle Partio-Integral Differential equations ?