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From a system of three coupled 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 the bc(s) as

The boundary conditions for the problem are as follows:

The PDE(s) needs to be solved on a rectangular region where $x$ varies between $0$ to $1$ and $y$ varies between $0$ to $1$.

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

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

$$\theta_h(0,y)=1 $$$$\theta_c(x,0)=0$$ The first two were solved and substituted in the following form as \begin{eqnarray} 0 &=& e^{-\beta_h x} \left( \lambda_h e^{\beta_h x} \frac{\partial^2 \theta_w}{\partial x^2} - \beta_h e^{\beta_h x} \theta_w + \beta_h^2 \int e^{\beta_h x} \theta_w \, \mathrm{d}x \right) +\\ && + V e^{-\beta_c y} \left( \lambda_c e^{\beta_c y} \frac{\partial^2 \theta_w}{\partial y^2} - \beta_c e^{\beta_c y} \theta_w + \beta_c^2 \int e^{\beta_c y} \theta_w \, \mathrm{d}y \right). \end{eqnarray}

The following assumption was made then $\theta_w(x,y) = e^{-\beta_h x} f(x) e^{-\beta_c y} g(y)$ where $F(x) := \int f(x) \, \mathrm{d}x$ and $G(y) := \int g(y) \, \mathrm{d}y$ to reach the following two separated third order ODEs

\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}

Can these ODEs be expressed in Sturm-Liouville form of EigenValue problems (All texts i see show examples for second order problem ?) My attempts at solving these two ODEs as eigen value problems were failing

1 Eigen values of a Third Order Linear Homogenous ODE 2 How to choose Eigenvaues for system extending in perpendicular direction? 3 Eigenvalues keep giving trivial solutions everytime.

I must mention here that user @Christoph ,@Cesareo and @LutzL have been extremely helpful in getting me this far.

So I have been trying other approaches and Sturm - Liouville problems was a form of problem i came across.

ATTEMPT

If i try the following form: $$ \lambda_h f'' - 2 \lambda_h \beta_h f' + ( (\lambda_h \beta_h - 1) \beta_h) f' + \beta_h^2 \int f \mathrm{d}x = \mu f $$ But this form has an integral operator on the LHS, Differentiating it w.r.t. $x$ makes it a third order problem.

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  • $\begingroup$ @Christoph Any guiding direction would be helpful. $\endgroup$ – Indrasis Mitra Jan 25 at 15:59

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