How to choose Eigenvaues for system extending in perpendicular direction?

Jan 2019
2
0
Kharagpur
I have two linear third order ODEs with a separation constant (eigenvalue parameter) on a rectangular domain where $x \in (0,1)$ and $y \in (0,1)$as follows:

I have two linear third order ODEs with a separation constant (eigenvalue parameter) on a rectangular domain where $x \in (0,1)$ and $y \in (0,1)$as follows:

$$
\lambda_h F''' - 2 \lambda_h \beta_h F'' + ( (\lambda_h \beta_h - 1) \beta_h ) F' + \beta_h^2 F = \mu F',\\
V \lambda_c G''' - 2 V \lambda_c \beta_c G'' +( (\lambda_c \beta_c - 1) V \beta_c ) G' + V \beta_c^2 G = -\mu G',
$$
$\mu \in \mathbb{R}$.

The b.c for $F(x)$ are $F(0) = 0, \frac{F''(0)}{F'(0)}=\beta_h,\frac{F''(1)}{F'(1)}=\beta_h $

For $G(y)$ they are $G(0) = 0, \frac{G''(0)}{G'(0)}=\beta_c,\frac{G''(1)}{G'(1)}=\beta_c$

For these bc(s) and $\lambda_h=0.02, \beta_h = 10$, I calculated the Eigenvalues using **chebfun** in *MATLAB* which came out to be

`F = -37.6413, -32.9463, -28.6002, -24.5873, -20.8885, -17.4846, -14.3643 -11.5367, -9.0383, -6.9287,......
`

`G = 6.9287, 9.0383, 11.5367, 14.3643, 17.4846, 20.8885, 24.5873, 28.6002, 32.9463, 37.6413, .....`

I now need to find the solutions to these two separated ODEs, to form my final solution.

What bothers me is that EVs of $F$ are in infinite numbers in the $-x$ direction, while of $G$ are in infinite numbers in $+y$ direction with the same magnitudes.

I cannot figure out what EVs should I choose and how many i should to build my solution ? I know that they are separated by the same constant. Basically, how should i proceed. haven't gotten any clue from texts.

*Steps after finding the EVs*

The general solution will be of the form

\(\displaystyle
F(x) = \sum_k C_k e^{-\delta_k(\mu)x}
\)

where $\delta_k(\mu)$ is a root of the characteristic equation dependent on $\mu$. There would be three roots of the char. eqn. So each EV when substituted in the characteristic equation should give three roots which would then be used to calculate $C_1,C_2,C_3$ constants using the b.c.

This is where i am facing problems in determining, how many and which EVs should be considered.