Understanding the Blow-Up Phenomenon in BVPs: y'' + ay' + e^{ax}y = 1

[rsq_a]

I'm looking at the BVP:

$$y'' + ay' + e^{ax}y = 1$$,

with y(0) = 0 and y(10) = 0.

The numerical solution blows up at certain values of $$a$$. For example, a near 0.089 and a near 0.2302. Why does this happen and how do I predict it?

 

[rsq_a]

I'm looking at the BVP:

$$y'' + ay' + e^{ax}y = 1$$,

with y(0) = 0 and y(10) = 0.

The numerical solution blows up at certain values of $$a$$. For example, a near 0.089 and a near 0.2302. Why does this happen and how do I predict it?

Erm. I found the problem. Near those values of 'a', there exists a zero eigenvalue of the linear operator. I guess that means that,

$$y'' + ay' + e^{ax}y = 0\cdot u^* = 1$$,

is a possible solution, and thus the eigenfunction $$u^* \to \infty$$ will cause the blowup.

Is this correct? It's been a while since I've done Sturm-Liouville stuff.

 

[jacophile]

What is x?

 

[rsq_a]

What is x?

$$y=y(x)$$