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)