How Can I Solve a Second-Order Differential Equation Using Laplace Transforms?

[mikeleeds]

Homework Statement

Find analytic solution of some kind:

$$0&=Y''(y)-\frac{\alpha^2 [u(y) + U]}{\epsilon}Y(y)$$

U eigenvalue, u(y) known, epsilon & alpha paramertres,and& Y thing to be found

Homework Equations

u(y) is a parallel flow of some kind and laplace transform given by:

$$\mathcal{L}[Y(y)]=\bar{Y}(s)=\int_0^{\infty} e^{-sy} Y(y) dy$$

The Attempt at a Solution

Do the transform:
$$-sY(0)+s^2 \bar{Y}(s) &= \frac{\alpha^2}{\epsilon} \left( \mathcal{L}[u(y)Y(y)] + U \bar{Y}(s) \right)$$
Do the u(y)Y(y) transform
$$\mathcal{L}[u(y)Y(y)]&=\int_0^{\infty} u(y)Y(y) e^{-sy}dy$$
by parts
$$&=\bar{Y}(s)u(y)|_0^{\infty}-\int_0^{\infty} u'(y)\mathcal{L}[Y(y)] dy$$
by parts again and again...
$$&=\bar{Y}(s)u(y)|_0^{\infty}-\frac{\bar{Y}(s) u'(y)|_0^{\infty}}{s} + \frac{\bar{Y}(s) u''(y)|_0^{\infty}}{s^2} - ...$$
this series might converge so we can truncate
Put it back together again and get a solutn. in Laplace space.

4. My problem:

I don't seem to get the right kind of answers, when I numerically undo my laplace transform they are either the wrong way round, or completely messed up (oscillating like a nutcase)! I don't know is you are allowed to do that kind of thing (the whole by parts business) or the validity of my series truncation... Could anyone clarify for me?

And if anyone is feeling REALLY helpful, they might tell me how to get the eigenvalue out and find what it is?!

 

[mighty2000]

Thank you for your post. I am a scientist who specializes in solving differential equations analytically. I will try to help you with your problem.

Firstly, your approach of using the Laplace transform is a good start. However, it seems that you are having trouble with the inverse Laplace transform. The inverse Laplace transform is a mathematical operation that converts a function from the s-domain (Laplace space) back to the time-domain. There are several methods for doing this, such as using tables of Laplace transforms or using partial fraction decomposition.

In your case, since you have a series in the s-domain, you can use the method of residues to find the inverse Laplace transform. This method involves using the Cauchy integral formula to evaluate the inverse Laplace transform as a contour integral. You can find more information about this method in any standard textbook on complex analysis.

Regarding your concerns about the validity of your series truncation, it is important to note that the Laplace transform is only valid for functions that are absolutely integrable. So if your series converges, then it is a valid solution. However, if the series does not converge, then it is not a valid solution and you may need to try a different approach.

Finally, to find the eigenvalue, you can use the boundary conditions of your problem. Since the equation you are trying to solve is a second-order differential equation, you will need two boundary conditions to determine the eigenvalue. Once you have the eigenvalue, you can substitute it back into your solution to get the complete solution.

I hope this helps you with your problem. If you have any further questions, please feel free to ask. Good luck with your work!
Scientist