Automate perturbation solution to a 4th order differential equation?

Questions
,
1

In the perturbation solution, each subsequent addition to the series is based on the previous addition solution. I have manually written the solution additions up to the 4th order but wish to automate for calculations up to the 10th order but do not see how to label each subsequent addition so it can be called into the next addition in an iterative method? Anyone have any ideas? The attached is my present code.

Version1-1.nb (41.2 KB)

2

Carroll,

Version 12 introduces AsymptoticSolve which I believe will help you.

Regards,

Neil

3

Thanks Neil, I will take a look at it.

4

Caroll,

Actually AsymptoticDSolveValue does exactly what you describe – it solves Differential Equations using perturbation methods.

Regards,

Neil

5

Neil,
Thanks for pointing that out. My problem is a little different in that the two simultaneous equations that I have to solve are related and are of different orders, one is fourth order and the other is second order. Correct me if I am wrong but it appears that the AsymptoticDSolveValue handles only a system of equations which are all of the same order and possibly not related. Maybe the AsymptoticDSolveValue could be modified to handle my problem. I will check. Again I appreciate you pointing this out to me. Thanks.

6

Carroll,

Please post your equations and boundary conditions. It was difficult to follow what you are doing in the file you posted.

Regards,

Neil

7

Neil,
Probably the easiest way to explain is to attach the technical paper that I am trying to simulate. In the paper the subject equations are items (2) and (3). Boundary conditions are Item (4). Rework of the equations to apply what is called the Homotopy Analysis Method (HAM) (no pun intended) is in items (20) thru (25). Hope that clears things. Again, thanks for your interest.

HAM somega .pdf (279 KB)