       Re: Boundary Value Problems

• To: mathgroup at yoda.physics.unc.edu
• Subject: Re: Boundary Value Problems
• From: leon at physics.su.oz.au
• Date: Fri, 26 Feb 1993 11:22:04 +1000

```Javid Attai writes
>
>I'd like to get a power series solution to the following differential equation:
>
> D[a[r],{r,2}] + (D[a[r],{r,1}])/r - 0.5 a[r] + (a[r])^3/(1+(a[r])^2) = 0
>
>The boundary conditions are
>
>a = 2.5
>a[Infinity] = 0
>
>da |
>-- |        = 0
>dr | r = 0
>
>
>da |
>-- |            = 0
>dr | r = Infinity
>
>My attempts to solve this problem have been unsuccessful.
>
>
>Any help will be most appreciated.
>

The following short sequence finds power series solutions to any number of
terms, limited only by memory and time. The boundary conditions at 0 have
been automatically included.

DE = D[a[r],{r,2}] + D[a[r],r]/r - a[r]/2 + a[r]^3/(1+a[r]^2);

nterms = 20;

vars = Table[ c[n], {n,2,nterms}];

a[r_] = 2.5 + Sum[ c[n] r^n, {n,2,nterms}];

seriesDE = LogicalExpand[Series[DE,{r,0,nterms-2}]==0];

soln = a[r] /. Solve[seriesDE, vars]

For this particular example, more efficiency can be obtained by realizing
that all the odd coefficients are identically zero.

Unfortunately, as a general technique for solving boundary value problems,
a series solution is unlikely to have a radius of convergence that extends
out to infinity and so will be of limited value.

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```

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