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[0] = 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. Leon Poladian ====== ====== Optical Fibre Technology Centre \\ // University of Sydney NSW 2006 ============ AUSTRALIA ============ PHONE +61 2 692 4670 // \\ FAX +61 2 692 4671 ====== ======