Advanced Search

Question: Change of variables involving square roots

Question:Change of variables involving square roots

esche0 5 Maple
differential-equation pde pdetools
July 05 2008
0
I have a particular system of partial differential equations to which I want to apply a change of variables involving square roots using PDEtools[dchange]. The result turns out to be not in a simple form due to the square roots. Why is this so ? Is there a better way solving the problem than the one I am suggesting below ? The system of partial differential equations of interest is the following: LLd_w:=[D[2](f)(w[1],w[2],w[3])+w[2] D[2,2](f)(w[1],w[2],w[3])-6 w[2] D[2](f)(w[1],w[2],w[3])-4 w[2]^2 D[2,2](f)(w[1],w[2],w[3])+4 w[2] w[3] D[2,3](f)(w[1],w[2],w[3])-w[3]^2 D[3,3](f)(w[1],w[2],w[3]),-6 w[3] D[3](f)(w[1],w[2],w[3])-2 w[2] D[2,3](f)(w[1],w[2],w[3])+D[3](f)(w[1],w[2],w[3])+w[3] D[3,3](f)(w[1],w[2],w[3])-4 w[3]^2 D[3,3](f)(w[1],w[2],w[3])-w[1]^2 D[1,1](f)(w[1],w[2],w[3])+4 w[1] w[3] D[1,3](f)(w[1],w[2],w[3]),-60 f(w[1],w[2],w[3])-864 w[1] D[1](f)(w[1],w[2],w[3])-432 w[1]^2 D[1,1](f)(w[1],w[2],w[3])+D[1](f)(w[1],w[2],w[3])+w[1] D[1,1](f)(w[1],w[2],w[3])-2 w[3] D[1,3](f)(w[1],w[2],w[3])] to the new variables ch := {w[1] = (2*(j[1]+j[2]-mu))/(j[2]*j[1]+sqrt(j[1]*(j[1]-mu))*sqrt(j[2]*(j[2]-mu))), w[2]=j[3],w[3] = (j[2]*j[1]+sqrt(j[1]*(j[1]-mu))*sqrt(j[2]*(j[2]-mu)))^2/(4*j[1]*j[2]*(j[1]+j[2]-mu)^2)} using LLd_j:=PDEtools[dchange](subs(mu=1728,op(ch)),LLd_w,{j[1],j[2],j[3]}): The result is terribly complicated, in particular there are lots of square root expressions that have not been simplified. Applying simplify to the result does not improve the result either. However, from other calculations I know that there must be simpler expressions, in particular the new differential operators should not contain any square roots. So first I tried to split up the change of variables into two steps as follows: ch1:={w[3]=t[1]*t[2]/(t[1]+t[2])^2,w[1]=-(t[1]+t[2])/(t[1]-1)/(t[2]-1)/432}; LLd_t:=factor(PDEtools[dchange](ch1,LLd_w,{t[1],t[2]})): ch2:={t[1]=(sqrt(1-1728/j[1])-1)/(sqrt(1-1728/j[1])+1),t[2]=(sqrt(1-1728/j[2])-1)/(sqrt(1-1728/j[2])+1)}; LLd_j:=factor(PDEtools[dchange](ch2,LLd_t,{j[1],j[2]})): This already provides simpler expressions which are, however, not yet simple enough. Next I split up the second change of variables into three steps as follows: LLd_s:=factor(PDEtools[dchange]({t[1]=(s[1]-1)/(s[1]+1),t[2]=(s[2]-1)/(s[2]+1),t[3]=s[3]},LLd_t)): LLd_r:=factor(PDEtools[dchange]({s[1]=sqrt(r[1]),s[2]=sqrt(r[2]),s[3]=r[3]},LLd_s)): LLd_j:=factor(PDEtools[dchange]({r[1]=1-1728/j[1],r[2]=1-1728/j[2],r[3]=j[3]},LLd_r)): This time the resulting expressions were as simple as expected. In particular, the transformed differential operators didn't contain any square roots anymore, they only appear in the prefactors, as expected. Why is it not possible to get this result directly with a single change of variables ? If it is nevertheless possible, then how ?
Please Wait...