desingularize a linear differential operator
Desingularize(L, Dx, x, func)
polynomial in Dx with coefficients that are polynomials in x
variable, denoting the differential operator w.r.t. x
Let L be a linear differential operator, given as a polynomial in Dx with univariate polynomial coefficients in x over a field k of characteristic zero. The command Desingularize(L,Dx,x) constructs a linear differential operator R such that any solution of L⁡y=0 is also a solution of R⁡y=0 and R has no apparent singularities. The operator R is said to maximally desingularize L, and will be right divisible by L over the field k⁡x.
An apparent singularity is a point p where the leading coefficient of L vanishes, yet p is not a pole of any holomorphic solution of L⁡y=0. In this case there will exist d linearly independent solutions at p where d is the order of L.
A function may be specified using the optional argument func. It is applied to the coefficients of the collected result. Often simplify or factor will be used.
For the given differential operator L
L ≔ 24⁢x3−18⁢x4+x8+6⁢x5−x6⁢Dx7+6⁢x5+72⁢x3−30⁢x4−8⁢x7−72⁢x2⁢Dx6+−144⁢x2+36⁢x6+72⁢x3−2⁢x7+144⁢x−18⁢x4⁢Dx5+24⁢x3+36⁢x6+144⁢x−144−72⁢x2−120⁢x5−8⁢x7−x10+x8⁢Dx4+−24⁢x5−x10−6⁢x7+x8+18⁢x6⁢Dx3+36⁢x5−6⁢x6−72⁢x4+2⁢x9⁢Dx2+−36⁢x4+12⁢x5−10⁢x8+2⁢x9⁢Dx+64⁢x7−12⁢x4−32⁢x8+8⁢x9+x12−x10
L ≔ x8−x6+6⁢x5−18⁢x4+24⁢x3⁢Dx7+−8⁢x7+6⁢x5−30⁢x4+72⁢x3−72⁢x2⁢Dx6+−2⁢x7+36⁢x6−18⁢x4+72⁢x3−144⁢x2+144⁢x⁢Dx5+−x10+x8−8⁢x7+36⁢x6−120⁢x5+24⁢x3−72⁢x2+144⁢x−144⁢Dx4+−x10+x8−6⁢x7+18⁢x6−24⁢x5⁢Dx3+2⁢x9−6⁢x6+36⁢x5−72⁢x4⁢Dx2+2⁢x9−10⁢x8+12⁢x5−36⁢x4⁢Dx+64⁢x7−12⁢x4−32⁢x8+8⁢x9+x12−x10
compute a desingularizing operator for L:
M ≔ Desingularize⁡L,Dx,x,factor
M ≔ 1728252⁢Dx8+54154⁢x7+161694⁢x6+263753⁢x5+452649⁢x4−324882⁢x3+1728252⁢Dx7+−433232⁢x6−1293552⁢x5−2218332⁢x4−2969808⁢x3+974646⁢x2+1728252⁢Dx6+−108308⁢x6+1626156⁢x5+5185170⁢x4+9891042⁢x3+9684162⁢x2−1949292⁢x+1728252⁢Dx5+−54154⁢x9−161694⁢x8−263753⁢x7−560957⁢x6+976266⁢x5−2924106⁢x4−10379136⁢x3−27461604⁢x2−45113328⁢x+3677544⁢Dx4−x⁢54154⁢x8+161694⁢x7+263753⁢x6+452649⁢x5−324882⁢x4+1728252⁢x+13826016⁢Dx3+108308⁢x8+323388⁢x7+635814⁢x6+253914⁢x5+974646⁢x4−1728252⁢x2−10369512⁢x−20739024⁢Dx2+108308⁢x8−218152⁢x7−981126⁢x6−2600232⁢x5−520824⁢x4−1728252⁢x2−6913008⁢x−10369512⁢Dx+54154⁢x11+161694⁢x10+263753⁢x9+560957⁢x8−759650⁢x7+538258⁢x6+2595900⁢x5+10387932⁢x4+31684620⁢x3−1728252⁢x2−3456504⁢x−3456504
Q,R ≔ op⁡DEtools['rightdivision']⁡M,L,Dx,x:
Hence, R=Q.L+R where
Tsai, H. "Weyl closure of a linear differential operator." Journal of Symbolic Computation Vol. 29 No. 4-5 (2000): 747-775.
Chyzak, F.; Dumas, P.; Le, H.Q.; Martins, J.; Mishna, M.; Salvy, B. "Taming apparent singularities via Ore closure." In preparation.
The DEtools[Desingularize] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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