- Tytuł:
- Reduction of differential equations
- Autorzy:
-
Skórnik, Krystyna
Wloka, Joseph - Powiązania:
- https://bibliotekanauki.pl/articles/1207630.pdf
- Data publikacji:
- 2000
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
differential algebra
linear differential equations
operational calculus - Opis:
- Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) $Ly = D^{n}y + a_{n-1}D^{n-1}y+...+ a_{0}y = 0$, where $a_0,... ,a_n ∈ F$, and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system $y_1,... ,y_n$ of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of E leaving the elements of F fixed. If E = PV(F,Ly=0) is a Picard-Vessiot extension, then the elements g ∈ G(E|F) are n × n matrices, n= ord L, with entries from C, the field of constants. Is it possible to solve an equation (1) by means of linear differential equations of lower order ≤ n-1? We answer this question by giving neccessary and sufficient conditions concerning the Galois group G(E|F) and its Lie algebra A(E|F).
- Źródło:
-
Banach Center Publications; 2000, 53, 1; 199-204
0137-6934 - Pojawia się w:
- Banach Center Publications
- Dostawca treści:
- Biblioteka Nauki
Artykuł