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Generalized solutions of boundary value problems for ordinary linear differential equations of second order in the Colombeau algebra. (English) Zbl 0837.34026

Let \(A_1\) be the set of all smooth functions \(\varphi : \mathbb{R} \to \mathbb{R}\) such that \(\int \varphi (x) dx = 1\), \(\int x^k \varphi (x) dx = 0\) \((0 < k \leq q)\). Let \(E\) be the ring of functions \(R : A_1 \to \mathbb{R}\) such that \(|R (\varphi (x/ \varepsilon)/ \varepsilon) |\leq c \varepsilon^{-N}\) for appropriate \(N \in \mathbb{N}\), \(c > 0\), \(\eta > 0\), and all \(\varphi \in A_q\) \((q \geq N)\), \(0 < \varepsilon < \eta\). Let \(T \subset E\) be the ideal of functions \(R\) satisfying \(|R (\varphi (x/ \varepsilon)/ \varepsilon |\leq c \varepsilon^{\alpha (q) - N}\) for appropriate function \(\alpha : \mathbb{N} \to \mathbb{R}^+\) such that \(\alpha (q) \to \infty\) as \(q \to \infty\). Denoting \(\overline \mathbb{R} = E/T\), the Colombeau algebra \(G\) includes (roughly speaking) all smooth functions \(\mathbb{R} \to \overline \mathbb{R}\) and may be regarded for a multiplicative theory of generalized functions, see J. F. Colombeau [New generalized functions and multiplication of distributions, North- Holland, Amsterdam (1984; Zbl 0532.46019)]. The author states conditions for the coefficients \(p,q,r \in G\) which ensure the existence and the uniqueness of solutions \(x \in G\) of the boundary value problem \(x''(t) + p(t)x'(t) + q(t)x(t) = r(t)\), \(x(a) = d_1\), \(x(b) = d_2\) \((d_1, d_2 \in \overline \mathbb{R})\).
Reviewer: J.Chrastina (Brno)

MSC:

34B05 Linear boundary value problems for ordinary differential equations
34B99 Boundary value problems for ordinary differential equations
46F99 Distributions, generalized functions, distribution spaces

Citations:

Zbl 0532.46019
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