Factorization method for solving systems of second-order linear ordinary differential equations

I. N. Parasidis, E. Providas

Department of Environmental Sciences, University of Thessaly, Gaiopolis Campus, 415 00 Larissa, Greece

Аннотация: We consider in a Banach space the following two abstract systems of first-order and second-order linear ordinary differential equations with general boundary conditions, respectively,
$$ X'(t)-A_0(t)X(t)=F(t), \quad \Phi(X)=\sum_{j=1}^n M_j\Psi_j(X), $$
and
\begin{gather*} X''(t)-S(t)X'(t)-Q(t)X(t)=F(t),\\ \Phi(X)=\sum_{i=j}^n M_j\Psi_j(X), \quad \Phi(X')=C\Phi(X)+\sum_{j=1}^r N_j\Theta_j(X), \end{gather*}
where $X(t) = \mathrm{col}(x_1(t), \dots,x_m(t))$ denotes a vector of unknown functions, $F(t)$ is a given vector and $A_0(t)$, $S(t)$, $Q(t)$ are given matrices, $\Phi$, $\Psi_1,\dots,\Psi_n$, $\Theta_1,\dots,\Theta_r$ are vectors of linear bounded functionals, and $M_1,\dots,M_n$, $C$, $N_1,\dots, N_r$ are constant matrices. We first provide solvability conditions and a solution formula for the first-order system. Then we construct in closed form the solution of a special system of $2m$ first-order linear ordinary differential equations with constant coefficients when the solution of the associated system of $m$ first-order linear ordinary differential equations is known. Finally, we construct in closed form the solution of the second-order system in the case in which it can be factorized into first-order systems.

Ключевые слова и фразы: systems of ordinary differential equations, nonlocal boundary value problems, multipoint boundary problems, integral boundary conditions, exact solution, correct problems, factorization.

MSC: 34A30, 34B10, 47A68

Поступила в редакцию: 12.12.2023
Принята в печать: 28.04.2025

Язык публикации: английский

DOI: 10.32523/2077-9879-2025-16-2-55-73