On linear differential equations and systems with reflection

Abstract

In this paper we develop a theory of linear differential systems analogous to the classical one for ODEs, including the obtaining of fundamental matrices, the development of a variation of parameters formula and the expression of the Green's functions. We also derive interesting results in the case of differential equations with reflection and generalize the Hyperbolic Phasor Addition Formula to the case of matrices.

A Liouville’s Formula for Systems with Reflection

“…We highlight the importance of research on qualitative aspects, such as the existence and uniqueness of solution [1][2][3], boundedness [4] or periodicity [5]. There has also been a search for Hilbert bases related to operator eigenfunction decompositions [6,7] and explicit solutions or the associated Green's functions [8][9][10][11][12][13]. Many of these works highlight the strong relation between linear analysis and linear algebra, either in the context of ordinary differential equations [10,12], systems [13,14], difference equations [15] or partial differential equations [16].…”

“…There has also been a search for Hilbert bases related to operator eigenfunction decompositions [6,7] and explicit solutions or the associated Green's functions [8][9][10][11][12][13]. Many of these works highlight the strong relation between linear analysis and linear algebra, either in the context of ordinary differential equations [10,12], systems [13,14], difference equations [15] or partial differential equations [16]. In particular, in [13], the authors developed an explicit fundamental matrix for the system of differential equations with reflection:…”

On the solvability of the main boundary value problems for a nonlocal Poisson equation

“…The results of studying the spectral properties for differential equations with involution were used in [1,14,31] to solve the related inverse problems. The series of papers by Cabada and Tojo (see [8,30] for an expanded list of citations) pioneered in creating a comprehensive theory of Green's functions for the one-dimensional differential equations with involution.…”

Differential systems with reflection and matrix invariants

“…In recent works regarding the solution and Green functions of differential equations with reflection (see, for instance, previous studies()), the strong relation between linear analysis and linear algebra is highlighted. In particular, in the most recent of the aforementioned works, the authors obtain an explicit fundamental matrix for the system of differential equations with reflection where and .…”

“…(Cabada and Tojo3 )Assume F − G and F + G are invertible. (F − G) −1 (A − B)(F + G) −1 (A + B)is a fundamental matrix of problem(1).…”