A Liouville’s Formula for Systems with Reflection

Codesido, Santiago; Tojo, F. Adrián F.

doi:10.3390/math9080866

Mathematics

2021

DOI: 10.3390/math9080866

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A Liouville’s Formula for Systems with Reflection

Santiago Codesido

F. Adrián F. Tojo

Abstract: In this work, we derived an Abel–Jacobi–Liouville identity for the case of two-dimensional linear systems of ODEs (ordinary differential equations) with reflection. We also present a conjecture for the general case and an application to coupled harmonic oscillators.

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On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution

Sarsenbi

2021

Symmetry

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We give a definition of Green’s function of the general boundary value problems for non-self-adjoint second order differential equation with involution. The sufficient conditions for the basis property of system of eigenfunctions are established in the terms of the boundary conditions. Uniform equiconvergence of spectral expansions related to the second-order differential equations with involution:−y″(x)+αy″(−x)+qxyx=λyx,−1<x<1, with the boundary conditions y′−1+b1y−1=0,y′1+b2y1=0, is obtained. As a corollary, it is proved that the eigenfunctions of the perturbed boundary value problems form the basis in L2(−1,1) for any complex-valued coefficient q(x)∈L1(−1,1).

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On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution

Sarsenbi

2021

Symmetry

View full text Add to dashboard Cite

show abstract

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A Liouville’s Formula for Systems with Reflection

Abstract: In this work, we derived an Abel–Jacobi–Liouville identity for the case of two-dimensional linear systems of ODEs (ordinary differential equations) with reflection. We also present a conjecture for the general case and an application to coupled harmonic oscillators.

Cited by 1 publication

References 17 publications

On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution

On Eigenfunctions of the Boundary Value Problems for Second Order Differential Equations with Involution

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