On linear ODEs with a time singularity of the first kind and unsmooth inhomogeneity

Jana Burkotová,Irena Rachůnková,Ewa B Weinmüller,Svatoslav Staněk

doi:10.1186/s13661-014-0183-6

Jana Burkotová, Irena Rachůnková + Show 2 more

Open Access

https://doi.org/10.1186/s13661-014-0183-6

Copy DOI

Journal: Boundary Value Problems	Publication Date: Sep 26, 2014
Citations: 4	License type: CC BY 2.0

Affiliation: Palacký University, Olomouc, TU Wien

Abstract

Abstract In this paper we investigate the analytical properties of systems of linear ordinary differential equations (ODEs) with unsmooth nonintegrable inhomogeneities and a time singularity of the first kind. We are especially interested in specifying the structure of general linear two-point boundary conditions guaranteeing existence and uniqueness of solutions which are continuous on a closed interval including the singular point. Moreover, we study the convergence behavior of collocation schemes applied to solving the problem numerically. Our theoretical results are supported by numerical experiments. MSC: 34A12, 34A30, 34B05.

Highlights

1 Introduction Singular boundary value problems (BVPs) arise in numerous applications in natural sciences and engineering and since many years, they have been in focus of extensive investigations
B and B are constant matrices and it turns out that they are subject to certain restrictions for a problem with a unique continuous solution
Before discussing the case of an arbitrary spectrum of M which enables to consider more general initial value problem (IVP), terminal value problem (TVP), and BVPs, we summarize here the results from the previous sections and point out the differences when compared to the framework given in [, ], where linear systems with smooth inhomogeneity, t ∈ (, ], ( )

Summary

Introduction

Singular boundary value problems (BVPs) arise in numerous applications in natural sciences and engineering and since many years, they have been in focus of extensive investigations. We recapitulate the case when all eigenvalues of M have positive real parts: For any f ∈ C [ , ] and any vector β ∈ Rn there exists a unique continuous solution y of TVP ( ) if and only if the matrix B ∈ Rn×n is nonsingular. For any B ∈ Rm×n such that the matrix B R ∈ Rm×m is nonsingular and for any f ∈ C[ , ] and β ∈ Rm, there exists a unique solution y ∈ C[ , ] of IVP ( ) This solution has the form y(t) = R (B R )– β + s–Ms– f (st) ds, t ∈ ( , ], and satisfies the initial condition My( ) = , which is necessary and sufficient for y ∈ C[ , ]. Estimates for the higher derivatives of y follow in an analogous manner

Differences between linear systems with smooth and unsmooth inhomogeneity

10 Conclusions