The reflectionless properties of Toeplitz waves and Hankel waves: An analysis via Bessel functions

Burrage, K; Burrage, P; MacNamara, S

The reflectionless properties of Toeplitz waves and Hankel waves: An analysis via Bessel functions

Burrage, K Burrage, P MacNamara, S

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Publisher:: Elsevier
Publication Type:: Journal Article
Citation:: Applied Mathematics and Computation, 2021, 389, pp. 1-17
Issue Date:: 2021-01-01

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Field	Value	Language
dc.contributor.author	Burrage, K
dc.contributor.author	Burrage, P
dc.contributor.author	MacNamara, S
dc.date.accessioned	2022-02-15T02:25:34Z
dc.date.available	2022-02-15T02:25:34Z
dc.date.issued	2021-01-01
dc.identifier.citation	Applied Mathematics and Computation, 2021, 389, pp. 1-17
dc.identifier.issn	0096-3003
dc.identifier.issn	1873-5649
dc.identifier.uri	http://hdl.handle.net/10453/154518
dc.description.abstract	We study reflectionless properties at the boundary for the wave equation in one space dimension and time, in terms of a well-known matrix that arises from a simple discretisation of space. It is known that all matrix functions of the familiar second difference matrix representing the Laplacian in this setting are the sum of a Toeplitz matrix and a Hankel matrix. The solution to the wave equation is one such matrix function. Here, we study the behaviour of the corresponding waves that we call Toeplitz waves and Hankel waves. We show that these waves can be written as certain linear combinations of even Bessel functions of the first kind. We find exact and explicit formulae for these waves. We also show that the Toeplitz and Hankel waves are reflectionless on even, respectively odd, traversals of the domain. Our analysis naturally suggests a new method of computer simulation that allows control, so that it is possible to choose — in advance — the number of reflections. An attractive result that comes out of our analysis is the appearance of the well-known shift matrix, and also other matrices that might be thought of as Hankel versions of the shift matrix. By revealing the algebraic structure of the solution in terms of shift matrices, we make it clear how the Toeplitz and Hankel waves are indeed reflectionless at the boundary on even or odd traversals. Although the subject of the reflectionless boundary condition has a long history, we believe the point of view that we adopt here in terms of matrix functions is new.
dc.language	en
dc.publisher	Elsevier
dc.relation.ispartof	Applied Mathematics and Computation
dc.relation.isbasedon	10.1016/j.amc.2020.125576
dc.rights	info:eu-repo/semantics/closedAccess
dc.subject	0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, 0802 Computation Theory and Mathematics
dc.subject.classification	Numerical & Computational Mathematics
dc.title	The reflectionless properties of Toeplitz waves and Hankel waves: An analysis via Bessel functions
dc.type	Journal Article
utslib.citation.volume	389
utslib.for	0102 Applied Mathematics
utslib.for	0103 Numerical and Computational Mathematics
utslib.for	0802 Computation Theory and Mathematics
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Science
pubs.organisational-group	/University of Technology Sydney/Faculty of Science/School of Mathematical and Physical Sciences
utslib.copyright.status	closed_access	*
pubs.consider-herdc	false
dc.date.updated	2022-02-15T02:25:30Z
pubs.publication-status	Published
pubs.volume	389

Abstract:

We study reflectionless properties at the boundary for the wave equation in one space dimension and time, in terms of a well-known matrix that arises from a simple discretisation of space. It is known that all matrix functions of the familiar second difference matrix representing the Laplacian in this setting are the sum of a Toeplitz matrix and a Hankel matrix. The solution to the wave equation is one such matrix function. Here, we study the behaviour of the corresponding waves that we call Toeplitz waves and Hankel waves. We show that these waves can be written as certain linear combinations of even Bessel functions of the first kind. We find exact and explicit formulae for these waves. We also show that the Toeplitz and Hankel waves are reflectionless on even, respectively odd, traversals of the domain. Our analysis naturally suggests a new method of computer simulation that allows control, so that it is possible to choose — in advance — the number of reflections. An attractive result that comes out of our analysis is the appearance of the well-known shift matrix, and also other matrices that might be thought of as Hankel versions of the shift matrix. By revealing the algebraic structure of the solution in terms of shift matrices, we make it clear how the Toeplitz and Hankel waves are indeed reflectionless at the boundary on even or odd traversals. Although the subject of the reflectionless boundary condition has a long history, we believe the point of view that we adopt here in terms of matrix functions is new.

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http://hdl.handle.net/10453/154518