Wave propagators for transient waves in one-dimensional media

Karlsson, Anders

doi:10.1016/0165-2125(96)00008-x

Cited by 16 publications

(15 citation statements)

References 41 publications

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“…(6.11), the wave propagator, P(z), can be written as 12) for some finite integer k. Since the functions P j = P j (z; t), j = 1, 2, 3, . .…”

Section: The Second Precursor-brillouin's Forerunnermentioning

confidence: 99%

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Time-domain theory of forerunners

Rikte

1998

J. Opt. Soc. Am. A

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The time-domain theory of forerunners (precursors) in temporally dispersive, nonmagnetic, isotropic materials is developed using the propagator technique. Specifically, the impulse response at a (comparatively) large propagation depth is expanded in two different ways: (a) with respect to the wave-front and (b) with respect to slowly varying field components. A few numerical examples illustrating the theory are given.

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“…(6.11), the wave propagator, P(z), can be written as 12) for some finite integer k. Since the functions P j = P j (z; t), j = 1, 2, 3, . .…”

Section: The Second Precursor-brillouin's Forerunnermentioning

confidence: 99%

“…There are, at least, three different time-domain methods that depend on optical wave splitting: the imbedding method [11], the Green functions method [9], and the propagator method [12]. A wave splitting is a change of the dependent variables which simplifies the propagation problem.…”

Section: Introductionmentioning

confidence: 99%

Time-domain theory of forerunners

Rikte

1998

J. Opt. Soc. Am. A

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“…Karlsson (Karlsson, 1996) has derived wave propagators for a dispersive electromagnetic problem. We derive the equations for the wave propagators for the advection problem under consideration in this paper.…”

Section: Wave Propagatorsmentioning

confidence: 99%

“…Differentiation with respect to x', the station where the wave is propagating to, leads to a form of the operators that will be required in the sequel. Differentiation with respect to x, the point where the wave has propagated from, leads to an equation suitable for invariant imbedding -these will not be discussed further here (see (Karlsson, 1996) for information on this case).Differentiation of equation (5.5), with respect to x', yields (5.7) (D1 + (Jx's(x,x'))D2)v+ (x',t+ ((x,x')Reconstruction in advective and diffusive media 21 Then use of the dynamics, (4.27), to rewrite terms on the left-hand-side of (5.7), and on interchanging the left-hand-side with the right-hand-side, leads to (5.8) ( ox'a(x,x') )v+ (x, t) where the operators A and ]Ill are defined by the convolution operators in (4.27). Furthermore, the use of (4.7) allows the right-hand-side of (5.8) to be written as…”

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confidence: 99%

“…There is no restriction on the relative magnitudes of x and x' in the definition of the wave propagators. When x' > x the propagators map the field v+ forward in the positive x-direction, along with the advection, and when x' < x the propagation is backwards, against the advection and in the negative x-direction.The propagators satisfy the properties of a group (see (Karlsson, 1996)), and the groups inverse operator is defined through, for example where p± is a kernel function, and the * operator denotes the temporal convolutionThe factor a modifies the wave front, and provides attenuation when x' > x, and amplification when x' < x, and its functional form is given in equation (A.6).Causality requires that v±(x,t) = 0 fort:::; ((O,x). In equation (5.5) the positive moving field at some point x' > 0 has been written in two parts.…”

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Signal restoration after transmission through an advective and diffusive medium

Shorten

Wall

2004

Proc. R. Soc. Lond. A

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ABSTRACT. Inverse problem, regularisation, singular perturbation, wave splitting, wave propagators, square root operator, inverse mass transport This paper considers an inverse problem associated with mass transport in a pipe. It illustrates how wave splitting techniques can be utilised for an inverse problem associated with one-dimensional mass transport processes. This is done by using a generalisation of Fick's law which introduces a relaxation parameter into the problem, so converting the parabolic partial differential equation by a singular perturbation into a hyperbolic one. This generalised law by ensuring finite mass flux propagation speeds, enables a stable equation to be utilised to reconstruct the interior boundary condition; so providing a regularised solution to the inverse problem. Theoretical results for the solution of the inverse problem are also developed. INTRODUCTIONThe perifusion apparatus is an experimental in vitro tool used by endocrinologists to model information transfer in endocrine systems (Mcintosh & Mcintosh, 1983;Mcintosh, Mcintosh, & Kean, 1984;Evans et al., 1985). The major drawback of the perifusion system derives from the dispersion, diffusion and mixing of the hormone within the apparatus which distort the original released hormone concentration profile. Recently mathematical techniques have been applied to improving this situation (Shorten & Wall, 2001). This paper presents another approach to the inverse problem of concentration signal restoration after signal transmission through an advective and diffusive medium, with applications to perifusion.The problem considered here has direct application to a related problem involving the estimation of secretion of adrenocorticotropic hormone (ACTH) from the pituitary gland. In this problem, assays of the blood flow are taken downstream from the pituitary in a horse, which is secreting ACTH, and it is then required to estimate the concentration of ACTH at the pituitary, (Alexander, Irvine, Liversey, & Donald, 1988). This in vivo experimental technique poses similar problems to the perifusion apparatus mentioned previously, but the flow situation is more complicated.When a mass concentration of a material is transported within a fluid of a different material, the estimation of the final temporal profile, downstream of the injection point, from the knowledge of the initial injection temporal profile, is a relatively straight-forward problem; it is required Date: September 9, 2002. 1991. to solve a well-posed parabolic partial differential equation. This is termed a direct problem. However, when a mass concentration of a material flowing and diffusing within an advecting fluid is measured downstream of the injection point, the estimation of the initial injection profile is a difficult problem. This is termed an inverse problem; in particular an inverse source problem. This problem is difficult because it is ill-posed and besides the difficulty of problem formulation, it must also be made well-posed.It is assumed throughout this paper ...

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