A numerical method for one-dimensional diffusion problem using Fourier transform and the B-spline Galerkin method

Addam, Mohamed; Bouhamidi, A.; Jbilou, Khalide

doi:10.1016/j.amc.2010.01.011

Cited by 9 publications

(13 citation statements)

References 7 publications

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“…To facilitate the readability of this paper, we recall some notations and results already given in [3]. Let φ(x, t) be the photon density at the position x ∈]a, b[⊂ R and at the time t ∈ [0, T ], where T is the time period.…”

Section: One-dimensional Diffusion Problemmentioning

confidence: 99%

“…Let us denote by c(x) = c 0 /ν(x) the speed of light in the domain [a, b], where c 0 is the speed of light in the vacuum [3,15] and ν(x) ≥ 1 for all x ∈ [a, b]. We assume that µ a , µ s and ν are real, bounded and nonnegative functions on [a, b].…”

Section: One-dimensional Diffusion Problemmentioning

confidence: 99%

“…A well-posedness of the problem (2.8)-(2.9) is established in [3,4] by the following theorem Theorem 2.1. For a given frequency ω ∈ R, the problem (2.8)-(2.9) has a unique solution u * (., ω) in H 1 satisfying the following upper bound 15) where the positive constants α and M (ω) are given in (2.4) and (2.12), respectively.…”

Section: Weak Variational Formulationmentioning

confidence: 99%

“…we denote by φ(x, t) the photon density located at position x at time t. The time-dependent diffusion equation is described by the following parabolic partial differential equation (see [3,4])…”

Section: One-dimensional Diffusion Problemmentioning

confidence: 99%

“…The radiative transfer transport equation (also derived from the Boltzmann equation) has been used successfully as a standard model for describing light transport in scattering media, see [11,13,14,17]. The variational method proves that the corresponding forward problem has a unique solution for the given optical properties and the sources [2,3]. In the literature, many numerical techniques have been proposed to solve such problem.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An inverse problem for one-dimensional diffusion equation in optical tomography

Addam¹

2017

B Soc Paran Mat

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In this paper, we study the one-dimensional inverse problem for the diffusion equation based optical tomography. The objective of the present work is a mathematical and numerical analysis concerning one-dimensional inverse problem. In the first stage, the forward diffusion equation with boundary conditions is solved using an intermediate elliptic equation. We give the existence and the uniqueness results of the solution. An approximation of the photon density in frequency-domain is proposed using a Splines Galerkin method. In the second stage, we give theoretical results such as the stability and lipschitz-continuity of the forward solution and the Fréchet differentiability of the Dirichlet-to-Neumann nonlinear map with respect to the optical parameters. The Fréchet derivative is used to linearize the considered inverse problem. The Newton method based on the regularization technique will allow us to compute the approximate solutions of the inverse problem. Several test examples are used to verify high accuracy, effectiveness and good resolution properties for smooth and discontinuous optical property solutions.

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Section: One-dimensional Diffusion Problemmentioning

confidence: 99%

Section: One-dimensional Diffusion Problemmentioning

confidence: 99%