Reconstruction of the initial function from the solution of the fractional wave equation measured in two geometric settings

Hyungyeong Jung; Sunghwan Moon; Hyungyeong Jung; Sunghwan Moon

doi:10.3934/era.2022225

Electronic Research Archive

2022, Volume 30, Issue 12: 4436-4446. doi: 10.3934/era.2022225

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Reconstruction of the initial function from the solution of the fractional wave equation measured in two geometric settings

Hyungyeong Jung ¹,
Sunghwan Moon ^{2
,
,}

1.
School of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
2.
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Republic of Korea

Academic Editor: Joel E. Restrepo

Received: 02 May 2022 Revised: 24 July 2022 Accepted: 01 September 2022 Published: 13 October 2022

Photoacoustic tomography (PAT) is a novel and rapidly developing technique in the medical imaging field that is based on generating acoustic waves inside of an object of interest by stimulating non-ionizing laser pulses. This acoustic wave was measured by using a detector on the outside of the object it was then converted into an image of the human body after several inversions. Thus, one of the mathematical problems in PAT is reconstructing the initial function from the solution of the wave equation on the outside of the object. In this study, we consider the fractional wave equation and assume that the point-like detectors are located on the sphere and hyperplane. We demonstrate a way to recover the initial function from the data, namely, the solution of the fractional wave equation, measured on the sphere and hyperplane.
- photoacoustic,
- tomography,
- wave equation,
- fractional derivative
Citation: Hyungyeong Jung, Sunghwan Moon. Reconstruction of the initial function from the solution of the fractional wave equation measured in two geometric settings[J]. Electronic Research Archive, 2022, 30(12): 4436-4446. doi: 10.3934/era.2022225

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Abstract

Photoacoustic tomography (PAT) is a novel and rapidly developing technique in the medical imaging field that is based on generating acoustic waves inside of an object of interest by stimulating non-ionizing laser pulses. This acoustic wave was measured by using a detector on the outside of the object it was then converted into an image of the human body after several inversions. Thus, one of the mathematical problems in PAT is reconstructing the initial function from the solution of the wave equation on the outside of the object. In this study, we consider the fractional wave equation and assume that the point-like detectors are located on the sphere and hyperplane. We demonstrate a way to recover the initial function from the data, namely, the solution of the fractional wave equation, measured on the sphere and hyperplane.

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