kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-2-108-123kaz29-2890Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESЧИСЛЕННОЕ РЕШЕНИЕ ОБРАТНОЙ КОЭФФИЦИЕНТНОЙ ЗАДАЧИ АКУСТИКИ С ПРИМЕНЕНИЕМ ПРЕОБРАЗОВАНИЯ ЛАПЛАСА И МЕТОДА ДИФФЕРЕНЦИАЛЬНОЙ ЭВОЛЮЦИИNUMERICAL SOLUTION OF THE INVERSE COEFFICIENT ACOUSTIC PROBLEM USING LAPLACE TRANSFORM AND DIFFERENTIAL EVOLUTION METHODhttps://orcid.org/0000-0002-5828-7820СиницаА. В.SinitsaA. V.

PhD, ассистент-профессор.

Алматы

PhD, Assistant Professor.

Almaty

a.sinitsa@kbtu.kzhttps://orcid.org/0009-0003-0772-4319ЦхайЮ. А.TskhayYu. A.

Магистр, лектор.

Алматы

MSc, Lecturer.

Almaty

y.tskhay@kbtu.kzhttps://orcid.org/0009-0003-0402-4720ШкоркоА. К.ShkorkoA. K.

Магистр, лектор.

Алматы

MSc, Lecturer.

Almaty

a.ukassova@kbtu.kzhttps://orcid.org/0000-0002-8142-6710КардукА. П.KarduckA. P.

PhD, профессор.

Фуртванген

PhD, Professor.

Furtwangen

achim.karduck@hs-furtwangen.deКазахстанско-Британский технический университетКазахстанKazakh-British Technical UniversityKazakhstanУниверситет ФуртвангенаГерманияFurtwangen UniversityGermany202627062026232108123Copyright © Синица А.В., Цхай Ю.А., Шкорко А.К., Кардук А.П., 20262026Синица А.В., Цхай Ю.А., Шкорко А.К., Кардук А.П.Sinitsa A.V., Tskhay Y.A., Shkorko A.K., Karduck A.P.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/2890

В данной статье рассматривается решение обратной коэффициентной задачи для волнового уравнения, направленной на восстановление пространственного распределения скорости звука в неоднородной среде. Для решения прямой задачи применяется преобразование Лапласа, позволяющее исключить временную зависимость и перейти к обыкновенным дифференциальным уравнениям в частотной области, что существенно снижает вычислительные затраты. Обратная задача формулируется как оптимизационная: минимизация функционала невязки между рассчитанными и измеренными значениями акустического давления осуществляется с помощью стохастического метода глобальной оптимизации, дифференциальной эволюции. Численные эксперименты проводились на модели многослойной среды (песок, грунт, камень, вода, воздух) с использованием синтетических данных, зашумленных случайным образом. Предложен адаптивный комбинированный метод восстановления, позволяющий снизить влияние ошибок на границах сред. Результаты показывают высокую точность метода: относительная погрешность восстановления профиля скорости звука составила около от 2,5 до 4,3%, что подтверждает эффективность подхода для задач акустической диагностики и томографии.

This paper addresses the solution of the inverse coefficient problem for the wave equation aimed at reconstructing the spatial distribution of the speed of sound in an inhomogeneous medium. The Laplace transform is applied to solve the direct problem, eliminating time dependence and reducing the problem to ordinary differential equations in the frequency domain, which significantly decreases computational costs. The inverse problem is formulated as an optimization task: minimizing the residual functional between calculated and measured acoustic pressure values using the stochastic global optimization method, Differential Evolution. Numerical experiments were conducted on a multilayer medium model (sand, soil, rock, water, air) using synthetic data with added random noise. An adaptive combined reconstruction method is proposed to reduce errors at medium boundaries. The results demonstrate high accuracy: the relative error of the sound speed profile reconstruction was approximately from 2.5 to 4.3%, confirming the approach's effectiveness for acoustic diagnostics and tomography applications.

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The authors declare that there are no conflicts of interest present.