Analyzing Approximate Solutions of the Cauchy Problem for Helmholtz Equation

Articles in Press

Document Type : Research Paper

Authors

1 Scientific Research Center, Baku Engineering University, Baku AZ0102, Azerbaijan.

2 Department of Mathematics, Baku Engineering University, Baku AZ0102, Azerbaijan.

3 Department of Mathematical Physics and Functional Analysis, Samarkand State University named after Sharof Rashidov, Samarkand 140104, Uzbekistan.

4 Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia.

5 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj, 11942, Saudi Arabia.

Abstract

In this article, we address the challenge of recovering solutions to the Helmholtz equation within a confined three-dimensional space, utilizing information gathered from a section of the boundary. This scenario pertains to the Cauchy problem. By employing the Carleman function, we derive an explicit approximate solution. To tackle this problem, we leverage the Carleman function, a powerful tool in the study of partial differential equations. The Carleman estimate allows us to transition from boundary data to an approximate solution in the interior of the domain.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 12 May 2026

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History

  • Receive Date: 29 January 2025
  • Revise Date: 11 April 2026
  • Accept Date: 11 May 2026

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