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Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations.

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Title: Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations.
Authors: Zewen Wang1, Shufang Qiua2 qiushufang@gzmtu.edu.cn, Xiuxing Ruib2, Wen Zhang2 wangzewen@gzmtu.edu.cn
Source: Mathematical Modelling & Analysis. 2025, Vol. 30 Issue 1, p74-96. 23p.
Subjects: Numerical differentiation, Inverse problems, Superposition principle (Physics), Equations, Noise
Abstract: In this paper, we mainly study the numerical differentiation problem of computing the fractional order derivatives from noise data of a single variable function. Firstly, the numerical differentiation problem is reformulated into an inverse source problem of first order hyperbolic equation, and the corresponding solvability and the conditional stability are provided under suitable conditions. Then, four regularization methods are proposed to reconstruct the unknown source of hyperbolic equation which is the numerical derivative, and they are implemented by utilizing the finite dimensional expansion of source function and the superposition principle of hyperbolic equation. Finally, Numerical experiments are presented to show effectiveness of the proposed methods. It can be conclude that the proposed methods are very effective for small noise levels, and they are simpler and easier to be implemented than the previous PDEs-based numerical differentiation method based on direct and inverse problems of parabolic equations. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Modelling & Analysis is the property of Vilnius Gediminas Technical University and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations.
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  Data: <searchLink fieldCode="JN" term="%22Mathematical+Modelling+%26+Analysis%22">Mathematical Modelling & Analysis</searchLink>. 2025, Vol. 30 Issue 1, p74-96. 23p.
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  Data: <searchLink fieldCode="DE" term="%22Numerical+differentiation%22">Numerical differentiation</searchLink><br /><searchLink fieldCode="DE" term="%22Inverse+problems%22">Inverse problems</searchLink><br /><searchLink fieldCode="DE" term="%22Superposition+principle+%28Physics%29%22">Superposition principle (Physics)</searchLink><br /><searchLink fieldCode="DE" term="%22Equations%22">Equations</searchLink><br /><searchLink fieldCode="DE" term="%22Noise%22">Noise</searchLink>
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  Data: In this paper, we mainly study the numerical differentiation problem of computing the fractional order derivatives from noise data of a single variable function. Firstly, the numerical differentiation problem is reformulated into an inverse source problem of first order hyperbolic equation, and the corresponding solvability and the conditional stability are provided under suitable conditions. Then, four regularization methods are proposed to reconstruct the unknown source of hyperbolic equation which is the numerical derivative, and they are implemented by utilizing the finite dimensional expansion of source function and the superposition principle of hyperbolic equation. Finally, Numerical experiments are presented to show effectiveness of the proposed methods. It can be conclude that the proposed methods are very effective for small noise levels, and they are simpler and easier to be implemented than the previous PDEs-based numerical differentiation method based on direct and inverse problems of parabolic equations. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematical Modelling & Analysis is the property of Vilnius Gediminas Technical University and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.3846/mma.2025.19339
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      – Code: eng
        Text: English
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        PageCount: 23
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      – SubjectFull: Numerical differentiation
        Type: general
      – SubjectFull: Inverse problems
        Type: general
      – SubjectFull: Superposition principle (Physics)
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      – SubjectFull: Equations
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      – SubjectFull: Noise
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              M: 01
              Text: 2025
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