Multiscale solution decomposition of nonlocal-in-time problems with application in numerical computation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914049735786496 |
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| author | Liu, Mengmeng Ma, Jie Qiu, Wenlin Zheng, Xiangcheng |
| author_facet | Liu, Mengmeng Ma, Jie Qiu, Wenlin Zheng, Xiangcheng |
| contents | This work develops a multiscale solution decomposition (MSD) method for nonlocal-in-time problems to separate a series of known terms with multiscale singularity from the original singular solution such that the remaining unknown part becomes smoother. We demonstrate that the MSD provides a scenario where the smoothness assumption for solutions of weakly singular nonlocal-in-time problems, a commonly encountered assumption in numerous literature of numerical methods that is in general not true for original solutions, becomes appropriate such that abundant numerical analysis results therein become applicable. From computational aspect, instead of handling solution singularity, the MSD significantly reduces the numerical difficulties by separating and thus circumventing the solution singularity. We consider typical problems, including the fractional relaxation equation, Volterra integral equation, subdiffusion, integrodifferential equation and diffusion-wave equation, to demonstrate the universality of MSD and its effectiveness in improving the numerical accuracy or stability in comparison with classical methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17020 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiscale solution decomposition of nonlocal-in-time problems with application in numerical computation Liu, Mengmeng Ma, Jie Qiu, Wenlin Zheng, Xiangcheng Numerical Analysis 65M12, 65M60 This work develops a multiscale solution decomposition (MSD) method for nonlocal-in-time problems to separate a series of known terms with multiscale singularity from the original singular solution such that the remaining unknown part becomes smoother. We demonstrate that the MSD provides a scenario where the smoothness assumption for solutions of weakly singular nonlocal-in-time problems, a commonly encountered assumption in numerous literature of numerical methods that is in general not true for original solutions, becomes appropriate such that abundant numerical analysis results therein become applicable. From computational aspect, instead of handling solution singularity, the MSD significantly reduces the numerical difficulties by separating and thus circumventing the solution singularity. We consider typical problems, including the fractional relaxation equation, Volterra integral equation, subdiffusion, integrodifferential equation and diffusion-wave equation, to demonstrate the universality of MSD and its effectiveness in improving the numerical accuracy or stability in comparison with classical methods. |
| title | Multiscale solution decomposition of nonlocal-in-time problems with application in numerical computation |
| topic | Numerical Analysis 65M12, 65M60 |
| url | https://arxiv.org/abs/2509.17020 |