| Project/Area Number |
24K06852
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Keio University |
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Principal Investigator |
彭 林玉 慶應義塾大学, 理工学部(矢上), 准教授 (90725780)
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| Project Period (FY) |
2024-04-01 – 2028-03-31
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| Project Status |
Granted (Fiscal Year 2024)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2027: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2026: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2025: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2024: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | Symmetries / Discrete equations / DDEs / Semi-discrete equations / Symmetry / Conservation law / Lie group / Numerical methods |
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| Outline of Research at the Start |
Symmetries have proven to be of great importance in various fields, owing to the versatile applications in elucidating solution properties to physical models. Many scholars made substantial contributions to the study of symmetry methods for discrete equations, giving rise to a plethora of subsequent research and applications. Despite these advancements, a multitude of unresolved questions continued to challenge the field. The primary objective of the current project is to tackle some of the unresolved questions concerning the symmetries of discrete equations and to explore their applications.
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| Outline of Annual Research Achievements |
We have established the general symmetry prolongation formula and applied it to a broad class of differential-difference equations. A key contribution is the introduction of an evolutionary representative that had previously been overlooked in the literature. This innovation enables the formulation and proof of a semi-discrete version of Noether-type theorems, providing a unified framework for deriving conservation laws in semi-discrete systems. These results have been well received and were presented at several invited international and domestic conferences and workshops, highlighting both their theoretical significance and potential for practical applications.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We have analytically derived the general symmetry prolongation formula for semi-discrete equations, thereby resolving an open problem that has remained unsolved for nearly three decades. This advancement provides a solid theoretical foundation for extending Noether's theorems to the semi-discrete setting, enabling the systematic construction of conservation laws. Furthermore, these results lay the groundwork for the development of symmetry-preserving numerical methods, particularly those based on discrete moving frames.
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| Strategy for Future Research Activity |
The research will continue in line with the original proposal. As a first step, we will focus on exploring several significant physical applications of the newly developed symmetry methods for discrete equations.
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