Theoretical research on the numerical analysis for differential equations based on the convergence theorem of Newton's method
| Project/Area Number |
17540103
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
KAWANAGO Tadashi Tokyo Institute of Technology, Graduate school of Science and Engineering, Associate Professor, 大学院理工学研究科, 助教授 (20214661)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2006: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | partial differential equations / numerical verification / The convergence theorem of Newton's method / The spectrum method / the computational efficiency / bifurcation phenomena / The estimate for the norm of the inverse of linearized operators / Newton法 / 有限要素法 |
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| Research Abstract |
In this project we carried out the theoretical research on the numerical analysis for differential equations by reformulating and optimizing the convergence theorem of Newton's method Banach spaces according to our need. To be more precise, we established an efficient algorithm on the numerical verification for the solutions of nonlinear partial differential equations, which is based in a new simplifies convergence theorem of Newton's method. We clarify by some verification examples that our method is more efficient in the verification for solutions than the other known methods. The convergence theorem of Newton's method is clear in principle and is very excellent from the theoretical view point. At the same time it is long believed by the related researchers that this theorem is not good from the view point of the computational efficiency and that therefore it is not well suited to the verification for solutions of partial differential equations. We are sure to override their fixes concept by our achievement. Our paper including the above results was published in J. Comput. Appl. Math.
The above convergence theorem is optimized in the numerical verification based on the finite element methods. the finite element methods is, however, inferior in general from the view point of the computational accuracy and is not well suited to the precise analysis for the complicated phenomena such as the bifurcation in dynamical systems. The spectrum method is spectrum methods. Moreover, we generalized the method on estimating the norm of the inverse of linearized operators (which plays an important rule in checking a condition in the convergence theorem of Newton's method) in order to apply it to the spectrum methods. We reported the above results and delivered a lecture at International conference of numerical analysis and applied mathematics 2006 held at Greece.
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Report
(3 results)
Research Products
(4 results)
