2002 Fiscal Year Final Research Report Summary
Nonlinear Analysis by Numerical Verification Methods
| Project/Area Number |
13640105
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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 | The University of Electro-Communications |
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Principal Investigator |
YAMAMOTO Nobito The University of Electro-Communications, Department of Electro-Communications, Associate Professor, 電気通信学部, 助教授 (30210545)
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| Co-Investigator(Kenkyū-buntansha) |
ONISHI Isamu Hiroshima University, Department of Science, Associate Professor, 大学院・理学研究科, 助教授 (30262372)
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| Project Period (FY) |
2001 – 2002
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| Keywords | verifind computation / spectrum method / nonlimear analysis / free boundary |
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| Research Abstract |
The objects of this research are partial differential equations (PDEs) with free boundaries. In 2001, we treated problems defined on the unit square in R^2. As the free boundaries are defined by potential contour which is obtained through some eigenvalue problem, we developed techniques of verified computation for eigenpairs of eigenvalue problems on partial differential operators. In these techniques, we adopted spectrum methods for approximation and error estimation
In order to treat other shapes of domains than the unit square, we improved existing verification methods to PDEs on nonconvex polygonal domains and obtained a more simple and accurate method
In 2002, we developed a methods of verified computation for PDEs defined on annuli. Using specrum method based on Fourier-Bessel functions, we needed coefficients of Bessel expansion with guaranteed accuracy. The coefficients are defined through an eigenvalue problem concerning a one-dimensional PDE with two points boundary values
We developed methods for verification of existence and nonexistence of eigenvalues in order to obtain the validated values of the coefficients. The method for nonexistence is simpler and more effective than existing methods, which we have shown by numerical calculations
Moreover, a software package to calculate the values of Bessel functions with guaranteed accuracy has been developed. It is constructed on INTLAB which is a library for interval calculation with verified computation on MATLAB
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