TY - JOUR

T1 - A computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem

AU - Cai, Shuting

AU - Watanabe, Yoshitaka

N1 - Funding Information:
This project was sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
Publisher Copyright:
© 2015, The JJIAM Publishing Committee and Springer Japan.

PY - 2015/3

Y1 - 2015/3

N2 - We propose a computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem. The method works in both the one-dimensional and the two-dimensional case. We begin by finding an approximate solution to a nonlinear problem, and we then enclose the solution by using Nakao’s numerical verification method. Instead of considering directly the eigenvalues for the elliptic operator linearized at the verified solution, we linearize the operator at the approximate solution. We present a theorem that allows us to determine under which conditions and in which disks there will be no eigenvalues. Thus, if any of those disks are contained in the enclosed area, we can exclude those eigenvalues. Next, we construct various computable criteria that allow us to use a computer program to find these disks. Finally, we use our results to determine which eigenvalues to exclude for the operator linearized at the verified solution. We present some verified results.

AB - We propose a computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem. The method works in both the one-dimensional and the two-dimensional case. We begin by finding an approximate solution to a nonlinear problem, and we then enclose the solution by using Nakao’s numerical verification method. Instead of considering directly the eigenvalues for the elliptic operator linearized at the verified solution, we linearize the operator at the approximate solution. We present a theorem that allows us to determine under which conditions and in which disks there will be no eigenvalues. Thus, if any of those disks are contained in the enclosed area, we can exclude those eigenvalues. Next, we construct various computable criteria that allow us to use a computer program to find these disks. Finally, we use our results to determine which eigenvalues to exclude for the operator linearized at the verified solution. We present some verified results.

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U2 - 10.1007/s13160-015-0167-7

DO - 10.1007/s13160-015-0167-7

M3 - Article

AN - SCOPUS:84925520075

VL - 32

SP - 263

EP - 294

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 1

ER -