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In this paper, spectral methods based on conformal mappings are proposed to solve the Steklov eigenvalue problem and its related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use a gradient ascent approach to find the optimal domain which maximizes k-th Steklov eigenvalue with a fixed area for a given k. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different k.
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Whittaker Fourier type solutions to differential equations arising from string theory
In this article, we find the full Fourier expansion for solutions of a certain differential equation. When such an f is fully automorphic these functions are referred to as generalized non-holomorphic Eisenstein series. We give a connection of the boundary condition on such Fourier series with convolution formulas on the divisor functions. Additionally, we discuss a possible relation with the differential Galois theory.
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- Award ID(s):
- 2001909
- PAR ID:
- 10538580
- Publisher / Repository:
- International Press of Boston
- Date Published:
- Journal Name:
- Communications in Number Theory and Physics
- Volume:
- 17
- Issue:
- 3
- ISSN:
- 1931-4523
- Page Range / eLocation ID:
- 583 to 641
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4310/CNTP.2023.v17.n3.a2
