Numerical Studies of the Steklov Eigenvalue Problem via Conformal Mappings


In this paper, a spectral method based on conformal mappings is proposed to solve Steklov eigenvalue problems and their 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 sothe discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape
optimization problem, we use the gradient ascent approach to find the optimal domain which maximizes kth 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. Joint work with Chiu-Yen Kao.

Brief Biography

A Saudi assistant professor in mathematics at Princess Nourah bint Abdulrahman University (PNU), Riyadh, Kingdom of Saudi Arabia. 
I got my bachelor’s degree with first degree honor from King Abdelaziz university, college of education, mathematics department, Jeddah, Saudi Arabia. I do have a tremendous experience in general mathematics, differential equations teaching and education. In 2009, I worked as a demonstrator and instructor for many courses at (PNU), before being awarded a scholarship to pursue my P.h.D. degree in the United States. I earned double masters degree in science from PNU in 2008 and from Claremont Graduate University (CGU), Claremont, CA, U.S.A. in 2014. In 2018 I honored a Doctor of Philosophy in mathematics from CGU. I am interested in applied mathematics that related to differential equations and its application, shape optimization, and Extremal eigenvalue problem.