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Full text release has been delayed at the author's request until January 01, 2026

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On Non-homogeneous Reflections via Dirichlet-to-Neumann and Robin-to-Neumann Operators

Aldawsari, Murdhy A.
http://orcid.org/0000-0002-1752-1756
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1667564038716025

Abstract Details

2022, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
This study is devoted to the construction of reflection formulae for solutions to elliptic differential equations subject to various boundary conditions on a real analytic arc about that arc. Most of the results known in the literature deal with the Dirichlet condition. One of the main results of this work is a derivation of the reflection law for a harmonic function w(x,y), defined in a neighborhood of a real-analytic curve in the plane subject to a non-homogeneous Robin condition, aw+b_nw=phi_w on that curve. Here a and b are constants, and phi_w is the restriction of a holomorphic function onto the curve. For the case of the homogeneous condition, when phi_w=0, while a and b are real-analytic functions, a reflection formula was derived in Belinskiy and Savina, using the reflected fundamental solution method. Here, we construct a Robin-to-Neumann mapping and use it for obtaining the reflection operator. Since the two formulae look different, we show their equivalence when a and b are constants and phi_w=0. As examples, we show reflection formulae for non-homogeneous Neumann and Robin conditions on the most important for applications boundaries, such as circles and lines. Construction of Neumann-to-Dirichlet and Neumann-to-Robin operators in the form that allows to obtain reflection formulae to solutions of elliptic equations subject to Neumann and Robin conditions given on a real analytic arc is also an important result by itself. In chapter 3, we discuss the simplest version of our approach, specifically what we are talking about is harmonic functions in a neighborhood of the unit circle. The results of this chapter are published in Analysis and Mathematical Physics (2019). In chapter 4, we discuss reflections for harmonic functions near a real-analytic arc. The results of this chapter are published in Applicable Analysis (2022). Chapter 5 is devoted to the reflections of solutions to the Helmholtz equation, where we expand our approach earlier developed for harmonic functions.
Tatiana Savin (Advisor)
Qiliang Wu (Committee Member)
Alexander Neiman (Committee Member)
Archil Gulisashvili (Committee Member)
120 p.
Applied Mathematics; Mathematics
Dirichlet-to-Neumann operator; Robin-to-Neumann operator; Schwarz function; Laplace equation; harmonic function; Helmholtz equation

Recommended Citations

Citations

  • Aldawsari, M. A. (2022). On Non-homogeneous Reflections via Dirichlet-to-Neumann and Robin-to-Neumann Operators [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1667564038716025

    APA Style (7th edition)

  • Aldawsari, Murdhy. On Non-homogeneous Reflections via Dirichlet-to-Neumann and Robin-to-Neumann Operators. 2022. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1667564038716025.

    MLA Style (8th edition)

  • Aldawsari, Murdhy. "On Non-homogeneous Reflections via Dirichlet-to-Neumann and Robin-to-Neumann Operators." Doctoral dissertation, Ohio University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1667564038716025

    Chicago Manual of Style (17th edition)

ohiou1667564038716025
© 2022, all rights reserved.
This open access ETD is published by Ohio University and OhioLINK.