Dirichlet Type Problem for 2D Quaternionic Time-Harmonic Maxwell System in Fractal Domains

Yudier Peña Pérez,Paul Bosch,Juan Bory Reyes,Ricardo Abreu Blaya

doi:10.1155/2020/4735357

Abstract

We investigate an electromagnetic Dirichlet type problem for the 2D quaternionic time-harmonic Maxwell system over a great generality of fractal closed type curves, which bound Jordan domains in R2. The study deals with a novel approach of h-summability condition for the curves, which would be extremely irregular and deserve to be considered fractals. Our technique of proofs is based on the intimate relations between solutions of time-harmonic Maxwell system and those of the Dirac equation through some nonlinear equations, when both cases are reformulated in quaternionic forms.

Highlights

A theory of hyperholomorphic functions of two real variables is the most natural and close generalization of complex analysis that preserves many of its important features
E Maxwell equations govern the behavior of the electromagnetic field
E outline of the paper is as follows: In Section 2 we provide an outlook to the basics of quaternionic analysis and elements of fractals geometry; a new hyperholomorphic Cauchy type integral for a domain with h−summable boundary in R2 is described in Section 3, where we state theoretical results on integral representation formulas in domains bounded by such curves

Summary

Introduction

A theory of hyperholomorphic functions of two real variables is the most natural and close generalization of complex analysis that preserves many of its important features. For pure and applied mathematical interest, in [10] some boundary value problems for time-harmonic electromagnetic fields on the more challenging case of domains with fractal boundaries are discussed. E main goal of this work is the study of an electromagnetic Dirichlet type problem for a domain with fractal boundary in R2.

Results

Conclusion