Leveraging Prior Known Vector Green Functions in Solving Perturbed Dirac Equation in Clifford Algebra
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Advances in Applied Clifford Algebras
Leveraging Prior Known Vector Green Functions in Solving Perturbed Dirac Equation in Clifford Algebra Morteza Shahpari∗
and Andrew Seagar
Communicated by Swanhild Bernstein Abstract. Solving boundary value problems with boundary element methods requires specific Green functions suited to the boundary conditions of the problem. Using vector algebra, one often needs to use a Green function for the Helmholtz equation whereas it is a solution of the perturbed Dirac equation that is required for solving electromagnetic problems using Clifford algebra. A wealth of different Green functions of the Helmholtz equation are already documented in the literature. However, perturbed Dirac equation is only solved for the generic case and only its fundamental solution is reported. In this paper, we present a simple framework to use known Green functions of Helmholtz equation to construct the corresponding Green functions of perturbed Dirac equation which are essential in finding the appropriate kernels for integral equations of electromagnetic problems. The procedure is further demonstrated in a few examples. Mathematics Subject Classification. 35J08, 65N80. Keywords. Green functions, Fundamental solutions, Maxwell’s equations, Dirac equation, Electromagnetism.
1. Introduction Electromagnetism is well developed using vector algebra and a great variety of problems are treated in-depth. Boundary value problems in electromagnetic theory are often formulated by the help of fundamental solutions and Green functions which can be written easily for the unbounded homogeneous M. Shahpari was with the Griffith University during the project. ∗ Corresponding
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M. Shahpari, A. Seagar
Adv. Appl. Clifford Algebras
medium (e.g. classical source in free space). However, if a problem has additional boundary conditions like infinite planar layers (stratified medium) or a wedge that is extended to infinity, we have to also incorporate those into the Green function [6,17,35]. Another algebraic tool chain to solve electromagnetic problems is the geometric algebra which is often also called Clifford algebra to honour William Kingdon Clifford [7]. Clifford generated his geometric algebra using two sets of mutually commutative quaternions producing what today would be called a four-dimensional Clifford algebra Cl(0, 4), supporting bivectors and trivectors in addition to vectors. Due to its flexibility, Clifford algebra has been used with different conventions and assumptions in the literature. A good review of early developments of using Clifford algebra for electromagnetic problems is provided in [34]. Development of new techniques in quaternion-valued functions and their associated boundary value problems (e.g. [11,16,21,22]) in around 1990s opened new horizons and made it possible to attempt solve new class of problems. For instance in [21], Laplace operator was extended to Helmoltz operator with complex quaternionic wave numbers and fundamental solutions for different types of wavenu
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