On the relation between the Maxwell system and the Dirac equation

A simple relation between the Maxwell system and the Dirac equation based on their quaternionic reformulation is discussed. We establish a close connection between solutions of both systems as well as a relation between the wave parameters of the electromagnetic field and the energy of the Dirac particle.Comment: Submitted the 18 of Dec., 2001 to Proceedings of 4th WSEAS Int. Conf. on Mathematics and Computers in Physics, Cancun, Mexico, May 12-16, 200

On a relation of pseudoanalytic function theory to the two-dimensional stationary Schroedinger equation and Taylor series in formal powers for its solutions

We consider the real stationary two-dimensional Schroedinger equation. With the aid of any its particular solution we construct a Vekua equation possessing the following special property. The real parts of its solutions are solutions of the original Schroedinger equation and the imaginary parts are solutions of an associated Schroedinger equation with a potential having the form of a potential obtained after the Darboux transformation. Using L. Bers theory of Taylor series for pseudoanalytic functions we obtain a locally complete system of solutions of the original Schroedinger equation which can be constructed explicitly for an ample class of Schroedinger equations. For example it is possible, when the potential is a function of one cartesian, spherical, parabolic or elliptic variable. We give some examples of application of the proposed procedure for obtaining a locally complete system of solutions of the Schroedinger equation. The procedure is algorithmically simple and can be implemented with the aid of a computer system of symbolic or numerical calculation

Quaternionic reformulation of Maxwell's equations for inhomogeneous media and new solutions

We propose a simple quaternionic reformulation of Maxwell's equations for inhomogeneous media and use it in order to obtain new solutions in a static case

Quaternionic equation for electromagnetic fields in inhomogeneous media

We show that the Maxwell equations for arbitrary inhomogeneous media are equivalent to a single quaternionic equation which can be considered as a generalization of the Vekua equation for generalized analytic functions.Comment: Submitted to Proceedings of the Third ISAAC congress in Berlin in August 200

Recent developments in applied pseudoanalytic function theory

We present recently obtained results in the theory of pseudoanalytic functions and its applications to elliptic second-order equations. The operator (divpgrad+q) with p and q being real valued functions is factorized with the aid of Vekua type operators of a special form and as a consequence the elliptic equation (divpgrad+q)u=0, (1) reduces to a homogeneous Vekua equation describing generalized analytic (or pseudoanalytic) functions. As a tool for solving the Vekua equation we use the theory of Taylor and Laurent series in formal powers for pseudoanalytic functions developed by L. Bers. The series possess many important properties of the usual analytic power series. Their applications until recently were limited mainly because of the impossibility of their explicit construction in a general situation. We obtain an algorithm which in a really broad range of practical applications allows us to construct the formal powers and hence the pseudoanalytic Taylor series in explicit form precisely for the Vekua equation related to equation (1). In other words, in a bounded domain this gives us a complete (in C-norm) system of exact solutions of (1)

Solution of parabolic free boundary problems using transmuted heat polynomials

A numerical method for free boundary problems for the equation $$u_{xx}-q(x)u=u_t$$ is proposed. The method is based on recent results from transmutation operators theory allowing one to construct efficiently a complete system of solutions for this equation generalizing the system of heat polynomials. The corresponding implementation algorithm is presented.Comment: 14 pages, 4 figures, contact details update

Biquaternions for analytic and numerical solution of equations of electrodynamics

We give an overview of recent advances in analysis of equations of electrodynamics with the aid of biquaternionic technique. We discuss both models with constant and variable coefficients, integral representations of solutions, a numerical method based on biquaternionic fundamental solutions for solving standard electromagnetic scattering problems, relations between different operators of mathematical physics including the Schrodinger, the Maxwell system, the conductivity equation and others leading to a deeper understanding of physics and mathematical properties of the equations.Comment: 1 figur

On Beltrami fields with nonconstant proportionality factor on the plane

We consider the equation rotB+aB=0 (1) in the plane with a being a real-valued function and show that it can be reduced to a Vekua equation of a special form. In the case when a depends on one Cartesian variable a complete system of exact solutions of the Vekua equation and hence of equation (1) is constructed based on L. Bers' theory of formal powers

Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems

A method for approximate solution of spectral problems for Sturm-Liouville equations based on the construction of the Delsarte transmutation operators is presented. In fact the problem of numerical approximation of solutions and eigenvalues is reduced to approximation of a primitive of the potential by a finite linear combination of generalized wave polynomials introduced in arXiv:1208.5984, arXiv:1208.6166. The method allows one to compute both lower and higher eigendata with an extreme accuracy.Comment: 32 pages, 9 figures, 4 tables; Sections 6 and 7 extended, runtimes adde

Modified spectral parameter power series representations for solutions of Sturm-Liouville equations and their applications

Spectral parameter power series (SPPS) representations for solutions of Sturm-Liouville equations proved to be an efficient practical tool for solving corresponding spectral and scattering problems. They are based on a computation of recursive integrals, sometimes called formal powers. In this paper new relations between the formal powers are presented which considerably improve and extend the application of the SPPS method. For example, originally the SPPS method at a first step required to construct a nonvanishing (in general, a complex-valued) particular solution corresponding to the zero-value of the spectral parameter. The obtained relations remove this limitation. Additionally, equations with "nasty" Sturm-Liouville coefficients $1/p$ or $r$ can be solved by the SPPS method. We develop the SPPS representations for solutions of Sturm-Liouville equations of the form $$(p(x)u')'+q(x)u=\sum_{k=1}^N \lambda^k R_k[u], \quad x\in(a,b)$$ where $R_k[u] :=r_k(x)u+s_k(x)u'$, $k=1,\ldots N$, the complex-valued functions $p$, $q$, $r_k$, $s_k$ are continuous on the finite segment $[a,b]$. Several numerical examples illustrate the efficiency of the method and its wide applicability.Comment: 28 pages, 5 figures, 7 table