Abstract. We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a Fischer decomposition for the discrete Laplacian.
We develop a constructive framework to define difference approximations of Dirac operators which factorize the discrete Laplacian. This resulting notion of discrete monogenic functions is compared with the notion of discrete holomorphic functions on quad-graphs. In the end Dirac operators on quad-graphs are constructed. (2000). Primary 30G35, 30G25; Secondary 05C78.
Mathematics Subject Classification
Abstract. The main goal of this paper is to adopt a multivector calculus scheme to study finite difference discretizations of Klein-Gordon and Dirac equations for which Chebyshev polynomials of the first kind may be used to represent a set of solutions. The development of a welladapted discrete Clifford calculus framework based on spinor fields allows us to represent, using solely projection based arguments, the solutions for the discretized Dirac equations from the knowledge of the solutions of the discretized Klein-Gordon equation. Implications of those findings on the interpretation of the lattice fermion doubling problem is briefly discussed.Mathematics Subject Classification (2010). Primary 30G35, 39A12; Secondary 33C05, 53Z05.
The aim of this work is to study the numerical solution of the nonlinear Schrödinger problem using a combination between Witt basis and finite difference approximations. We construct a discrete fundamental solution for the nonstationary Schrödinger operator and we show the convergence of the numerical scheme. Numerical examples are given at the end of the article.
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