@inproceedings{ConstalesKrausshar, author = {Constales, Denis and Kraußhar, Rolf S{\"o}ren}, title = {ON THE KLEIN-GORDON EQUATION ON THE 3-TORUS}, editor = {G{\"u}rlebeck, Klaus and K{\"o}nke, Carsten}, organization = {Bauhaus-Universit{\"a}t Weimar}, issn = {1611-4086}, doi = {10.25643/bauhaus-universitaet.2863}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20170314-28639}, pages = {10}, abstract = {In this paper we consider the time independent Klein-Gordon equation on some conformally flat 3-tori with given boundary data. We set up an explicit formula for the fundamental solution. We show that we can represent any solution to the homogeneous Klein-Gordon equation on the torus as finite sum over generalized 3-fold periodic elliptic functions that are in the kernel of the Klein-Gordon operator. Furthermore we prove Cauchy and Green type integral formulas and set up a Teodorescu and Cauchy transform for the toroidal Klein-Gordon operator. These in turn are used to set up explicit formulas for the solution to the inhomogeneous version of the Klein-Gordon equation on the 3-torus.}, subject = {Angewandte Informatik}, language = {en} } @inproceedings{GrobConstalesKrausshar, author = {Grob, Dennis and Constales, Denis and Kraußhar, Rolf S{\"o}ren}, title = {THE HYPERCOMPLEX SZEG{\"O} KERNEL METHOD FOR 3D MAPPING PROBLEMS}, editor = {G{\"u}rlebeck, Klaus and K{\"o}nke, Carsten}, organization = {Bauhaus-Universit{\"a}t Weimar}, issn = {1611-4086}, doi = {10.25643/bauhaus-universitaet.2846}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20170314-28464}, pages = {7}, abstract = {In this paper we present rudiments of a higher dimensional analogue of the Szeg{\"o} kernel method to compute 3D mappings from elementary domains onto the unit sphere. This is a formal construction which provides us with a good substitution of the classical conformal Riemann mapping. We give explicit numerical examples and discuss a comparison of the results with those obtained alternatively by the Bergman kernel method.}, subject = {Angewandte Informatik}, language = {en} } @inproceedings{CacaoConstalesKrausshar, author = {Cacao, Isabel and Constales, Denis and Kraußhar, Rolf S{\"o}ren}, title = {A UNIFIED APPROACH FOR THE TREATMENT OF SOME HIGHER DIMENSIONAL DIRAC TYPE EQUATIONS ON SPHERES}, editor = {G{\"u}rlebeck, Klaus and K{\"o}nke, Carsten}, organization = {Bauhaus-Universit{\"a}t Weimar}, issn = {1611-4086}, doi = {10.25643/bauhaus-universitaet.2834}, url = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20170314-28343}, pages = {8}, abstract = {Using Clifford analysis methods, we provide a unified approach to obtain explicit solutions of some partial differential equations combining the n-dimensional Dirac and Euler operators, including generalizations of the classical time-harmonic Maxwell equations. The obtained regular solutions show strong connections between hypergeometric functions and homogeneous polynomials in the kernel of the Dirac operator.}, subject = {Angewandte Informatik}, language = {en} }