Modeling Electrostatic Field in Complex Non-Axisymmetric Domains With a Non-Compact Boundary

Using the example of modeling the electrostatic field of a system of charged parallel circular disks located near a conductive grounded screen in theform of a parabolic cylinder, this paper proposes a method for solving a whole class of problems. This class simultaneously combines the complexityof the geometric domain with algorithmic and computational complexities. A generalized Fourier method was used for this purpose, the mathematicalapparatus of which was further developed in this paper. As a result, the problem of reconstructing the potential field was reduced to a resolving systemwith a Fredholm operator, which guaranteed a correct approximate solution with sufficient accuracy. The article presents the results of an extensivecomputer experiment in which the geometric parameters of the problem, the number of disks considered, and the planes in which the potential was calculated were varied. The graphical information is supplemented by numerical results, summarized in tables. To verify the adequacy of the model, anotherproblem was also solved: the potential of a system of point charges located on the axis of a conductive screen at points coinciding with the centersof the disks. The magnitudes of the charges were chosen to be equal to the charges of the corresponding disks. As a result, good consistency was achieved between both models. For comparison, another problem about the potential field of a system of charged parallel circular disks without ascreen was solved separately. Calculations showed that the presence of a screen, located at a relative distance of 0.5 from the five disks, reduces thefield potential by an average of 33 % relative to the potential of the disks without a screen. The article derives formulas for the charge density distributionon the disks, as well as the magnitudes of these charges and their numerical values. The capacitance of the disk-screen conductor system was alsodetermined separately as a function of the relative distance between them. For the first time, the Green's function for the Laplace equation in a paraboliccylinder was obtained explicitly. To this end, a new addition theorem was proven for basis harmonic functions in arbitrarily shifted, identicallydirected coordinate systems of the parabolic cylinder. All problems solved in the paper are new, the theoretical modeling results are rigorously substantiated by five theorems proven in the paper, and the numerical results have guaranteed accuracy. Thus, the modeling method proposed in the articlehas proven its high effectiveness.