On one numerical method of fractal antenna synthesis

Background. At the present time the theory and technology of antennas are one of most rapidly developing fields of radio engineering. Modern progress in the theory and technology of antennas is based on latest achievements in physics and mathematics. Due to the need for antenna miniaturization for mobile devices, the methods of fractal geometry are being adopted in radio engineering. In recent decades there has been observed a growing interest in construction and research of fractal and genetic antennas. In this regard there rises a necessity to develop analytical and numerical methods of analysis and synthesis of fractal and genetic antennas. From the mathematical point of view the problem is complicated by the description of antenna synthesis through operator equations compact operators, i.e. ill-conditioned problems. The article is devoted to construction of a numerical method of solution of Fredholm's equations of first kind that model antennas and illustrations of this method at fractal antenna synthesis. Materials and methods. The authors used methods of optimization, functional analysis and linear algebra. Results. The article suggests a modification of the local correction method to solve Fredholm's integral equations of first kind. The suggested method is used for solving the problem of fractal antenna synthesis on pre-fractals of the Cantor's perfect set and pre-fractals of the Sierpinski's carpet. The work describes the constructed computational methods of mechanical quadratures to solve Fredholm's integral equations of first kind and offers their implementation algorithms (the method local corrections). Conclusions. The authors have constructed and implemented on software the approximate methods for solving Fredholm's equations of first kind that simulate antenna synthesis. It is shown that the modifications of the local correction method are efficient methods for solving Fredholm's equations of first kind. The article demonstrates the efficiency of the suggested numerical algorithms by the example of synthesizing antennas with apertures on pre-fractals of the Cantor's perfect set (in a one-dimensional case) and pre-fractals of the Sierpinski's carpet (in a multi-dimensional case). The suggested algorithms may be applied when solving many ill-conditioned problems.