Человек, опередивший время

213 Âîñïîìèíàíèÿ äðóçåé è ñîðàòíèêîâ impedance of the system electrode/electrolyte is proportional to p -1/2 , where p is the Laplace operator. This was a technical realization of the mathematical operations of fractional order (semi-order) integro-differentiation, that can be found in his 1964 papers [3, 4]. He proposed a polarography method for construction of fractional-differentiated polarograms, by means of which it was possible to find, with enough speed and exactness (1-2%), the form of the polarogram after a fractional differentiation. Another proposal for FC applications came from analysis of the integral equation given by Nigmatullin to relate the surface concentration and the density of the substance flow through the electrode. He showed that differentiating by d 1/2 /dt 1/2 the time variance of the surface concentration C(0,t) it is possible to find directly the gradient of concentration or density, thus avoiding to solve the boundary value problem for the diffusion equation. This property became the base of electrical modelling of cells (see details in [5]). In the mid-1980s, FC was related with the so-called constant phase elements (CPE) and also with the objects of the fractal geometry, thus finding new horizons for applications. Nigmatullin expected such a close relationship to exist, since any arbitrary mathematical operation has, at its physical realization, a definite geometry or topology. He thought about the representation of the diffusion impedance operator via a cascade-model of involved RC-elements. And should the frequency for preserving the equivalence of the diffusion impedance increase, it would be necessary to increase as well the number of cascades by a geometrical division of the initial RC- element into smaller RC-elements, self-similar in a topological sense. These ideas and relationships were later developed and mathematically proved by his son Raoul, see e.g. [6] and many earlier his papers. One of the last projects of Rashid Nigmatullin in developing the trend of use of FC operators was the transition from problems in analysis to problems in synthesis of electrical chains with an arbitrary fractional order Laplace operator and system with a fixed phase. For such a synthesis, he introduced a new type of elements in the theory of electrical chains, combining resistor-condensator and resistor-induction properties, in 1982. These new elements Íàó÷íûå ðàçðàáîòêè