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In this paper, the theory of the Lagrange polynomial approximation of a directly non-measurable characteristic of an element of a mechanical structure by means of an information-measuring system of numerical differentiation is refined. It is shown how to obtain the maximal accuracy of approximation on the basis of the theory of inverse problems and the method of reduction of measurements. For a cantilever beam loaded with concentrated force at its free end, a method has been developed for the experimental and calculated determination of the bending moment at the fixed end of the beam. The basis of the method is the procedure for the optimal placement of sensors and transformation of sensor output signals, if the length of the beam and the initial parameters of the elastic line (lateral displacement and the angle of rotation on the support) are not accurate.

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Purpose of research. The work is devoted to a range of questions related to Cauchy problem on the segment of real axis with the application of the inverse Cauchy problem, in which real constants are initial conditions which are optimally restored according to experimental or tabular values of the solution of the differential equation. The object of the study is an information-measuring system, in which approximate values of initial conditions are calculated from discrete function values of Cauchy problem solving.Methods. The following problems are solved for this purpose: parameters of measuring section placement on the investigated object and approximation grid on measuring section are developed. Characteristics of recovery accuracy of initial conditions of the task are formulated.Results. An experimental-calculated method of determining initial conditions in the inverse Cauchy problem is proposed. It is based on the concept of objective function of regularization of the problem. Task regularization parameter in the form of minimum value by Lebesgue function is proposed.Conclusion. The reaction of uniformly approximating method of the initial conditions of the inverse Cauchy problem to the deviation of the approximation grid coordinates nodes from the coordinates of Chebyshev alternance was described. Graphs of method reaction to deviation of grid pitch from optimal pitch are given.

Purpose of research is to create a method for determining the source of messages in the receiver, which allows, based on the analysis of the characteristics of the distribution of the time of receipt of messages, to increase the reliability of determining the source.Methods. Authentication of the message source is based on statistical analysis of metadata values, which in this study are the time intervals between messages. The metadata processing model is based on the well-known model of receiving messages from the target source in LoRaWAN networks. At the same time, the source was determined using coding methods in the block coupling mode, which provide higher identification reliability for short-length messages characteristic of the specified type of networks. With the help of numerical modeling, the patterns of changes in the characteristics of the time of receipt of messages in the event of an identification error were determined. The decision-making rule is also formulated in case of impossibility of authentication based on processing the contents of identification fields. Studies have shown the effectiveness of the application of the source detection model in various ranges of parameters for the formation of message sequences.Results. The result of the conducted research is the development of an authentication method based on the analysis of the time of receipt of messages to the receiver, which differs in that it uses the characteristics of the distribution of moments of high orders for a series of time delays as the analyzed data. Its use in combination with coding methods in the message concatenation mode reduces the probability of a single message replacement error in a sequence by 4-6 times, compared with methods that perform identification only based on the results of processing the identifiers of the messages themselves.Conclusion. The result of experimental studies has shown the possibility of using the developed method to increase the reliability of determining the authenticity of the source of messages, the possibility of reducing the number of reinquiries that occur when errors are detected, the possibility of reducing the size of additional identifier fields in each message. The resulting effect will be expressed in a general increase in the bandwidth of the communication channel between remote components of a distributed system.