A method for numerical study of different integral problems with sufficiently strongly localized solutions is considered. Approximations of the solution with different degrees of accuracy were constructed and, using the existing approximation, an approach to an increase of the solution accuracy is proposed. Numerical solutions are obtained for the problem of free oscillations of laser cavities the optical surface of which was approximated by polynomials of the second and fourth degree inclusive. The obtained numerical solutions were stable.

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The subject matter of the research in the article is the process of restoring the quality and accuracy of the working surface of the wheelsets of the main electric locomotives, electric sections. The purpose of the work is to develop a modeling method for a special shaped cutter with increased productivity, designed to process the profile of locomotive railway wheels taking into account the shaping processes, which is based on the theory of modeling the high-speed milling process through the shaping efficiency coefficient. The article solves the following tasks: substantiation of the importance of upgrading the most promising designs of "duplex" mills, taking into account the reduction of vibrations due to chip crushing, to increase tool wear resistance, machining accuracy, reduce tool readjustment time and, accordingly, reduce equipment downtime, increase processing productivity, determine the mathematical apparatus for calculating model parameters; development of a method for modeling and optimization of shaping processes in the processing of locomotive wheel sets. Such methods were used: fundamentals of system analysis, simulation method. The following results were obtained: the design of a "duplex" milling cutter with a predicted increase in resistance and increase in processing productivity due to modeling of forming processes with a decrease in the sliding angle by three to five times, which was not previously taken into account when designating and selecting cutting parameters, is provided. It is concluded that the model design of the cutter, taking into account "duplex" and the tribological process of modeling shaping during processing, can increase the resistance of the cutters and ensure the full operation of all cups of the cutter. The forming component of the cutting conditions was evaluated taking into account the coefficient of efficiency of the process of forming the surface layer of the machined wheel. The conclusion was made about the need to create new processes of shaping the design and modernization of tools. Various ways of modeling tool systems are considered. A method for creating a "duplex" cutter model is defined. A "duplex" mill is considered and design features of the algorithm for creating such tools are given. The functioning of the shaping parameters during the processing of locomotive wheelsets is formalized in the form of separate mathematical models and formulas, on the basis of which a method for modeling the process of increasing productivity and increasing the resistance of mills is described. Findings: Thus, a new method was created for modeling the process of shaping when machining wheel sets of locomotives with a "duplex" cutter with minimizing the cost of operating the tool and reducing the sliding angle on the arc of cutting the cutter cup with an increase in the quality and accuracy of processing, which makes it possible to significantly reduce the cost of preparing the production and increase processing efficiency.