Authors: D.Yu. Sirota, M.A. Babushkin

Title of the article: The solving of some inverse problems for ordinary differential equation in engineering applications

Year: 2018, Issue: 2, Pages: 65-74

Branch of knowledge: MINING AND GEOTECHNICAL ENGINEERING

Index UDK: 622.822.2: 622.271: 519.6

DOI: 10.26730/2618-7434-2018-2-65-74

Abstract: Ordinary differential equations (ODE) are one of the fundamental methods of physical and mathematical modeling for various engineering problems. For example: the Newton's second law in mechanics, the Guldberg's law of masses action in chemistry, the Förster's law of the population growth in ecology, the Newton's law of cooling in thermodynamics. The result of such modeling (solution a direct problem) is a function that satisfies both the ODE itself and additional initial-boundary conditions. In applied problems it is required to solve inverse problems on definition of various elements for ODE provided that discrete values of the specified function are already known as result of observations and full-scale measurements. In this paper let us introduce three inverse problems of determination the initial and boundary values for ordinary differential equation by the known experimental values. This problem let us solve by the A.N. Tikhonov’s regularization method which is to find the minimum of the modified residual functional. Let us find this minimum by two zero-order sub methods: the Hook-Jives pattern search method and «simulated annealing» method. We will compare these methods with each other and with the analytical exact solution. The aim of the using these methods are: don’t calculate the Frechet gradient of the Tikhonov functional and effective movement on the ravine bottom.

Key words: The differential equation the inverse initial-boundary value problem the AN Tikhonov regularization method simulated annealing method

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