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Application of HAM for Nonlinear Integro-differential equations of Order Two

Journal of Applied Mathematics and Physics 2023年第1期11卷 55-68页

作者： Zainidin Eshkuvatov Davron Khayrullaev Muzaffar Nurillaev Shalela Mohd Mahali Anvar Narzullaev Faculty of Ocean Engineering Technology and Informatics University Malaysia Terengganu (UMT) Terengganu Malaysia Faculty of Applied Mathematics and Intellectual Technologies National University of Uzbekistan Tashkent Uzbekistan Faculty of Physics and Mathematics Tashkent State Pedagogical University (TSPU) Tashkent Uzbekistan Faculty of Science and Technology University Sains Islam Malaysia (USIM) Bandar Baru Nilai Malaysia

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable;therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type;then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

关键词： integral-differential equations Homotopy Analyses Method Iterative System Algebraic equations Gauss-Legendre Quadrature Formula

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Mathematical Modeling Algorithms for Creating New Materials with Desired Properties Using Nano-Hierarchical Structures

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Advances in Materials Physics and Chemistry 2019年第11期9卷 211-217页

作者： Olga Hachay Andrey Khachay Oleg Khachay Department Name of Organization Institute of GeophysicsUral Branch of Russian Academy of SciencesEkaterinburgRussia Institute of Natural Sciences and Mathematics Ural Federal UniversityEkaterinburgRussia

In the enormous and still poorly mastered gap between the macro level, where well developed continuum theories of continuous media and engineering methods of calculation and design operate, and atomic, subordinate to the laws of quantum mechanics, there is an extensive meso-hierarchical level of the structure of matter. At this level unprecedented previously products and technologies can be artificially created. Nano technology is a qualitatively new strategy in technology: it creates objects in exactly the opposite way—large objects are created from small ones [1]. We have developed a new method for modeling acoustic monitoring of a layered-block elastic medium with several inclusions of various physical and mechanical hierarchical structures [2]. An iterative process is developed for solving the direct problem for the case of three hierarchical inclusions of l, m, s-th ranks based on the use of 2D integro-differential equations. The degree of hierarchy of inclusions is determined by the values of their ranks, which may be different, while the first rank is associated with the atomic structure, the following ranks are associated with increasing geometric sizes, which contain inclusions of lower ranks and sizes. Hierarchical inclusions are located in different layers one above the other: the upper one is abnormally plastic, the second is abnormally elastic and the third is abnormally dense. The degree of filling with inclusions of each rank for all three hierarchical inclusions is different. Modeling is carried out from smaller sizes to large inclusions;as a result, it becomes possible to determine the necessary parameters of the formed material from acoustic monitoring data.

关键词： Nano Hierarchic Objects Algorithms of Modeling integral-differential equations Iterative Process Block Layered Medium Combined Hiererachical

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