A hybrid Harris hawks-Nelder-Mead optimization for practical nonlinear ordinary differential equations
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RESEARCH PAPER
A hybrid Harris hawks‑Nelder‑Mead optimization for practical nonlinear ordinary differential equations Rizk M. Rizk‑Allah1,3 · Aboul Ella Hassanien2,3 Received: 9 April 2020 / Revised: 17 September 2020 / Accepted: 19 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Differential equations can often be seen in many fields of scientific research and engineering. Typically, finding the analytical (exact) solution is expensive task in terms of computational effort and may not an attainable task for some complex tasks. To effectively handle a wide variety of linear and nonlinear differential equations, this paper presents an approximate methodology based on hybrid Harris hawks–Nelder–Mead optimization algorithm with the aim to achieve accurate and reliable solution. The proposed methodology is introduced on basis of Fourier series expansion and Harris hawks–Nelder–Mead optimization algorithm. In this sense, the differential equation is represented as an optimization model by the means of the weighted residual function (cost function) that needed to be minimized, where the boundary and initial conditions of the differential equation are considered as the constraints of the optimization model. The practicality and efficiency of the proposed algorithm are demonstrated through six differential equations with different nature as well as four mechanical engineering differential equations. The comparison against different algorithms, by using the generational distance metric and Wilcoxon sign rank test, showed the effectiveness of the proposed algorithm. Keywords Harris hawks optimization · Nelder–Mead optimization · Analytical and approximate solutions · Residual function · Fourier series
1 Introduction The differential equations (DEs) often appear in many disciplines including engineering, science, physics, economics, and other areas [1]. The DEs can be classified into two categories: ordinary DEs (ODEs) and partial DEs (PDEs). Some prominent examples of the DEs are the Maxwell’s equations in electromagnetism, the Newton’s Second Law in dynamics, the heat equation in thermodynamics, Schrödinger equation in quantum mechanics, Einstein’s field equation in general Rizk M. Rizk-Allah, Aboul Ella Hassanien: Scientific Research Group in Egypt http://www.egyptscience.net. * Aboul Ella Hassanien [email protected]; [email protected] 1
Faculty of Engineering, Menoufia University, Shebin El‑Kom, Egypt
2
Faculty of Computers and Information, Cairo University, Giza, Egypt
3
Scientific Research Group in Egypt (SRGE)
http://www.egyptscience.net
relativity, the Navier–Stokes equations in fluid dynamics [2] and other disciplines [3, 4]. Analytical methods are often inefficient in handling the ODEs due to their dependence on the orders, forms and the given conditions. In order to avoid these limitations, the numerical approximation methods have been introduced to obtain the approximate solutions of the ODEs. Some of the established methods have been proposed incl
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