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New Analytical Approximate Solutions of Fractional Differential Equations
Published in Santanu Saha Ray, Subhadarshan Sahoo, Generalized Fractional Order Differential Equations Arising in Physical Models, 2018
Santanu Saha Ray, Subhadarshan Sahoo
Here, the homotopy perturbation transform method [38–40] and homotopy analysis method [41–48] with modification have been applied for solving fractional coupled Klein–Gordon–Zakharov equations which play an important role in plasma physics. The homotopy perturbation transform method is a combined form of the Laplace transform method with the homotopy perturbation method. Recently, an analytical method, namely, homotopy analysis method (HAM) [41–48] has been developed and successfully applied for getting the approximate solutions for various non-linear problems in science and engineering. The HAM contains an auxiliary parameter ℏ which provides us with a simple way to adjust the convergence region and rate of the series solution. Moreover, by means of the so-called ℏ-curve, it is easy to find the valid region of ℏ to gain a convergent series solution. Thus, through the HAM, explicit analytic solutions of non-linear problems are possible to obtain. The above methods find the solution without any discretization or restrictive assumptions and avoid the round-off errors.
Lane–Emden Equations
Published in Snehashish Chakraverty, Susmita Mall, Artificial Neural Networks for Engineers and Scientists, 2017
Snehashish Chakraverty, Susmita Mall
Exact solutions of Equation 7.3 for m = 0, 1, and 5 have been obtained by Chandrasekhar [5] and Datta [6]. For m = 5, only one parameter family of solution is obtained in [7]. For other values of m, the standard Lane–Emden equations can only be solved numerically. The singularity behavior at the origin, that is, at x = 0, gives rise to a challenge to the solution of not only the Lane–Emden equations but also various other linear and nonlinear initial value problems (IVPs) in astrophysics and quantum mechanics. Different analytical approaches based on either series solutions or perturbation techniques have been used to handle the Lane–Emden equations. In this regard, the Adomian decomposition method (ADM) has been used by Wazwaz [8–10] and Shawagfeh [11] to handle the Lane–Emden equations. Chowdhury et al. [12,13] employed the homotopy perturbation method (HPM) to solve singular IVPs of time-independent equations. Liao [14] presented an algorithm based on ADM for solving Lane–Emden-type equations. An approximate solution of a differential equation arising in astrophysics using the variational iteration method has been developed by Dehghan and Shakeri [15]. The Emden–Fowler equation has been solved by utilizing the techniques of Lie and Painleve analysis proposed by Govinder and Leach [16]. An efficient analytical algorithm based on the modified homotopy analysis method has been presented by Singh et al. [17]. Muatjetjeja and Khalique in [18] provided an exact solution of the generalized Lane–Emden equations of the first and second kind. Recently, Mall and Chakraverty [19] have developed ChNN model for solving second-order singular initial value problems of Lane–Emden type.
Nonlinear dynamic behaviour of a cable with multiple intra-span elastic supports
Published in European Journal of Environmental and Civil Engineering, 2020
Wang Tian-Qi, Wang Rong-Hui, Ma Niu-Jing
Homotopy analysis method is a universal analytical technique for both weakly and strongly nonlinear problems presented by Liao (2003). He systematically dealt with homotopy analysis method in his monograph, involving its basic concepts, principle together with its difference and relationship to other methods. Hoseini, Pirbodaghi, Asghari, Farrahi, and Ahmadian (2008) derived a precise analytic solution for the nonlinear vibration of a conservative oscillator by means of homotopy analysis method, showing this technique was effective for parametric analysis. Pirbodaghi, Ahmadian, and Fesanghary (2009) adopted homotopy analysis method to analyse the nonlinear dynamics of Euler-Bernoulli beams under axial loads. Through comparison with other commonly used methods, it demonstrated this method was highly accurate and effective for a variety of vibration amplitudes as well. Hassan and El-Tawil (2012) proposed to solve second-order nonlinear differential equations via homotopy analysis method. The results demonstrated accuracy and validity of the developed technique. Ma, Wang, and Han (2015) employed homotopy analysis method to investigate primary parametric resonance-primary resonance response of stiffened plates with moving boundary conditions. Since it differs from perturbation approaches, homotopy analysis method is totally irrelevant to small parameters, thus it is suitable for most nonlinear problems. Apart from that, it also offers an easy means to guarantee the convergence of solution, so as to obtain precise enough analytical approximations. This method is also applied in this research.
A Laguerre spectral method for quadratic optimal control of nonlinear systems in a semi-infinite interval
Published in Automatika, 2020
Mojtaba Masoumnezhad, Mohammadhossein Saeedi, Haijun Yu, Hassan Saberi Nik
The homotopy analysis method is an analytical technique for solving nonlinear differential equations. The HAM [19,20] was first proposed by Liao in 1992 to solve lots of nonlinear problems. This method has been successfully applied to many nonlinear problems, such as physical models with an infinite number of singularities [21], nonlinear eigenvalue problems [22], fractional Sturm–Liouville problems [23], optimal control problems [24,25], Cahn–Hilliard initial value problem [26], semi-linear elliptic boundary value problems [27] and so on [28]. The HAM contains a certain auxiliary parameter ℏ which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution. Moreover, by means of the so-called ℏ-curve, it is easy to determine the valid regions of ℏ to gain a convergent series solution. The HAM, however, suffers from a number of restrictive measures, such as the requirement that the solution sought ought to conform to the so-called rule of solution expression and the rule of coefficient ergodicity. These HAM requirements are meant to ensure that the implementation of the method results in a series of differential equations can be solved analytically.