The homotopy analysis method is a semianalytical technique to solve nonlinear ordinarypartial differential equations. It is worth pointing out that this method presents a. Pdf a new spectralhomotopy analysis method for solving a. A fast, spectrally accurate homotopy based numerical method for. A fast, spectrally accurate homotopy based numerical. Pdf advances in the homotopy analysis method researchgate. On the piecewisespectral homotopy analysis method and its. Dehghan and salehi 21 used vim and adm to solve the delay logistic equation which has been extensively.
The spectral homotopy analysis method sham is a numerical version of the homotopy analysis method ham which has only been previously used to solve nonlinear ordinary differential equations. Pdf on the bivariate spectral homotopy analysis method. A combination of the hybrid spectral collocation technique and the homotopy analysis method is used to construct an iteration algorithm for solving a class of nonlinear optimal control problems nocps. Homotopy analysis method for nonlinear differential. These methods are used for solving deformation equation. Solving nonlinear boundary value problems using the homotopy. A new spectralhomotopy analysis method for the mhd. Examples of spectral methods a concrete, linear example. The current solutions are compared to those in refs. The spectral homotopy analysis method is applied to solve the system of differential equations 8 to 11 subject to the boundary conditions equation.
This article presents an improved spectralhomotopy analysis method isham for solving nonlinear differential equations. They used hankelpade and homotopy analysis methods for the derivation of the solutions. A note on the solution of general falknerskan problem by. On the solution of doublediffusive convective flow due to a.
The spectral homotopy analysis method extended to systems. Sep 12, 20 the spectral homotopy analysis method sham is a numerical version of the homotopy analysis method ham which has only been previously used to solve nonlinear ordinary differential equations. The spectral homotopy analysis method is extended to solutions of systems of nonlinear partial differential equations. On the solution of doublediffusive convective flow due to.
Mar 01, 2020 homotopy analysis method is a nonperturbation method that is valid for linear and nonlinear problems with or without smalllarge parameters and the method is dependent upon four factors, namely, initial approximation, convergencecontrol parameter, auxiliary function, and auxiliary linear operator while spectral homotopy analysis method is a coupling technique of the traditional homotopy analysis method and chebyshev spectral collocation method. Numerical solution of nonlinear fredholm integrodifferential. The results show that the isham converges faster and gives accurate results. Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the fast fourier transform. The advantage of this approach is that it eliminates the restriction in the standard ham of searching for prescribed solu tions that conform to the rule of. The method is a hybrid numerical scheme that combines the underlying ideas of the homotopy analysis method ham and the chebyshev spectral collocation method. Create free account to access unlimited books, fast download and ads free. Processes free fulltext numerical solutions of heat transfer for. This scheme is based upon applying the homotopy analysis method ham to decompose a nonlinear differential equation into a series of linear differential equations that can be solved using a sparse, spectrally accurate gegenbauer discretisation. The spectral quasilinearisation method sqlm was used to validate the results. An improved spectral homotopy analysis method for mhd flow in a. Yildirim 31 applied hes homotopy perturbation method to solve the cauchy reaction. Pdf spectral homotopy analysis method for pdes that. A note on the solution of general falknerskan problem by two.
Sep 01, 2010 the spectral homotopy analysis method is more efficient as it does not depend on the rule of solution expression and the rule of ergoticity unlike the standard ham approach. This paper expands the ideas of the spectral homotopy analysis method to apply them, for the first time, on nonlinear partial differential equations. Pdf composition methods in homotopy groups of spheres. An analytical solution is obtained using the homotopy analysis method ham and compared with the numerical results and those obtained using the new hybrid method. The homotopy analysis method in this section we extend the homotopy analysis method proposed by liao 15 to differential equations with fractional derivatives. Ham provides a simple way to ensure the convergence of the solution series and is able to combine with other techniques employed in nonlinear differential equations such as spectral methods, pade approximants, and so on. Spectral homotopy analysis method sham as a modification of homotopy analysis method ham is applied to obtain solution of highorder nonlinear fredholm integrodifferential problems. We demonstrate the applicability of the method by solving the problem of steady conduction in a slab and the convective fin equation with variable thermal conductivity. Some recent studies on nanofluid used the homotopy analysis method to solve the conservation equations. The homotopy analysis method is developed in 1992 by liao 18. Dec 20, 2011 the hybrid method converges rapidly and is an enhancement of the utility of the original spectral homotopy analysis method motsa et al. Application of homotopy analysis method for solving non. In this work, the modified version of the sham is used to solve a partial differential equation pde that models the problem of unsteady boundary layer flow caused by an impulsively stretching plate.
The ham was first devised in 1992 by liao shijun of shanghai jiaotong university in his phd dissertation and further modified in 1997. It however remains to be generalized and verified for more complicated nonlinear problems. By means of generalizing the traditional homotopy method, liao constructed the socalled zeroorder deformation. A modification of the homotopy analysis method based on. In thi thesi we give a urvey ofthe different series methods available to solve initial and boundary value problems. Pdf solving fractional twopoint boundary value problems. The implementation of this new technique is shown by solving the falknerskan and magnetohydrodynamic boundary layer problems. They applied the chebyshev spectral collocation differentiation matrix to define the auxiliary linear operator. The spectral homotopy analysis method extended to systems of. Falknerskan flow, mhd flow, improved spectral homotopy analysis method. These iterative methods may sometimes fail to converge or give slow convergence for strongly nonlinear problems or problems involving large parameters. Analytic algorithms for some models of nonlinear age. When combined with a construction called postnikov towers, which generalize the construction of covering spaces, these sequences can also be used in an iterative procedure to compute homotopy groups.
Numerical solution of deformation equations in homotopy. Unlike ham, these methods dont need convergence control parameter. This is enabled by utilizing a homotopy maclaurin series to deal with the nonlinearities in the system. Modeling and analysis of modern fluid problems, 2017. In this work, we demonstrate the efficiency of the newly developed spectral homotopy analysis method sham in solving nonlinear heat transfer equations. Pdf unlike other analytic techniques, the homotopy analysis method ham is independent of smalllarge physical parameters. Pdf an improved spectral homotopy analysis method for mhd. In this paper a novel hybrid spectralhomotopy analysis technique developed by motsa et al. Pdf a new spectralhomotopy analysis method for solving.
On the practical use of the spectral homotopy analysis. An improved spectral homotopy analysis method for solving boundary layer problems. Jun 22, 2011 this article presents an improved spectralhomotopy analysis method isham for solving nonlinear differential equations. Spectral sequences spectral sequences are a tool for computing homology, another algebraic invariant. An improved spectral homotopy analysis method for mhd flow. Pdf analytic solution for mhd falknerskan flow over a. It is apparently seen that ham is a very powerful and efficient technique in finding analytical solutions for wide classes of and integral equations. The results obtained are compared to numerical solutions in the literature and matlabs bvp4c solver. Pdf a combination of the hybrid spectral collocation technique and the homotopy analysis method is used to construct an iteration algorithm. The difference with the other perturbation methods is that this method is independent of smalllarge physical parameters. The slm is see 9, 38, 39, 44, 45, 50 a recent method that has been successfully utilized in solving several nonlinear problems. Pdf the spectral homotopy analysis method extended to. Pdf an improved spectral homotopy analysis method for. Spectral homotopy analysis method and its convergence for.
Particular attention is paid to the applications of spectral methods to nonlinear problems arising in fluid dynamics, quantum mechanics, weather prediction, heat conduction and other fields. In this paper, the homotopy analysis method has been successfully applied to find the solution of integral and integro differential equations. Application of homotopy analysis method for solving. Application of homotopy analysis method for solving various. This is an open access article under the cc byncnd license. Homotopy analysis method in nonlinear differential. Comparison of the differential transform method dtm and shooting results with the spectral homotopy analysis method sham approximate results for f 0 when. The idea is to write the solution of the differential equation as a sum of certain basis functions for example, as a fourier series which is a sum of sinusoids and then to choose the. Homotopy analysis method, deformation equation, spectral method, finite difference method 1 introduction. May 08, 2012 results are compared with finite difference method and spectral method. The partial sum of solution series is determined using finite difference method and spectral method. Spectral collocation methods, such as the spectral relaxation, spectral local linearization, spectral quasilinearization, successive linearization, spectral perturbation, piecewise successive linearization, and the spectral homotopy analysis. Massoudi and ramezan 6 extended the idea of rajagopal et al.
Homotopy analysis method an overview sciencedirect topics. In 1992, liao employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely homotopy analysis method 7. Application of homotopy analysis method for solving non linear dynamical system g. The existence and uniqueness of the solution and convergence of the proposed method are proved. The implementation of this new technique is shown by solving the falkner. In this paper we report on a novel method for solving systems of highly nonlinear differential equations by blending two recent seminumerical. The homotopy analysis method is based on replacing a nonlinear equation by a system of ordinary. An improved spectral homotopy analysis method for solving.
A new spectralhomotopy analysis method for solving a. Pdf spectral homotopy analysis method and its convergence for. The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems. Geophysical journal international nonlinear problems. The new approach, termed bivariate spectral homotopy analysis method bisham, is based on the use of bivariate lagrange interpolation in the socalled rule of solution expression of the ham. On the piecewise spectral homotopy analysis method and its convergence. The spectral homotopy analysis method sham was first introduced for the solution of nonlinear ordinary differential equation. A spectralhomotopy analysis method for heat transfer flow of. A fast, spectrally accurate homotopy based numerical method. Shijun liao homotopy analysis method in nonlinear differential equations monograph march 31, 2011 springer. On the practical use of the spectral homotopy analysis method.
A hybrid collocation method for solving highly nonlinear. Gharib mathematics department, college of science and information technology, zarqa university, jordan abstract. Some examples are given to approve the efficiency and the accuracy of the proposed method. In fact, the nonlinear twopoint boundary value problem tpbvp, derived from the pontryagins maximum principle pmp, is solved by spectral homotopy analysis method sham.
Dec 01, 2015 the falknerskan equations were solved recently using the homotopy analysis method, liao, the homotopy perturbation method, alizadehpahlavan and borjianboroujeni and the spectral homotopy analysis method, motsa et al. Solving nonlinear boundary value problems using the. Here we presume an understanding of basic multivariate calculus and fourier series. Homotopy analysis method we consider the following differential equations, where are nonlinear operators that the represents the whole equations, x and t are independent variables and are unknown functions respectively. Spectral sequences and higher homotopy groups of spheres. Dec 03, 2019 the main objective is to find and compare approximate solutions of these equations found using optimal q homotopy analysis method oqham, homotopy analysis transform method hatm, varitional iteration method vim and adomian decomposition method adm. We present an algorithm for constructing numerical solutions to onedimensional nonlinear, variable coefficient boundary value problems.
The above results will be applied in 2 to establish for that spectral sequence smash and composition pairings and whitehead products. Click get books and find your favorite books in the online library. We demonstrate the applicability of the method by solving the problem of steady conduction in a slab and the. The spectral homotopy analysis method was introduced by motsa et al. We illustrate the application of the method by solving a system of nonlinear differential equations that govern the problem of laminar viscous flow in a semiporous channel subject to a transverse. Numerical experiments show efficiency and performance of proposed methods. Pdf spectral homotopy analysis method for pdes that model. It is an analytical approach to get the series solution of linear and nonline arpartial differential equations.
A new spectralhomotopy analysis method for the mhd jeffery. We first use the domain truncation method to approximate the domain of the problem from 0, to 0,l, where l is chosen to be sufficiently large. The caputo fractional derivative spectral method for solving equation d1. On the piecewisespectral homotopy analysis method and its convergence.
In this paper, the nonlinear dynamical systems are solved by using the homotopy analysis method ham. Jan 06, 2012 the spectral modification of the homotopy analysis method is a new procedure that has been shown to work efficiently for fluid flow problems in bounded domains. Recently methods like adomian decomposition method adm, homotopy perturbation method hpm, homotopy analysis method ham have been used successfully to solve a variety of non linear problems 1720. Our prime example is the unstable adams spectral sequence obtained in l by a different method. Download full composition methods in homotopy groups of spheres book or read online anytime anywhere, available in pdf, epub and kindle. A spectral homotopy analysis method sham is used to find numerical solutions for the unsteady viscous flow problem due to an infinite rotating disk. Non parametric spectral analysis summary of fourierbased spectral analysis properties of fourierbased methods robust methods which require very few assumptions about the signal, hence applicable to a very large class of signals. Homotopy analysis method for nonlinear differential equations. Homotopy analysis method in nonlinear differential equations. Numerical solution of deformation equations in homotopy analysis. A relatively new numerical method called the spectral homotopy analysis method sham was used to solve the governing nonlinear differential equations.
This book presents the basic algorithms, the main theoretical results, and some applications of spectral methods. Homotopy analysis method of nonlinear 5 miller ks, ross b. In this paper, the nonlinear dynamical systems are solved by using the homotopy analysis method. The homotopy analysis method ham is an analytic approximation method for highly nonlinear problems, proposed. The sham is more flexible than ham since it allows for a wider range of linear operators and one is not restricted to using the method of higher order differential mapping for solving bvps in bounded domains, unlike the ham. Good performance, even at low signal to noise ratio. The improved spectral homotopy analysis method is a hybrid method that blends the spectral homotopy analysis method and the successive linearisation method slm. An improved spectral homotopy analysis method for mhd flow in.
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