The Experts below are selected from a list of 9294 Experts worldwide ranked by ideXlab platform
Saeid Abbasbandy - One of the best experts on this subject based on the ideXlab platform.
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The Homotopy Analysis Method and the Lienard equation
International Journal of Computer Mathematics, 2011Co-Authors: Saeid Abbasbandy, Jose-luis Lopez, Ricardo López-ruizAbstract:In this article, Lienard equations are considered. The limit cycles of these systems are studied by applying the Homotopy Analysis Method (HAM). The amplitude and frequency obtained with this Methodology are in good agreement with those calculated by computational Methods. This puts in evidence that HAM is a useful tool to solve nonlinear differential equations.
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Analytic Solution of the Sharma-Tasso-Olver Equation by Homotopy Analysis Method
Zeitschrift für Naturforschung A, 2010Co-Authors: Saeid Abbasbandy, Mahnaz Ashtiani, Esmail BabolianAbstract:An analytic technique, the Homotopy Analysis Method (HAM), is applied to obtain the kink solution of the Sharma-Tasso-Olver equation. The Homotopy Analysis Method is one of the analytic Methods and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter h which gives us a simple way to adjust and control the convergence region of series solution. “Due to this reason, it seems reasonable to rename h the convergence-control parameter” [1].
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Homotopy Analysis Method for the Kawahara equation
Nonlinear Analysis: Real World Applications, 2010Co-Authors: Saeid AbbasbandyAbstract:Abstract The Homotopy Analysis Method (HAM) is used to find a family of travelling-wave solutions of the Kawahara equation. This approximate solution, which is obtained as a series of exponentials, has a reasonable residual error. The Homotopy Analysis Method contains the auxiliary parameter ħ , which provides us with a simple way to adjust and control the convergence region of series solution. This Method is reliable and manageable.
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The Homotopy Analysis Method and the Lienard equation
arXiv: Pattern Formation and Solitons, 2008Co-Authors: Saeid Abbasbandy, Jose-luis Lopez, Ricardo López-ruizAbstract:In this work, Lienard equations are considered. The limit cycles of these systems are studied by applying the Homotopy Analysis Method. The amplitude and frequency obtained with this Methodology are in good agreement with those calculated by computational Methods. This puts in evidence that the Homotopy Analysis Method is an useful tool to solve nonlinear differential equations.
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Homotopy Analysis Method for quadratic Riccati differential equation
Communications in Nonlinear Science and Numerical Simulation, 2008Co-Authors: Y. Tan, Saeid AbbasbandyAbstract:Abstract In this paper, the quadratic Riccati differential equation is solved by means of an analytic technique, namely the Homotopy Analysis Method (HAM). Comparisons are made between Adomian’s decomposition Method (ADM), Homotopy perturbation Method (HPM) and the exact solution and the Homotopy Analysis Method. The results reveal that the proposed Method is very effective and simple.
Shijun Liao - One of the best experts on this subject based on the ideXlab platform.
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advances in the Homotopy Analysis Method
2014Co-Authors: Shijun LiaoAbstract:A Short Review of Homotopy Analysis Method: Change and Challenge Predictor Homotopy Analysis Method Spectral Homotopy Analysis Method Stability of Auxiliary Linear Operator and Convergence - Control Parameter On the Convergence of the Homotopy Analysis Method Homotopy Analysis Method for Some Boundary Layer Flows of Nanofluids Homotopy Analysis Method for Fractional Swift - Hohenberg Equation Homotopy Analysis Method-based Package NOPH for Periodic Oscillations Homotopy Analysis Method-based Package BVPh 2.0 for Nonlinear BVPs.
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Advances in the Homotopy Analysis Method - Advances in the Homotopy Analysis Method
2013Co-Authors: Shijun LiaoAbstract:A Short Review of Homotopy Analysis Method: Change and Challenge Predictor Homotopy Analysis Method Spectral Homotopy Analysis Method Stability of Auxiliary Linear Operator and Convergence - Control Parameter On the Convergence of the Homotopy Analysis Method Homotopy Analysis Method for Some Boundary Layer Flows of Nanofluids Homotopy Analysis Method for Fractional Swift - Hohenberg Equation Homotopy Analysis Method-based Package NOPH for Periodic Oscillations Homotopy Analysis Method-based Package BVPh 2.0 for Nonlinear BVPs.
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Optimal Homotopy Analysis Method
Homotopy Analysis Method in Nonlinear Differential Equations, 2012Co-Authors: Shijun LiaoAbstract:In this chapter, we describe and compare the different optimal approaches of the Homotopy Analysis Method (HAM). A generalized optimal HAM is proposed, which logically contains the basic optimal HAM with only one convergence-control parameter and also the optimal HAM with an infinite number of parameters. It is found that approximations given by the optimal HAMs converge fast in general. Especially, the basic optimal HAM mostly gives good enough approximations. Thus, the optimal HAMs with a couple of convergence-control parameters are strongly suggested in practice.
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On the relationship between the Homotopy Analysis Method and Euler transform
Communications in Nonlinear Science and Numerical Simulation, 2010Co-Authors: Shijun LiaoAbstract:A new transform, namely the Homotopy transform, is defined for the first time. Then, it is proved that the famous Euler transform is only a special case of the so-called Homotopy transform which depends upon one non-zero auxiliary parameterand two convergent series P þ1 k¼1a1;k ¼ 1 and Pþ1 k¼1b1;k ¼ 1. In the frame of the Homotopy Analysis Method, a gen- eral analytic approach for highly nonlinear differential equations, the so-called Homotopy transform is obtained by means of a simple example. This fact indicates that the famous Euler transform is equivalent to the Homotopy Analysis Method in some special cases. On one side, this explains why the convergence of the series solution given by the homot- opy Analysis Method can be guaranteed. On the other side, it also shows that the Homotopy Analysis Method is more general and thus more powerful than the Euler transform.
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Analysis of nonlinear fractional partial differential equations with the Homotopy Analysis Method
Communications in Nonlinear Science and Numerical Simulation, 2009Co-Authors: Shijun Liao, Xiangcheng YouAbstract:Abstract In this paper, the time fractional partial differential equations are investigated by means of the Homotopy Analysis Method. This technique is extended to study the partial differential equations of fractal order for the first time. The accurate series solutions are obtained. This indicates the validity and great potential of the Homotopy Analysis Method for solving nonlinear fractional partial differential equations.
Ishak Hashim - One of the best experts on this subject based on the ideXlab platform.
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on convergence of Homotopy Analysis Method and its application to fractional integro differential equations
Quaestiones Mathematicae, 2013Co-Authors: S. Abbasbandy, M S Hashemi, Ishak HashimAbstract:In this paper, we have used the Homotopy Analysis Method (HAM) to obtain approximate solution of fractional integro-differential equations (FIDEs). Convergence of HAM is considered for this kind of equations. Also some examples are given to illustrate the high efficiency and precision of HAM. Keywords: Fractional integro-differential equation, Homotopy Analysis Method, convergence control parameter Quaestiones Mathematicae 36(2013), 93–105
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On a new reliable modification of Homotopy Analysis Method
Communications in Nonlinear Science and Numerical Simulation, 2009Co-Authors: A. Sami Bataineh, Mohd. Salmi Md. Noorani, Ishak HashimAbstract:Abstract In this paper, a new modification of the Homotopy Analysis Method (HAM) is presented and applied to homogeneous or non-homogeneous differential equations with constant or variable coefficients. A comparative study between the new modified Homotopy Analysis Method (MHAM) and the classical HAM is conducted. The main advantage of MHAM is that one can avoid the uncontrollability problems of the non-zero endpoint conditions encountered in the traditional HAM. Several illustrative examples are given to demonstrate the effectiveness and reliability of MHAM.
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Homotopy Analysis Method FOR FRACTIONAL IVPS
Communications in Nonlinear Science and Numerical Simulation, 2009Co-Authors: Ishak Hashim, O. Abdulaziz, Shaher MomaniAbstract:Abstract In this paper, the Homotopy Analysis Method is applied to solve linear and nonlinear fractional initial-value problems (fIVPs). The fractional derivatives are described by Caputo’s sense. Exact and/or approximate analytical solutions of the fIVPs are obtained. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the approach.
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solving systems of odes by Homotopy Analysis Method
Communications in Nonlinear Science and Numerical Simulation, 2008Co-Authors: Sami A Bataineh, Mohd. Salmi Md. Noorani, Ishak HashimAbstract:Abstract This paper applies the Homotopy Analysis Method (HAM) to systems of ordinary differential equations (ODEs). The systems investigated include stiff systems, the chaotic Genesio system and the matrix Riccati-type differential equation. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge–Kutta Method (RK78).
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the Homotopy Analysis Method for cauchy reaction diffusion problems
Physics Letters A, 2008Co-Authors: Sami A Bataineh, Mohd. Salmi Md. Noorani, Ishak HashimAbstract:Abstract In this Letter, the Homotopy Analysis Method (HAM) is employed to obtain a family of series solutions of the time-dependent reaction–diffusion problems. HAM provides a convenient way of controlling the convergence region and rate of the series solution.
Xiangcheng You - One of the best experts on this subject based on the ideXlab platform.
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Analysis of nonlinear fractional partial differential equations with the Homotopy Analysis Method
Communications in Nonlinear Science and Numerical Simulation, 2009Co-Authors: Shijun Liao, Xiangcheng YouAbstract:Abstract In this paper, the time fractional partial differential equations are investigated by means of the Homotopy Analysis Method. This technique is extended to study the partial differential equations of fractal order for the first time. The accurate series solutions are obtained. This indicates the validity and great potential of the Homotopy Analysis Method for solving nonlinear fractional partial differential equations.
Mohd. Salmi Md. Noorani - One of the best experts on this subject based on the ideXlab platform.
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Homotopy Analysis Method for solving fractional Lorenz system
Communications in Nonlinear Science and Numerical Simulation, 2010Co-Authors: A. K. Alomari, Mohd. Salmi Md. Noorani, Roslinda Mohd. NazarAbstract:Abstract In this paper, a new reliable algorithm called the step Homotopy Analysis Method (SHAM) based on an adaptation of the standard Homotopy-Analysis Method (HAM) is presented to solve the fractional Lorenz system. This modified Method yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. The accuracy of the present solution is found to be in excellent agreement with previously published solution.
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On a new reliable modification of Homotopy Analysis Method
Communications in Nonlinear Science and Numerical Simulation, 2009Co-Authors: A. Sami Bataineh, Mohd. Salmi Md. Noorani, Ishak HashimAbstract:Abstract In this paper, a new modification of the Homotopy Analysis Method (HAM) is presented and applied to homogeneous or non-homogeneous differential equations with constant or variable coefficients. A comparative study between the new modified Homotopy Analysis Method (MHAM) and the classical HAM is conducted. The main advantage of MHAM is that one can avoid the uncontrollability problems of the non-zero endpoint conditions encountered in the traditional HAM. Several illustrative examples are given to demonstrate the effectiveness and reliability of MHAM.
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solving systems of odes by Homotopy Analysis Method
Communications in Nonlinear Science and Numerical Simulation, 2008Co-Authors: Sami A Bataineh, Mohd. Salmi Md. Noorani, Ishak HashimAbstract:Abstract This paper applies the Homotopy Analysis Method (HAM) to systems of ordinary differential equations (ODEs). The systems investigated include stiff systems, the chaotic Genesio system and the matrix Riccati-type differential equation. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge–Kutta Method (RK78).
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Solution of Delay Differential Equation by Means of Homotopy Analysis Method
Acta Applicandae Mathematicae, 2008Co-Authors: A. K. Alomari, Mohd. Salmi Md. Noorani, Roslinda Mohd. NazarAbstract:The algorithm of approximate analytical solution for delay differential equations (DDE) is obtained via Homotopy Analysis Method (HAM) and modified Homotopy Analysis Method (MHAM). Various examples of linear, nonlinear and system of initial value problems of DDE are solved and the results obtained show that these algorithms are accurate and efficient for the DDE. The convergence of this algorithm is also proved.
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the Homotopy Analysis Method for cauchy reaction diffusion problems
Physics Letters A, 2008Co-Authors: Sami A Bataineh, Mohd. Salmi Md. Noorani, Ishak HashimAbstract:Abstract In this Letter, the Homotopy Analysis Method (HAM) is employed to obtain a family of series solutions of the time-dependent reaction–diffusion problems. HAM provides a convenient way of controlling the convergence region and rate of the series solution.