Showing papers on "Homotopy analysis method published in 2006"
TL;DR: In this paper, a survey of recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones, is presented.
Abstract: This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.
2,135 citations
TL;DR: Wazwaz et al. as mentioned in this paper applied homotopy perturbation method to nonlinear boundary value problems and compared the result obtained by the present method with that obtained by Adomian method.
1,112 citations
TL;DR: In this paper, the homotopy analysis method (HAM) is compared with the numerical and HPM in the heat transfer file and the auxiliary parameter ℏ, which provides a simple way to adjust and control the convergence region of solution series.
643 citations
TL;DR: In this paper, homotopy perturbation method (HPM), which does not need small parameters in the equations, is compared with the perturbations and numerical methods in the heat transfer field.
496 citations
TL;DR: In this article, a guided tour through the mathematics needed for a proper understanding of homotopy perturbation method as applied to various nonlinear problems is presented, and a new interpretation of the concept of constant expansion is given.
Abstract: The present work constitutes a guided tour through the mathematics needed for a proper understanding of homotopy perturbation method as applied to various nonlinear problems. It gives a new interpretation of the concept of constant expansion in the homotopy perturbation method.
483 citations
Journal Article•
473 citations
TL;DR: In this paper, the homotopy perturbation method was used to solve an unsteady nonlinear convective-radiative equation and a nonlinear conduction equation containing two small parameters of e1 and e2.
251 citations
TL;DR: In this article, He's homotopy perturbation method (HPM) is implemented for solving the nonlinear Hirota-Satsuma coupled KdV partial differential equation.
231 citations
TL;DR: In this paper, the non-linear equations governing the flow under discussion are reduced to an ordinary differential equation with the help of homotopy analysis method (HAM), the analytical solution is obtained in the series form convergence of the series is explicitly discussed.
226 citations
TL;DR: In this paper, He's homotopy perturbation method (HPM) is implemented to solve the linear Helmholtz partial differential equation and some nonlinear fifth-order Korteweg-de Vries (FKdV) partial differential equations with specified initial conditions.
Abstract: In this article, He's homotopy perturbation method (HPM), which does not need small parameter in the equation, is implemented to solve the linear Helmholtz partial differential equation and some nonlinear fifth-order Korteweg-de Vries (FKdV) partial differential equations with specified initial conditions. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary or initial conditions after few iterations. Comparison of the results with those obtained by Adomian's decomposition method reveals that HPM is very effective, convenient and quite accurate to both linear and nonlinear problems. It is predicted that HPM can be widely applied in engineering.
221 citations
TL;DR: A homotopy perturbation method, like Adomian's decomposition method, is proposed for solving the non-singular integral equations of the first kind and the results reveal that the proposed method is very effective and simple.
TL;DR: Comparisons are made between Adomian’s decomposition method (ADM) and the exact solution and the proposed homotopy perturbation method to solve quadratic Riccati differential equation.
TL;DR: In this article, the homotopy analysis method was used to investigate the flow of a fourth grade fluid past a porous plate, and the non-linear effects on the velocity distribution were discussed.
TL;DR: In this paper, the momentum and heat transfer in a laminar liquid film on a horizontal stretching sheet is analyzed by the Homotopy analysis method (HAM), and analytic series solutions are given and compared with numerical results given by other authors.
Abstract: The momentum and heat transfer in a laminar liquid film on a horizontal stretching sheet is analyzed by the Homotopy analysis method (HAM). Analytic series solutions are given and compared with numerical results given by other authors. The good agreement between them shows the effectiveness of HAM to the problem of liquid film on an unsteady stretching surface.
TL;DR: In this paper, a two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell fluid is investigated in a channel and the walls of the channel are taken as porous.
Abstract: Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell fluid is investigated in a channel. The walls of the channel are taken as porous. Using the similarity transformations and boundary layer approximations, the nonlinear partial differential equations are reduced to an ordinary differential equation. The developed nonlinear equation is solved analytically using the homotopy analysis method. An expression for the analytic solution is derived in the form of a series. The convergence of the obtained series is shown. The effects of the Reynolds number Re, Deborah number De and Hartman number M are shown through graphs and discussed for both the suction and injection cases.
TL;DR: In this article, an iterated He's homotopy perturbation method is proposed to solve quadratic Riccati differential equation, and the results reveal that the method is very effective and simple.
TL;DR: In this article, the authors analyzed the flow of a power-law fluid film on an unsteady stretching surface by means of homotopy analysis method and compared the numerical results with the good agreement between them.
Abstract: Flow of a power-law fluid film on an unsteady stretching surface is analyzed by means of homotopy analysis method (HAM [1] ). For real power-law index and the unsteadiness parameter in wide ranges, analytic series solutions are given and compared with the numerical results. The good agreement between them shows the effectiveness of HAM to this problem. Additionally, unlike previous studies, the value of the critical unsteadiness parameter S 0 , above which no solution exists, is determined analytically in this paper.
TL;DR: In this article, the authors applied the homotopy perturbation method to obtain analytic approximations of the nonlinear equations modeling thin film flow of a fourth grade fluid falling on the outer surface of an infinitely long vertical cylinder.
TL;DR: In this article, the homotopy analysis method is applied to give series solution of the unsteady boundary-layer flows over an impermeable stretching plate, and the series solutions are convergent in the whole time region 0 ≤ r < +∞.
Abstract: An analytic technique, namely, the homotopy analysis method, is applied to give series solution of the unsteady boundary-layer flows over an impermeable stretching plate. Different from all previous perturbation solutions, our series solutions are convergent in the whole time region 0 ≤ r < +∞. To the best of our knowledge, such kind of series solution has never been reported for this problem. Besides, two kinds of new similarity transformations about dimensionless time are proposed. Using these two different similarity transformations, we obtain the same convergent solution valid in the whole time region 0 < r < +∞. Furthermore, it is shown that a nonlinear initial/boundary-value problem can be replaced by an infinite number of linear boundary-value subproblems.
TL;DR: In this article, an application of He's homotopy perturbation method is proposed to compute Laplace transform and the results reveal that the method is very effective and simple.
Abstract: In this paper, an application of He’s homotopy perturbation method is proposed to compute Laplace transform. The results reveal that the method is very effective and simple.
TL;DR: In this paper, a homotopy analysis method was applied to solve Von Karman swirling viscous flow, governed by a set of two fully coupled differential equations with strong nonlinearity.
TL;DR: In this paper, the numerical resolution of the low thrust orbital transfer problem around the Earth with the maximization of the final mass or minimization of consumption is investigated, and a homotopic method is proposed to deal with these difficulties for which convergence properties are established.
Abstract: The numerical resolution of the low thrust orbital transfer problem around the Earth with the maximization of the final mass or minimization of the consumption is investigated. This problem is difficult to solve by shooting method because the optimal control is discontinuous and a homotopic method is proposed to deal with these difficulties for which convergence properties are established. For a thrust of 0.1 Newton and a final time 50% greater than the minimum one, we obtain 1786 switching times.
TL;DR: In this article, an analytic approach based on the homotopy analysis method is proposed to solve a nonlinear model of combined convective and radiative cooling of a spherical body, and an explicit series solution is given, which agrees well with the exact or numerical solutions.
TL;DR: In this paper, a convergent series solution for the viscous flow of non-Newtonian fluids near the forward stagnation point of a two-dimensional body is obtained, which is valid for all dimensionless time in the whole spatial region 0 ≤ η ∞.
Abstract: In this paper, the unsteady viscous flow of non-Newtonian fluids near the forward stagnation point of a two-dimensional body is studied analytically. By using the homotopy analysis method, a convergent series solution is obtained, which is uniformly valid for all dimensionless time in the whole spatial region 0 ≤ η ∞ . Besides, the effects of integral power-law index of the non-Newtonian fluids on the flow are investigated. To the best of our knowledge, such kind of series solutions have never been reported for this problem.
TL;DR: In this article, a new perturbation technique coupled with the iteration method was applied to derive approximate analytical solutions, valid for small as well as large values of oscillation amplitude, for nonlinear oscillations with a single degree of freedom.
Abstract: In this paper, we apply a new perturbation technique coupled with the iteration method. This procedure is obtained by combining the iteration methods of J. H. He and Mickens into a new iteration procedure such that excellent approximate analytical solutions, valid for small as well as large values of oscillation amplitude, can be determined for nonlinear oscillations with a single degree of freedom. Four examples are given to illustrate the validity and accuracy of this procedure. We compare the approximate periods obtained by our procedure with the exact known periods. The results show that the approximations are of extreme accuracy.
TL;DR: Some efficient numerical algorithms for solving nonlinear equations based on Newton–Raphson method by using modified homotopy perturbation method are presented.
TL;DR: In this paper, the homotopy analysis method was used for the flow of a third grade fluid bounded by two parallel porous plates, and a comparison was made with the exact numerical solution for the various values of the physical parameters.
Abstract: The solution for the flow of a third grade fluid bounded by two parallel porous plates is given using homotopy analysis method (HAM). A comparison is made with the exact numerical solution for the various values of the physical parameters. It is found that a proper choice of the auxiliary parameter occurring in HAM solution gives very close results.
TL;DR: In this paper, a general linear product (GLP) polynomial system structure for homotopy path tracking is presented. But the GLP structure is intermediate between the partitioned linear product structure used by POLSYS_PLP (Algorithm 801) and the BKK-based structure used for PHCPACK.
Abstract: Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked and handling singular solutions have made probability-one homotopy methods even more practical. POLSYS_GLP consists of Fortran 95 modules for finding all isolated solutions of a complex coefficient polynomial system of equations. The package is intended to be used on a distributed memory multiprocessor in conjunction with HOMPACK90 (Algorithm 777), and makes extensive use of Fortran 95-derived data types and MPI to support a general linear product (GLP) polynomial system structure. GLP structure is intermediate between the partitioned linear product structure used by POLSYS_PLP (Algorithm 801) and the BKK-based structure used by PHCPACK. The code requires a GLP structure as input, and although finding the optimal GLP structure is a difficult combinatorial problem, generally physical or engineering intuition about a problem yields a very good GLP structure. POLSYS_GLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem definition both at a high level and via hand-crafted code. Different GLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding.
TL;DR: In this paper, the inverse static analysis of planar compliant mechanisms in polynomial form is presented, where the goal is to find the equilibrium configurations of the system in response to a known force/moment applied to the mechanism.
Abstract: This paper formulates the inverse static analysis of planar compliant mechanisms in polynomial form. The goal is to find the equilibrium configurations of the system in response to a known force/moment applied to the mechanism. The geometric constraint of the linkage defines a set of kinematics equations which are combined with equilibrium equations obtained from partial derivatives of the potential-energy function. In order to apply polynomial homotopy solver to these equations, we approximate the linear torsion spring torque at each joint by using sine and cosine functions. The results obtained from the homotopy solver are then refined using Newton-Raphson iteration. To demonstrate the analysis steps, we study two example planar compliant mechanisms, a four-bar linkage with two torsional springs, and a parallel platform supported by three linear springs. Numerical examples are provided together with plots of the potential energy during a movement between selected equilibrium positions.
TL;DR: In this paper, the unsteady boundary-layer flow of a micropolar fluid started impulsively from rest near the forward stagnation point of a two-dimensional plane surface is studied by means of an analytic approach, namely homotopy analysis method.
Abstract: In this paper, the unsteady boundary-layer flow of a micropolar fluid started impulsively from rest near the forward stagnation point of a two-dimensional plane surface is studied by means of an analytic approach, namely homotopy analysis method. This approach gives accurate approximations uniformly valid for all dimensionless time. Besides, analytic results are given for the reduced velocity and microrotation profiles, as well as for the skin friction coefficient when the material parameter K takes the value K=0 (Newtonian fluid), 1, 3, 5 and 10. To the best of our knowledge, such a kind of series solutions has been never reported.