Showing papers on "Homotopy analysis method published in 2005"
TL;DR: In this article, the homotopy perturbation method is applied to the search for traveling wave solutions of nonlinear wave equations and some examples are given to illustrate the determination of the periodic solutions or the bifurcation curves of the nonlinear Wave equations.
Abstract: The homotopy perturbation method is applied to the search for traveling wave solutions of nonlinear wave equations. Some examples are given to illustrate the determination of the periodic solutions or the bifurcation curves of the nonlinear wave equations.
1,202 citations
TL;DR: In this article, a simple homotopy is constructed by the modified Lindstedt-Poincare method, by the solution and the coefficient of linear term are expanded into series of the embedding parameter.
Abstract: A simple homotopy is constructed, by the modified Lindstedt-Poincare method(He,J.H. International Journal of Non-Linear Mechanics , 37, 2002, 309-314 ), the solution and the coefficient of linear term are expanded into series of the embedding parameter. Only one iteration leads to accurate solution.
907 citations
TL;DR: In this paper, the boundary-layer flows over a stretched impermeable wall are solved by means of an analytic technique, namely the homotopy analysis method, and two branches of solutions are found.
450 citations
TL;DR: It is shown that the so-called ''homotopy perturbation method'' is only a special case of the homotopy analysis method, which contains the auxiliary parameter @?
419 citations
TL;DR: In this paper, He's homotopy perturbation method is implemented to the model and the solution is obtained, the results reveal that the method is very effective and convenient.
Abstract: In this paper, He's Homotopy Perturbation Method is proposed for solving Volterra's Integro-differential Equation. The Volterra's population model is converted to a nonlinear ordinary differential equation and the solution of which is then approximated by using the differential transform method. He's homotopy perturbation method is implemented to the model and the solution is obtained. The results reveal that the method is very effective and convenient.
290 citations
TL;DR: In this article, a modified model of second-grade fluid that has shear-dependent viscosity and can predict the normal stress difference is used, and the differential equations governing the flow are solved using homotopy analysis method.
Abstract: The flow of a second-grade fluid past a porous plate subject to either suction or blowing at the plate has been studied. A modified model of second-grade fluid that has shear-dependent viscosity and can predict the normal stress difference is used. The differential equations governing the flow are solved using homotopy analysis method (HAM). Expressions for the velocity have been constructed and discussed with the help of graphs. Analysis of the obtained results showed that the flow is appreciably influenced by the material and normal stress coefficient. Several results of interest are deduced as the particular cases of the presented analysis.
176 citations
TL;DR: In this article, the effects of the integral power-law index ( n = 1, 2, 3) of the non-Newtonian fluids and the magnetic parameter M = 0, 1, 2 on the flows are investigated.
Abstract: In this paper, the unsteady magnetohydrodynamic viscous flows of non-Newtonian fluids caused by an impulsively stretching plate are studied by means of an analytic technique, namely the homotopy analysis method We give the analytic series solutions which are accurate and uniformly valid for all dimensionless time in the whole spatial region 0 ≤ η ∞ To the best of authors’ knowledge, such kind of analytic solutions have been never reported Besides, the effects of the integral power-law index ( n = 1 , 2, 3) of the non-Newtonian fluids and the magnetic parameter M = 0 , 1, 2 on the flows are investigated
156 citations
TL;DR: In this paper, a totally analytic solution of the nonhomogeneous Blasius problem is obtained using the homotopy analysis method (HAM), which converges for 0 = l"c.
116 citations
TL;DR: In this article, an analytic method, namely the homotopy analysis method (HAM), is applied to solve solitary waves governed by the Camassa-Holm equation, and purely analytic solutions are given for soliton waves with and without continuity at crest.
Abstract: An analytic method, namely the homotopy analysis method (HAM), is applied to solve solitary waves governed by Camassa–Holm equation. Purely analytic solutions are given for soliton waves with and without continuity at crest. This provides with a new analytic approach to solve soliton waves with discontinuity. 2005 Elsevier Ltd. All rights reserved.
115 citations
TL;DR: Based on a new kind of analytic method, namely the homotopy analysis method, an analytic approach to solve multiple solutions of strongly nonlinear problems is described by using Gelfand equation as an example.
111 citations
TL;DR: In this paper, a homotopy analysis method was applied to solve the Vakhnenko equation, a nonlinear equation with loop soliton solutions governing the propagation of high-frequency waves in a relaxing medium.
Abstract: A powerful, easy-to-use analytic technique for nonlinear problems, namely the Homotopy analysis method, is applied to solve the Vakhnenko equation, a nonlinear equation with loop soliton solutions governing the propagation of high-frequency waves in a relaxing medium. By means of the transformation of independent variables, an analysis one-loop soliton solution expressed by a series of exponential functions is obtained, which agrees well with the exact solution. This indicates the validity and great potential of the Homotopy analysis method in solving complicated solitary wave problems.
TL;DR: In this paper, the homotopy perturbation method is adopted for solving a complex-valued second-order strongly nonlinear differential equation, and the solution is compared with the exact one and shows good agreement.
TL;DR: In this paper, the homotopy analysis method was used to solve the unsteady mixed convection flow near the stagnation point on a heated vertical flat plate embedded in a Darcian fluid-saturated porous medium.
Abstract: In this paper, we solve the unsteady mixed convection flow near the stagnation point on a heated vertical flat plate embedded in a Darcian fluid-saturated porous medium by means of an analytic technique, namely the Homotopy Analysis Method. Different from previous perturbation results, our analytic series solutions are accurate and uniformly valid for all dimensionless times and for all possible values of mixed convection parameter, and besides agree well with numerical results. This provides us with a new analytic approach to investigate related unsteady problems.
TL;DR: In this article, the effects of slip at the wall when an Oldroyd 6-constant fluid is considered in a channel were examined and three non-linear problems were solved using homotopy analysis method (HAM).
Abstract: The assumption that a fluid adheres to a solid boundary (‘no-slip’ boundary condition) is one of the central tenets of the Navier-Stokes theory. However, there are situations wherein this assumption does not hold. In this communication we examine the effects of slip at the wall when an Oldroyd 6-constant fluid is considered in a channel. The slip assumed depends on the shear stress at the wall. The three non-linear problems are solved using homotopy analysis method (HAM). The results for the velocity profiles are presented and discussed.
01 Dec 2005
TL;DR: A new method for global optimization, the Homotopy Optimization Method (HOM), which differs from previous homotopy and continuation methods in that its aim is to find a minimizer for each of a set of values of thehomotopy parameter, rather than to follow a path of minimizers.
Abstract: We define a new method for global optimization, the Homotopy Optimization Method (HOM). This method differs from previous homotopy and continuation methods in that its aim is to find a minimizer for each of a set of values of the homotopy parameter, rather than to follow a path of minimizers. We define a second method, called HOPE, by allowing HOM to follow an ensemble of points obtained by perturbation of previous ones. We relate this new method to standard methods such as simulated annealing and show under what circumstances it is superior. We present results of extensive numerical experiments demonstrating performance of HOM and HOPE.
01 Jan 2005
TL;DR: In this article, calculus-based formulas for the continuous Euler and homotopy operators are introduced for symbolically computing local conservation laws of nonlinear systems of partial differential equations in multi-dimensions.
Abstract: We introduce calculus-based formulas for the continuous Euler and homotopy operators. The 1D continuous homotopy operator automates integration by parts on the jet space. Its 3D generalization allows one to invert the total divergence operator. As a practical application, we show how the operators can be used to symbolically compute local conservation laws of nonlinear systems of partial differential equations in multi-dimensions.
TL;DR: In this article, an analytic technique, namely the homotopy analysis method, is applied to study the flow and heat transfer characteristics in an electrically conducting fluid near an isothermal sheet.
27 May 2005
TL;DR: In this article, a numerical continuation for tracing the double bounded homotopy (DBH) for obtaining DC solutions of nonlinear circuits is proposed, which is used to find multiple DC solutions with the advantage of having a stop criterion which is based on the property of a double bounded trajectory.
Abstract: A numerical continuation for tracing the double bounded homotopy (DBH) for obtaining DC solutions of nonlinear circuits is proposed. The double bounded homotopy is used to find multiple DC solutions with the advantage of having a stop criterion which is based on the property of having a double bounded trajectory. The key aspects of the implementation of the numerical continuation are presented in this paper. Besides, in order to trace and apply the stop criterion some blocks of the numerical continuation are modified and explained.
TL;DR: In this paper, a general homotopy inversion strategy is proposed for the continuous inverse model to overcome the defects of local convergence of conventional methods, and a practical algorithm is constructed by introducing Tikhonov regularization method for the nonlinear operator equation which is obtained from the discretization of the SINR model.
Abstract: This article considers the problem of estimating the velocity in a two-dimensional acoustic wave equation. In order to overcome the defects of local convergence of conventional methods, a general homotopy inversion strategy is proposed for the continuous inverse model. On the basis of this strategy, a practical algorithm is constructed by introducing Tikhonov regularization method for the nonlinear operator equation which is obtained from the discretization of the continuous inverse model. Numerical simulations show that the algorithm is a fast and widely convergent inversion method.
TL;DR: In this article, an analytic technique, namely the Homotopy Analysis Method (HAM), is applied to solve the nonlinear mKdV equation, which shows the validity of the HAM for nonlinear periodic wave problems.
23 May 2005
TL;DR: A homotopy method for obtaining DC solutions of nonlinear circuits is proposed which presents a novel stop criterion which is based on the property of tracing a double bounded trajectory.
Abstract: A homotopy method for obtaining DC solutions of nonlinear circuits is proposed. The homotopy method is called double-bounded homotopy and it is used to find multiple DC solutions. This method presents a novel stop criterion which is based on the property of tracing a double bounded trajectory. The main properties of the homotopy are explained by using the Lambert-W function.
TL;DR: The proposed homotopy method utilizes the nonlinear function closely related to circuit equations to be solved, so that it efficiently finds DC operating points of practical transistor circuits.
Abstract: Finding DC operating points of transistor circuits is a very important and difficult task. The Newton-Raphson method employed in SPICE-like simulators often fails to converge to a solution. To overcome this convergence problem, homotopy methods have been studied from various viewpoints. For efficiency of homotopy methods, it is important to construct an appropriate homotopy function. In conventional homotopy methods, linear auxiliary functions have been commonly used. In this paper, a homotopy method for solving transistor circuits using a nonlinear auxiliary function is proposed. The proposed method utilizes the nonlinear function closely related to circuit equations to be solved, so that it efficiently finds DC operating points of practical transistor circuits. Numerical examples show that the proposed method is several times more efficient than conventional three homotopy methods.
TL;DR: An analytic technique, namely the homotopy analysis method, is applied to solve the nonlinear travelling waves governed by the Klein-Gordon equation.
TL;DR: In this paper, the steady flow of a fluid, called a fourth grade fluid, between two parallel plates is considered, and depending upon the relative motion of the plates, the nonlinear differential equation describing the velocity field is solved using perturbation technique and homotopy analysis method.
Abstract: The steady flow of a fluid, called a fourth grade fluid, between two parallel plates is considered. Depending upon the relative motion of the plates we analyze four types of flows: Couette flow, plug flow, Poiseuille flow and generalized Couette flow. In each case, the nonlinear differential equation describing the velocity field is solved using perturbation technique and homotopy analysis method. The pressure distribution is also found. It is observed that the homotopy analysis method is more efficient and flexible than the perturbation technique.
TL;DR: The steady flow of a rotating third grade fluid past a porous plate has been analyzed and the resulting nonlinear boundary value problem has been solved using homotopy analysis method.
TL;DR: This diagonal homotopy method to compute a numerical representation of all positive dimensional components in the intersection of two irreducible algebraic sets is rewritten in intrinsic coordinates, which reduces the number of variables, typically in half, to save a significant amount of computation.
24 Jul 2005
TL;DR: A hybrid symbolic-numeric differential-elimination method for identifying and including missing constraints arising in differential systems is constructed, exploiting the fact that a system once differentiated becomes linear in its highest derivatives.
Abstract: Two ideas are combined to construct a hybrid symbolic-numeric differential-elimination method for identifying and including missing constraints arising in differential systems. First we exploit the fact that a system once differentiated becomes linear in its highest derivatives. Then we apply diagonal homotopies to incrementally process new constraints, one at a time. The method is illustrated on several examples, combining symbolic differential elimination (using rifsimp) with numerical homotopy continuation (using phc).
TL;DR: A modified formula, iteration representation, of ancient Chinese algorithm by combining the homotopy continuation technique is presented and the new iteration formula and the technique for the choice of the auxiliaryhomotopy function are developed.
TL;DR: Wang et al. as mentioned in this paper combined the Tikhonov regularization method for ill-posed problems with the widely convergent homotopy method applied to the inversion process of operator identification.
Abstract: Combining the Tikhonov regularization method for ill-posed problems with the widely convergent homotopy method applied to the inversion process of operator identification, a new strategy−regularization-homotopy method is proposed for the inversion of 2-D wave equation, which is suitable for the nonlinear, ill-posed and multi-extremum seismic inverse problem. In order to restrain noises and improve the quality of inversion, a well log constraint regularization-homotopy method is further constructed. Lots of numerical simulations indicate effectiveness of our methods.