Homotopy Analysis Method

A scheme is developed for the numerical study of the Korteweg-de Vries (KdV) and the Korteweg-de Vries Burgers (KdVB) equations with initial conditions by a homotopy approach. Numerical solutions obtained by homotopy analysis method are compared with exact solution. The comparison shows that the obtained solutions are in excellent agreement. Full text

In this paper, we consider the nonlinear boundary value problem (BVP) for the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. We present analytical solutions based on the homotopy analysis method (HAM) for various values of the relevant parameters and discuss the convergence of these solutions. We also compare our results with numerical solutions. The results provide another example of a highly nonlinear problem in which HAM is the only known analytical method that yields convergent solutions for all values of the relevant parameters.

The asymptotic behavior of the memristor at dc yields a nonlinear 𝑖-𝑢 branch relationship, i.e. the memristor equivalent at dc is a nonlinear resistor. In this way, the resulting equilibrium equation of memristive circuits in dc regime has the form of system of nonlinear algebraic equations, which may have one solution, multiple solutions or no solution at all. In this work, the phenomenon of multiple dc operating points (MOPs) in memristor circuits is studied and the conditions for the occurrence of MOPs in a simple two-memristor circuit are established. Besides, homotopy methods are applied in order to find them.

We describe, very briefly, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers. Definitions of some new concepts such as the homotopy-derivative, the convergence-control parameter and so on, are given to redescribe the method more rigorously. Some lemmas and theorems about the homotopy-derivative and the deformation equation are proved. Besides, a few open questions are discussed, and a hypothesis is put forward for future studies.

The purpose of the present paper is to introduce a method, probably for the first time, to predict the multiplicity of the solutions of nonlinear boundary value problems. This procedure can be easily applied on nonlinear ordinary differential equations with boundary conditions. This method, as will be seen, besides anticipating of multiplicity of the solutions of the nonlinear differential equations, calculates effectively the all branches of the solutions (on the condition that, there exist such solutions for the problem) analytically at the same time. In this manner, for practical use in science and engineering, this method might give new unfamiliar class of solutions which is of fundamental interest and furthermore, the proposed approach convinces to apply it on nonlinear equations by today's powerful software programs so that it does not need tedious stages of evaluation and can be used without studying the whole theory. In fact, this technique has new point of view to well-known powerful analytical method for nonlinear differential equations namely homotopy analysis method (HAM). Everyone familiar to HAM knows that the convergence-controller parameter plays important role to guarantee the convergence of the solutions of nonlinear differential equations. It is shown that the convergence-controller parameter plays a fundamental role in the prediction of multiplicity of solutions and all branches of solutions are obtained simultaneously by one initial approximation guess, one auxiliary linear operator and one auxiliary function. The validity and reliability of the method is tested by its application to some nonlinear exactly solvable differential equations which is practical in science and engineering.

This study explores the effects of heat transfer on the Williamson fluid over a porous exponentially stretching surface. The boundary layer equations of the Williamson fluid model for two dimensional flow with heat transfer are presented. Two cases of heat transfer are considered, i.e., the prescribed exponential order surface temperature (PEST) case and the prescribed exponential order heat flux (PEHF) case. The highly nonlinear partial differential equations are simplified with suitable similar and non-similar variables, and finally are solved analytically with the help of the optimal homotopy analysis method (OHAM). The optimal convergence control parameters are obtained, and the physical features of the flow parameters are analyzed through graphs and tables. The skin friction and wall temperature gradient are calculated.

This work presents HomotopyDC, a package that carries out the DC analysis of a circuit by using homotopy methods, i.e. methods that are able to find more than one DC solution. The capabilities of MAPLE have been used to their full extent in order to formulate the equilibrium equation of the circuit in a fully symbolic form, which is then used to formulate the homotopic equation. Not only are the DC solutions calculated, but also their stability is assessed. Several homotopy schemes have been implemented within the package

The nonlinear Boussinesq's equation for infiltration phenomenon in unsaturated porous media is nonlinear partial differential equation and it has been solved by using Homotopy analysis method. the solution gives height of free surface of infiltrated water mound in unsaturated porous media by using appropriate initial guess value of the solution. The solution is physically interpreted and it is concluded that the height of free surface of infiltrated water in unsaturated porous media is decreasing and graph of the solution is given by using Maple coding.

A new kind of analytic technique, namely the homotopy analysis method, is employed to give an explicit analytic solution of the Thomas-Fermi equation and the related recurrence formulae of constant coefficients. This solution can be regarded as the definition of the exact solution of the Thomas-Fermi equation.

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2, 2), Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like B(m, n) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

System of coupled second-order ordinary differential equations Boundary layer flow Boundary-value problems (BVPs) Numerical method Collocation method Similarity transformations a b s t r a c t Based on Haar wavelets an efficient numerical method is proposed for the numerical solution of system of coupled Ordinary Differential Equations (ODEs) related to the natural convection boundary layer fluid flow problems with high Prandtl number (Pr). The numerical study of these flow models is necessary as the existing literature is more focused on the flow problems with small values of Pr. In this work, the problem of natural convection which consists of coupled nonlinear ODEs is solved simultaneously. The ODEs are obtained from the Navier Stokes equations through the similarity transformations. The effects of variation of Pr on heat transfer are investigated. Performance of the Haar Wavelets Collocation Method (HWCM) is compared with the finite difference method (FDM), RungeeKutta Method (RKM), homotopy analysis method (HAM) and exact solution for the last problem. More accurate solutions are obtained by wavelets decomposition in the form of a multi-resolution analysis of the function which represents solution of the given problems. Through this analysis the solution is found on the coarse grid points and then refined towards higher accuracy by increasing the level of the Haar wavelets. Neumann's boundary conditions which are problematic for most of the numerical methods are automatically coped with. A distinctive feature of the proposed method is its simple applicability for a variety of boundary conditions. Efficiency analysis of HWCM versus RKM is performed using Timing command in Mathematica software. A brief convergence analysis of the proposed method is given. Numerical tests are performed to test the applicability, efficiency and accuracy of the method.

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many 9 Extensions 96 10 Conclusions 98 Acknowledgements 101

Effects of porous medium have been investigated on the steady flow of a third grade fluid between two stationary porous plates. The continuity and momentum equations along with modified Darcy's law are used for the development of mathematical problem. The governing nonlinear problem is solved by a homotopy analysis method. The dimensionless velocity and shear stresses at the plates are analyzed.

This paper investigates the Magnetohydrodynamic flow and heat transfer towards a stationary/moving plate with convective boundary conditions in presence of thermal radiation. The governing equations are simplified by similarity transformations which are then solved by homotopy analysis method (HAM). Effects of emerging parameters such as Prandtl number, local Grashof number, Magnetic parameter, Radiation parameter and the Biot number on the velocity field and temperature field are plotted and discussed. It is found that the temperature of the stationary flat plate is higher than the temperature of the moving flat plate. The obtained results are compared with other results obtain in previous works so that the high accuracy of the present result is apparent. Keywords MHD boundary layer flow · Inclined plate · Grashof number · Radiation · HAM

In this paper, the boundary-layer natural convection flow on a permeable vertical plate with thermal radiation and mass transfer is studied when the plate moves in its own plane. A uniform temperature with uniform species concentration at the plate is affected and the fluid is considered to be a gray, absorbing–emitting. A viscous flow model is presented using boundary-layer theory comprising the momentum, energy, and concentration equations, which is solved analytically by means of an excellent method called homotopy analysis method (HAM). First, a comparison between HAM results and those obtained by means of a higher-order numerical method, namely differential quadrature method (DQM), is done. Close agreement of two sets of results indicates the accuracy of the HAM. The velocity, temperature, and concentration distributions are displayed graphically, and a parametric study is performed in which the effect of various parameters on the skin friction, the local Nusselt number (Nn), and the local Sherwood number (Mu) are investigated.

In this paper, the homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourth-order Sturm-Liouville problems. These eigenvalues are calculated by starting the HAM algorithm with one initial guess. In this paper, it can be observed that the auxiliary parameter h, which controls the convergence of the HAM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more important qualitative difference in analysis between HAM and other methods.

In the present paper, we have examined the two-dimensional flow of Williamson fluid over a stretching sheet under the effects of nano-sized particle also described as nano Williamson fluid. The boundary layer equations of nano Williamson fluid model along with energy and nanoparticle volume fraction are presented and simplified with the help of useful transformations. Governing equations are somewhat different from the ones present in literature (reason is explained in the introduction section). The expressions for coefficients of skin friction and Nusselt number have been computed. The physical features of nondimensional Williamson parameter, Lewis number, Schmidt number and nano particle parameters (diffusivity ratio and heat capacities ratio) have been discussed by plotting the graphs of velocity, temperature and nanoparticle volume fraction.

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.

In this paper, an optimal homotopy-analysis approach is described by means of the nonlinear Blasius equation as an example. This optimal approach contains at most three convergence-control parameters and is computationally rather efficient. A new kind of averaged residual error is defined, which can be used to find the optimal convergence-control parameters much more efficiently. It is found that all optimal homotopy-analysis approaches greatly accelerate the convergence of series solution. And the optimal approaches with one or two unknown convergence-control parameters are strongly suggested. This optimal approach has general meanings and can be used to get fast convergent series solutions of different types of equations with strong nonlinearity.

The construction of C 2 Pythagorean-hodograph (PH) quintic spline curves that interpolate a sequence of points p 0 , . . . , p N and satisfy prescribed end conditions incurs a "tridiagonal" system of N quadratic equations in N complex unknowns. Albrecht and Farouki [1] invoke the homotopy method to compute all 2 N+k solutions to this system, among which there is a unique "good" PH spline that is free of undesired loops and extreme curvature variations (k ∈ {−1, 0, +1} depends on the adopted end conditions). However, the homotopy method becomes prohibitively expensive when N 10, and efficient methods to construct the "good" spline only are desirable. The use of iterative solution methods is described herein, with starting approximations derived from "ordinary" C 2 cubic splines. The system Jacobian satisfies a global Lipschitz condition in C N , yielding a simple closed-form expression of the Kantorovich condition for convergence of Newton-Raphson iterations, that can be evaluated with O(N 2 ) cost. These methods are also generalized to the case of non-uniform knots.

The homotopy analysis method (HAM) is used to develop an analytical solution for the thermal performance of a radial fin of rectangular and various convex parabolic profiles mounted on a rotating shaft and losing heat by convection to its surroundings. The convection heat transfer coefficient is assumed to be a function of both the radial coordinate and the angular speed of the shaft. Results are presented for the temperature distribution, heat transfer rate, and the fin efficiency illustrating the effect of thickness profile, the ratio of outer to inner radius, and the angular speed of the shaft. Comparison of HAM results with the direct numerical solutions shows that the analytic results produced by HAM are highly accurate over a wide range of parameters that are likely to be encountered in practice.

In this paper, approximate and/or exact analytical solutions of singular initial value problems (IVPs) of the Emden-Fowler type in the second-order ordinary differential equations (ODEs) are obtained by the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. It is shown that the solutions obtained by the Adomian decomposition method (ADM) and the homotopy-perturbation method (HPM) are only special cases of the HAM solutions.

An inversion code in Matlab is constructed to recover the parameters of Cole-Cole model from spectral induced polarization (SIP) data in a 1D earth. In a spectral induced polarization survey the impedances at various frequencies are recorded. Both induced polarization and electromagnetic coupling effects occur simultaneously in this frequency bandwidth, the latter being more and more dominate when frequency increases. We used CR1Dmode code published by as forward modelling. This code solves electromagnetic responses in the presence of complex resistivity effects in a 1D earth. A homotopy method is designed to overcome the local convergence of normal iterative methods. In addition, to further condition the inverse problem, we incorporate standard Gauss-Newton (or Quasi Newton) methods. The graphical user interfaces allow for easy data entry and the a priori model and also cable configuration. We present two synthetic examples to illustrate that the spectral parameters can be recovered from multifrequency complex resistivity data.

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.

Adomian decomposition method has been used intensively to solve nonlinear boundary and initial value problems. It has been proved to be very efficient in generating series solutions of the problem under consideration under the assumption that such series solution exits. However, very little has been done to address the mathematical foundation of the method and its error analysis.

This article studies the motion of temperature dependent plastic dynamic viscosity and thermal conductivity of steady
incompressible laminar free convective magnetohydrodynamic (MHD) Casson fluid flow over an exponentially stretching surface with suction and exponentially decaying internal heat generation. It is assumed that the natural convection is driven by buoyancy
and space dependent heat generation. The viscosity and thermal conductivity of Casson fluid is assumed to vary as a linear function
of temperature. By using suitable transformation, the governing partial differential equations corresponding to the momentum and energy equations are converted into non-linear coupled ordinary differential equations and solved by the Homotopy analysis method. A new kind of averaged residual error is adopted and used to find the optimal convergence control parameter. A parametric study is performed to illustrate the influence of Prandtl number, Casson parameter, temperature dependent viscosity,
temperature dependent thermal conductivity, Magnetic parameter and heat source parameter on the fluid velocity and temperature profiles within the boundary layer. The flow controlling parameters are found to have a profound effect on the resulting flow profiles.

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.

In this study, the homotopy analysis method is used to obtain an accurate analytical solution for fundamental non-linear natural frequency and corresponding displacement of tapered beams. Comparison between the obtained results and numerical solutions shows that the first-order approximation of present study leads to accurate solutions with a maximum relative error less than 0.5%.

In this paper a novel hybrid spectral-homotopy analysis technique developed by Motsa et al. (2009) and the homotopy analysis method (HAM) are compared through the solution of the nonlinear equation for the MHD Jeffery–Hamel problem. 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 results show that the spectral-homotopy analysis technique converges at least twice as fast as the standard homotopy analysis method.

In this study, the problem of unsteady squeezing flow between circular parallel plates is performed. The similarity transformation is applied to reduce a governing partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE) in dimensionless form. A new method, called the homotopy analytical method (HAM), is applied to obtain approximate analytical solution of this nonlinear equation. The obtained solutions, in comparison with the numerical solutions (fourth-order Runge–Kutta scheme), admit a remarkable accuracy.

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257-62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37-43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527-39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73-79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287-92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695-700] is applied to solve linear and nonlinear systems of integro-differential equations.

Based on a new kind of analytic method, namely the Homotopy analysis method, an analytic approach to solve nonlinear, chaotic system of ordinary differential equations is presented. The method is applied to Lorenz system; this system depends on the three parameters: r, b and the so-called bifurcation parameter R are real constants. Two cases are considered. The first case is when R = 20.5 which corresponds to the transition region and the second case corresponds to R = 23.5 which corresponds to the chaotic region.

In this study, an accurate analytical solution for Duffing equations with cubic and quintic nonlinearities is obtained using the Homotopy Analysis Method (HAM) and Homotopy Pade technique. Novel and accurate analytical solutions for the frequency and displacement are derived. Comparison between the obtained results and numerical solutions shows that only the first order approximation of the Homotopy Pade technique leads to accurate solution with a maximum relative error less than 0.4%.

A fundamental task in geodesy is solving systems 1 of equations. Many geodetic problems are represented as sys-2 tems of multivariate polynomials. A common problem in 3 solving such systems is improper initial starting values for 4 iterative methods, leading to convergence to solutions with 5 no physical meaning, or to convergence that requires global 6 methods. Though symbolic methods such as Groebner bases 7 or resultants have been shown to be very efficient, i.e., provid-8 ing solutions for determined systems such as 3-point prob-9 lem of 3D affine transformation, the symbolic algebra can be 10 very time consuming, even with special Computer Algebra 11 Systems (CAS). This study proposes the Linear Homotopy 12 method that can be implemented easily in high-level com-13 puter languages like C++ and Fortran that are faster than CAS 14 B. Paláncz (B) by at least two orders of magnitude. Using Mathematica, the 15 power of Homotopy is demonstrated in solving three nonlin-16 ear geodetic problems: resection, GPS positioning, and affine 17 transformation. The method enlarging the domain of conver-18 gence is found to be efficient, less sensitive to rounding of 19 numbers, and has lower complexity compared to other local 20 methods like Newton-Raphson. 21 Keywords Homotopy · Nonlinear systems of equations · 22 GPS positioning · Resection · Affine transformation 23 Journal: 190 MS: 0346 TYPESET DISK LE CP Disp.:2009/9/15 Pages: 17 Layout: Large Author Proof u n c o r r e c t e d p r o o f B. Paláncz et al. (Bates et al. 2008), HOM4PS (Lee et al. 2008), HOMPACK 47 (Watson et al. 1997), PHCpack (Verschelde 1999), and PHoM 48 (Gunji et al. 2004). Other codes are written in Maple (see, 49 e.g., Leykin and Verschelde 2004) and Matlab (Decarolis 50 et al. 2002). Recently, codes for fixed point and Newton 51 homotopy were developed in Mathematica, too, see Binous 52 (2007a,b). 53 In geodesy, several algebraic procedures have been put 54 forward for solving nonlinear systems of equations (see, e.g., 55 Awange and Grafarend 2005; Paláncz et al. 2008a). The pro-56 cedures suggested in these studies, such as Groebner bases 57 and resultant approaches, are, however, normally restricted 58 by the size of the nonlinear systems of equations involved. 59 In most cases the symbolic computations are time consum-60 ing. These symbolic computations were necessitated by the 61 failure to obtain suitable starting values for numerical itera-62 tive procedures. In situations where large systems of equa-63 tions are to be solved (e.g., affine transformation), and where 64 symbolic methods are insufficient, there exists the need to 65 investigate the suitability of other alternatives. One such alter-66 native is the linear homotopy. 67 In this study linear homotopy continuation methods are 68 introduced (see Kotsireas 2001). The definitions and basic 69 ideas are considered and illustrated. The general algorithms 70 of linear homotopy method are considered: iterated solution 71 of homotopy equations via Newton-Raphson method; and as 72 an initial value problem of a system of ordinary differential 73 equations. The efficiency of these methods are illustrated here 74 by solving polynomial equation systems in geodesy, namely 75 solution of 3D resection, GPS navigation, and 3D affine trans-76 formation problems. Our computations were carried out with 77 Mathematica and Fermat computer algebra systems on HP 78 workstation xw 4100 with XP operation system, 3 GHz P4 79 Intel processor and 1 GB RAM. The details can be found 80 in MathSource (Wolfram Research Inc.) in Paláncz (2008). 81 Although the mathematical background of the algorithms is 82 discussed in this paper, it can be also found in the literature, 83 e.g., Sommese et al. (2005). The application of the imple-84 mented Mathematica functions is illustrated in the Appendix. 85 The study is organized as follows: In Sect. 2, we pres-86 ent the definition and basic concepts of homotopy required 87 to understand the solution of nonlinear equations presented 88 in Sect. 3. For simplicity purposes, a quadratic and a third-89 degree polynomial equation are used to illustrate the 90 approach. Once the example has been made clear, Sect. 4 91 demonstrates how the technique is applied to solve three non-92 linear geodetic problems. The results are discussed in Sect. 5 93 and concluded in Appendix.

The problem of micropolar fluid flow in a channel subject to a chemical reaction is presented. The effect of small and large Peclet numbers on the temperature and concentration profiles is determined while the effects of various parameters such as the Reynolds number, the coupling parameter and the spin gradient viscosity parameter on the fluid properties are determined and shown graphically. The study uses the homotopy analysis method to find approximate analytical series solutions for the governing system of nonlinear differential equations. The analytical results are validated using the Matlab bvp4c numerical routine.

We consider the non-linear partial differential equation of time-fractional type describing the spontaneous imbibition of water by an oil-saturated rock (double phase flow through porous media). The fact that oil and water form two immiscible liquid phases and water represents preferentially wetting phase are the basic assumption of this work. The Homotopy Analysis Method is used to obtain the saturation of injected water. We obtain the graphical representation of solution using MATLAB R2007b and Microsoft Excel 2010 with different fractional order (α > 0) and the comparison is made with the solution obtained in using Adomian Decomposition Method when α = 1 including numerical values.

Consider a one-dimensional (1D) flow in a confined porous medium shown in . Before t=0, the piezometric head is a constant h I . After t=0, the piezometric head at left end raises Δh, while the piezometric head at the far right end remains unchanged.

The problem of wire coating by withdrawal from a bath of a magnetohydrodynamic Oldroyd 8-constant fluid is investigated. The fluid is electrically conducting in the presence of a uniform applied magnetic field. The obtained non-linear differential equation has been solved using homotopy analysis method. The solution is given in the form of a series. The convergence of the series is explicitly discussed. The effects of emerging non-Newtonian parameters and the Hartman number is seen. The results are presented graphically and discussed.

Many problems of physical and engineering sciences are described by singular boundary value problems (SBVPs). Due to the presence of singularity, these problems pose difficulties in obtaining their solutions, and various solution schemes have been proposed to overcome these difficulties. The present work is concerned with the application of one such recently developed method, namely optimal homotopy analysis method (OHAM), to solve SBVPs. The OHAM has certain advantages: (i) it is a general method, (ii) contains an adjustable parameter to control the convergence of solution, and (iii) for certain choices of auxiliary quantities, its working resembles with those of other similar methods.

We give an analytic solution at the 10th order of approximation for the steady-state laminar viscous #ows past a sphere in a uniform stream governed by the exact, fully non-linear Navier}Stokes equations. A new kind of analytic technique, namely the homotopy analysis method, is applied, by means of which Whitehead's paradox can be easily avoided and reasonably explained. Di!erent from all previous perturbation approximations, our analytic approximations are valid in the whole "eld of #ow, because we use the same approximations to express the #ows near and far from the sphere. Our drag coe$cient formula at the 10th order of approximation agrees better with experimental data in a region of Reynolds number R B (30, which is considerably larger than that (R B (5) of all previous theoretical ones.

The limit cycle of the van der Pol oscillator, $\ddot{x}+ \epsilon (x^2-1) \dot{x} + x =0$, is studied in the plane $(x,\dot{x})$ by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit cycle for the whole range of the parameter $\epsilon$ is obtained. These formulas generate the amplitude with an error less than 0.1%. To our knowledge, this is the first time where an analytical approximation of the amplitude of the van der Pol limit cycle, with validity from the weakly up to the strongly nonlinear regime, is given.

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of

In this paper, we investigate the accuracy of the Homotopy Analysis Method (HAM) for solving the problem of the spread of a non-fatal disease in a population. The advantage of this method is that it provides a direct scheme for solving the problem, i.e., without the need for linearization, perturbation, massive computation and any transformation. Mathematical modeling of the problem leads to a system of nonlinear ODEs. MATLAB 7 is used to carry out the computations. Graphical results are presented and discussed quantitatively to illustrate the solution.