Laplace transform for the solution of higher order deformation equations arising in the homotopy analysis method

A new analytic approach for solving nonlinear ordinary differential equations with initial conditions is proposed. First, the homotopy analysis method is used to transform a nonlinear differential equation into a system of linear differential equations; then, the Laplace transform method is applied to solve the resulting linear initial value problems; finally, the solutions to the linear initial value problems are employed to form a convergent series solution to the given problem. The main advantage of the new approach is that it provides an effective way to solve the higher order deformation equations arising in the homotopy analysis method.     

The explicit series solution of SIR and SIS epidemic models

In this paper the SIR and SIS epidemic models in biology are solved by means of an analytic technique for nonlinear problems, namely the homotopy analysis method (HAM). Both of the SIR and SIS models are described by coupled nonlinear differential equations. A one-parameter family of explicit series solutions are obtained for both models. This parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models. Our analytic results agree well with the numerical ones. This analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology.     

An analytical solution for a nonlinear time-delay model in biology

In this paper, the homotopy analysis method is applied to develop a analytic approach for nonlinear differential equations with time-delay. A nonlinear model in biology is used as an example to show the basic ideas of this analytic approach. Different from other analytic techniques, the homotopy analysis method provides a simple way to ensure the convergence of the solution series, so that one can always get accurate approximations. A new discontinuous function is defined so as to express the piecewise continuous solutions of time-delay differential equations in a way convenient for symbolic computations. It is found that the time-delay has a great influence on the solution of the time-delay nonlinear differential equation. This approach has general meanings and can be applied to solve other nonlinear problems with time-delay.     

On the practical use of the spectral homotopy analysis method and local linearisation method for unsteady boundary-layer flows caused by an impulsively stretching plate

This paper expands the ideas of the spectral homotopy analysis method to apply them, for the first time, on non-linear partial differential equations. The spectral homotopy analysis method (SHAM) is a numerical version of the homotopy analysis method (HAM) which has only been previously used to solve non-linear ordinary differential equations. 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 robustness of the SHAM approach is demonstrated by its flexibility to allow linear operators that are partial derivatives with variable coefficients. This is seen to significantly improve the convergence and accuracy of the method. To validate accuracy of the the present SHAM results, the governing PDEs are also solved using a novel local linearisation technique coupled with an implicit finite difference approach. The two approaches are compared in terms of accuracy, speed of convergence and computational efficiency.     

Series Solutions of Unsteady Boundary-Layer Flows over a Stretching Flat Plate

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 ≤τ < +∞. 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 ≤τ < +∞. Furthermore, it is shown that a nonlinear initial/boundary-value problem can be replaced by an infinite number of linear boundary-value subproblems.     

A one-step optimal homotopy analysis method for nonlinear differential equations

In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A generalized homotopy equation is first expressed by means of a unknown embedding function in Taylor series, whose coefficient is then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach, which indicate that convergent series solution can be obtained by the optimal homotopy analysis method with much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations with strong nonlinearity.     

Investigation of a powerful analytical method into natural convection boundary layer flow

The natural convection boundary layer flow modeled by a system of nonlinear differential equations is considered. By means of similarity transformation, the non-linear partial differential equations are reduced to a system of two coupled ordinary differential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). Convergence of the obtained series solution is discussed. Finally some figures are illustrated to show the accuracy of the applied method and assessment of various prandtl numbers on the temperature and the velocity is undertaken.     

Solving nonlinear fractional partial differential equations using the homotopy analysis method

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. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Series solutions for unsteady laminar MHD flow near forward stagnation point of an impulsively rotating and translating sphere in presence of buoyancy forces

The similarity solution for the unsteady laminar incompressible boundary layer flow of a viscous electrically conducting fluid in stagnation point region of an impulsively rotating and translating sphere with a magnetic field and a buoyancy force gives a system of non-linear partial differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the homotopy analysis method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the magnetic parameter, buoyancy parameter and rotation parameter on the surface shear stresses and surface heat transfer. It is noted that the behavior of the HAM solution for the surface shear stresses and surface heat transfer is in good agreement with the numerical solution given in reference [H. S. Takhar, A. J. Chamkha, G. Nath, Unsteady laminar MHD flow and heat transfer in the stagnation region of an impulsively spinning and translating sphere in the presence of buoyancy forces, Heat Mass Transfer 37 (2001) 397].     

Rational approximation solution of the fractional Sharma–Tasso–Olever equation

In the paper, we implement relatively new analytical techniques, the variational iteration method, the Adomian decomposition method and the homotopy perturbation method, for obtaining a rational approximation solution of the fractional Sharma–Tasso–Olever equation. The three methods in applied mathematics can be used as alternative methods for obtaining an analytic and approximate solution for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The numerical results demonstrate the significant features, efficiency and reliability of the three approaches.     

A General Approach to Obtain Series Solutions of Nonlinear Differential Equations

Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adjust and control the convergence of solution series. Especially, it provides us with great freedom to replace a nonlinear differential equation of order n into an infinite number of linear differential equations of order k , where the order k is even unnecessary to be equal to the order n . In this paper, a nonlinear oscillation problem is used as example to describe the basic ideas of the homotopy analysis method. We illustrate that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)th-order linear differential equations, where κ≥ 1 can be any a positive integer. Then, the homotopy analysis method is further applied to solve a high-dimensional nonlinear differential equation with strong nonlinearity, i.e., the Gelfand equation. We illustrate that the second-order two or three-dimensional nonlinear Gelfand equation can be replaced by an infinite number of the fourth or sixth-order linear differential equations, respectively. In this way, it might be greatly simplified to solve some nonlinear problems, as illustrated in this paper. All of our series solutions agree well with numerical results. This paper illustrates that we might have much larger freedom and flexibility to solve nonlinear problems than we thought traditionally. It may keep us an open mind when solving nonlinear problems, and might bring forward some new and interesting mathematical problems to study.     

Series solution for unsteady axisymmetric flow and heat transfer over a radially stretching sheet

This paper deals with the unsteady axisymmetric flow and heat transfer of a viscous fluid over a radially stretching sheet. The heat is prescribed at the surface. The modelled non-linear partial differential equations are solved using an analytic approach namely the homotopy analysis method. Unlike perturbation technique, this approach gives accurate analytic approximation uniformly valid for all dimensionless time. The explicit expressions for velocity, temperature and skin friction coefficient are developed. The influence of time on the velocity, temperature and skin friction coefficient is discussed.     

On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations.     

Natural frequencies of thin rectangular plates using homotopy-perturbation method

The homotopy perturbation method (HPM) was developed to search for asymptotic solutions of nonlinear problems involving parabolic partial differential equations with variable coefficients. This paper illustrates that HPM be easily adapted to solve parabolic partial differential equations with constant coefficients. Natural frequencies of a rectangular plate of uniform thickness, simply-supported on all sides, are obtained with minimum amount of computation. The solution is shown to converge rapidly to a combination of sine and cosine functions. Truncating the series solution by using only the first three terms of the sine and cosine functions as compared to the exact solution results in an absolute error not exceeding 2 × 10⁻⁴ and 9×10⁻⁴ for the trigonometric functions respectively. HPM is then applied to solve the nonlinear problem of a rectangular plate of variable thickness. A direct expression for the eigenvalues (natural frequencies) of the rectangular plate is obtained as compared to determining its eigenvalues by solving the characteristic equation using the conventional method. Comparison of results for the frequency parameter with existing literature show that HPM is highly efficient and accurate. Natural frequencies of a simply-supported guitar soundboard were obtained using an equivalent rectangular plate with the same boundary condition.     

无限深水中具有指数密度变率的周期性永形内波渐进解析解

采用同伦分析方法研究了一系列有限振幅的周期深水驻波问题.水密度在垂直方向的分布可以是变化的,假设为指数连续分布.提出一种新形式的偏微分方程作为辅助方程,获得解的新的表达形式来满足底部的边界条件和无限大的刚性假设.给出了解的表达式中系数的递推关系和周期海洋内波形成的永久驻波的显式表达式.得到垂直方向和水平方向的全局收敛解,揭示了密度变量和内波幅度间的关系.同伦分析方法对求解具有指数密度率周期性的永形波解是一致有效的.     

A Series Solution of the Unsteady Von Kármán Swirling Viscous Flows

A new analytic technique is applied to solve the unsteady viscous flow due to an infinite rotating disk, governed by a set of two fully coupled nonlinear partial differential equations deduced directly from the exact Navier-Stokes equations. The system of coupled nonlinear partial differential equations is replaced by a sequence of uncoupled systems of linear ordinary differential equations. Different from all other previous analytic results, our series solution is accurate and valid for all time in the whole spatial region. Accurate expressions for skin friction coefficients are given, which are valid for all time. Such kind of series solutions have not been reported, to the best of our knowledge.      

Application of homotopy analysis method for fractional Swift Hohenberg equation – Revisited

In this article, the homotopy analysis method is used to obtain the approximate analytical solutions of the non-linear Swift Hohenberg equation with fractional time derivative. The fractional derivative is described in Caputo sense. Numerical results reveal that the method is easy to implement, reliable and accurate when applied to time fractional nonlinear partial differential equations. Effects of parameters of physical importance on the probability density function and the convergence of the approximate series solution using residual error formula with the proper choices of auxiliary parameter for various fractional Brownian motions and standard motion are depicted through graphs and tables for different particular cases.     

Heat transfer in unsteady axisymmetric second grade fluid

This work looks at the heat transfer effects on the flow of a second grade fluid over a radially stretching sheet. The axisymmetric flow of a second grade fluid is induced due to linear stretching of a sheet. Mathematical analysis has been carried out for two heating processes, namely (i) with prescribed surface temperature (PST case) and (ii) prescribed surface heat flux (PHF case). The modelled non-linear partial differential equations in two dependent variables are reduced into a partial differential equation with one dependent variable. The resulting non-linear partial differential equations are solved analytically using homotopy analysis method (HAM). The series solutions are developed and the convergence is properly discussed. The series solutions and graphs of velocity and temperature are constructed. Particular attention is given to the variations of emerging parameters such as second grade parameter, Prandtl and Eckert numbers.     

Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled KdV equation by an analytical technique

In this work, we implement a relatively analytical technique, the homotopy perturbation method (HPM), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. He’s homotopy perturbation method (HPM) which does not need small parameter is implemented for solving the differential equations. It is predicted that HPM can be found widely applicable in engineering.     

An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate

In this paper, the unsteady boundary-layer flows caused by an impulsively stretching flat plate is solved by means of an analytic approach. Unlike perturbation techniques, this approach gives accurate analytic approximations uniformly valid for all dimensionless time. Besides, a simple but accurate analytic formula for the local skin friction is given, which agrees well with numerical results and thus is useful in the related industries. To the best of our knowledge, this type of analytic solutions has been never reported. Furthermore, the proposed analytic approach has general meaning and therefore may be applied in the similar way to other unsteady boundary-layer flows to get accurate analytic solutions valid for all time.