Numerically solving non-linear problems by the homotopy analysis method (original) (raw)
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The application of homotopy analysis method to nonlinear equations arising in heat transfer
Physics Letters A, 2006
Here, the homotopy analysis method (HAM), which is a powerful and easy-to-use analytic tool for nonlinear problems and dose not need small parameters in the equations, is compared with the perturbation and numerical and homotopy perturbation method (HPM) in the heat transfer filed. The homotopy analysis method contains the auxiliary parameterh, which provides us with a simple way to adjust and control the convergence region of solution series.
Homotopy analysis method for heat radiation equations
International Communications in Heat and Mass Transfer, 2007
Here, the homotopy analysis method (HAM), one of the newest analytical methods which is powerful and easy-to-use, is applied to solve heat transfer problems with high nonlinearity order. Also, the results are compared with the perturbation and numerical Runge-Kutta methods and homotopy perturbation method (HPM). Here, homotopy analysis method is used to solve an unsteady nonlinear convective-radiative equation containing two small parameters of ϵ 1 and ϵ 2 . The homotopy analysis method contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of solution series.
HOMOTOPY ANALYSIS METHOD: A NEW ANALYTIC METHOD FOR NONLINEAR PROBLEMS
In this paper, the basic ideas of a new (malytic techniq,te, ntmwly the llomotopy Analysis:Method (HAM). are described. Diff'erent from perturl~ttioJt methods, the ralidit.r of the tlAM is huh'l)ende, t o, whether or not there exist sin,l/pammwters h~ considered notdinear equations. Therefore, it provides tls with a poweJ]'zd alttdytic tool Jbr strongly nonlhtear problems. A typical no,lhwar problem is used as an example to ver(])' the validity and the great potential of the HAM.
Mathematical modeling in microwave heating in an infinite slab with isothermal walls is discussed. In this paper , the approximate analytical solution of steady state non-linear reaction-diffusion equation is obtained using Homotopy analysis method (HAM). In particular, we consider the case where the source term decreases spatially and increases with temperature. A simple form of an approximate analytical expression of temperature profiles in terms of dimensionless parameter λ and kare also reported. These approximate results are compared with the numerical results. A good agreement with simulation data is noted. The present approach is less computational and is applicable for solving other non-linear boundary value problem.
APPLICATION OF HOMOTOPY ANALYSIS METHOD FOR SOLVING NONLINEAR PROBLEMS
In this project we introduced an analytic approximation method for nonlinear problem in general, namely the homotopy analysis method. The homotopy analysis method (HAM) is an analytic approximation method for highly nonlinear problems, proposed by the Liao in 1992.Unlike perturbation techniques; the HAM is independent of any small/large physical parameters at all: one can always transfer a nonlinear problem into an infinite number of linear sub problems by means of the HAM. Secondly, different from all of other analytic techniques, the HAM provides us a convenient way to guarantee the convergence of solution series so that it is valid even if nonlinearity becomes rather strong. Besides, based on the homotopy in topology, it provides us extremely large freedom to choose equation type of linear sub-problems, base function of solution, initial guess and so on, so that complicated nonlinear ODEs and PDEs can often be solved in a simple way. In this project, the homotopy analysis method is employed to solve non linear problems; the results reveal that the proposed method is effective.
Homotopy analysis method: A new analytical technique for nonlinear problems
Communications in Nonlinear Science and Numerical Simulation, 1997
In this paper, the basic ideas of a new kind of analytical technique, namely the Homotopy Analysis Method (HAM), are briefly described. Different from perturbation techniques, the HAM does not depend on whether or not there exist small parameters in nonlinear equations under consideration. Therefore, it provides us with a powerful tool to analyse strongly nonlinear problems. A simple but typical example is used to illustrate the validity and the great potential of the HAM. Moreover, a pure mathematical theorem, namely the General Taylor Theorem, is given in appendix, which provides us with some rational knowledge for the validity of this new analytical technique.
Homotopy Analysis Method for Solving Non-linear Various Problem of Partial Differential Equations
2014
In this paper, solve several important equations such as korteweg-devries (kdv) problem, Boussinesq equation of non-homogeneous problem and non-homogeneous system Hirota-Satsuma problem of partial differential equation by Homotopy analysis method (HAM). Studied comparison exact solution with numerical results , this method have shown that is very effective and convenient and gives numerical solutions in the form of convergent series with easily computable components for solving non-linear various problem of partial differential equation .
Homotopy analysis method (HAM) for solving 3D Heat equations
River Publishers Series in Proceedings, 2022
In this research paper, we present a classical analytical method known as homotopy analysis method for solving three-dimensional heat equations arising in several applications of sciences and engineering. Some numerical experiments have been presented to illustrate the simplicity and accuracy of the presented technique.
Thermal Science, 2011
Analytical solutions play a very important role in heat transfer. In this paper, the He's homotopy perturbation method (HPM) has been applied to nonlinear convective-radiative non-Fourier conduction heat transfer equation with variable specific heat coefficient. The concept of the He's homotopy perturbation method are introduced briefly for applying this method for problem solving. The results of HPM as an analytical solution are then compared with those derived from the established numerical solution obtained by the fourth order Runge-Kutta method in order to verify the accuracy of the proposed method. The results reveal that the HPM is very effective and convenient in predicting the solution of such problems, and it is predicted that HPM can find a wide application in new engineering problems.
Homotopy Analysis Method for Solving System of Non-Linear Partial Differential Equations
International Journal of Emerging Multidisciplinaries: Mathematics
This paper applies Homotopy Analysis Method (HAM) to obtain analytical solutions of system of non-linear partial differential equations. Numerical results clearly reflect complete compatibility of the proposed algorithm and discussed problems. Moreover, the validity of the present solution and suggested scheme is presented and the limiting case of presented findings is in excellent agreement with the available literature. The computed solution of the physical variables against the influential parameters is presented through graphs. Several examples are presented to show the efficiency and simplicity of the method.