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In this paper, we propose new technique for solving stiff system of ordinary differential equations. This algorithm is based on Laplace transform and homotopy perturbation methods. The new technique is applied to solving two mathematical models of stiff problem. We show that the present approach is relatively easy, efficient and highly accurate. 相似文献
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In this paper, the homotopy analysis method (HAM) has been employed to obtain analytical solution of a two reaction–diffusion systems of fractional order (fractional Schnakenberg systems) which has been modeling morphogen systems in developmental biology. Different from all other analytic methods, HAM provides us with a simple way to adjust and control the convergence region of solution series by choosing proper values for auxiliary parameter h. The fractional derivative is described in the Caputo sense. The reason of using fractional order differential equations (FOD) is that FOD are naturally related to systems with memory which exists in most biological systems. Also they are closely related to fractals which are abundant in biological systems. The results derived of the fractional system are of a more general nature. Respectively, solutions of FOD spread at a faster rate than the classical differential equations, and may exhibit asymmetry. However, the fundamental solutions of these equations still exhibit useful scaling properties that make them attractive for applications. 相似文献
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A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations 总被引:1,自引:0,他引:1
In this paper, based on the homotopy analysis method (HAM), a powerful algorithm is developed for the solution of nonlinear ordinary differential equations of fractional order. The proposed algorithm presents the procedure of constructing the set of base functions and gives the high-order deformation equation in a simple form. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ?. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus. 相似文献
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S. Abbasbandy E. Babolian M. Ashtiani 《Communications in Nonlinear Science & Numerical Simulation》2009,14(12):4114-4121
In this paper, an analytic technique, namely the homotopy analysis method (HAM) is applied to obtain approximations to the analytic solution of the generalized Zakharov equation. The HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of the solution series. 相似文献
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In this article, the homotopy analysis method has been applied to solve nonlinear differential equations of fractional order. The validity of this method has successfully been accomplished by applying it to find the solution of two nonlinear fractional equations. The results obtained by homotopy analysis method have been compared with those exact solutions. The results show that the solution of homotopy analysis method is good agreement with the exact solution. 相似文献
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《Applied Mathematical Modelling》2014,38(19-20):4614-4624
In this paper we combined the homotopy analysis method (HAM) and the method of integral manifold (MIM) to investigate the problem of thermal explosion in two-phases polydisperse combustible mixtures of gas with fuel droplets. The size distribution of the fuel droplets is assumed to be continuous in the form of an exponential distribution and is found from the solution of the kinetic equation for the probability density function. The system of the polydisperse fuel spray takes into account the effects of the thermal radiation and convection. By applying the HAM and the MIM, we derived an analytical solution of the system and we compared our results with the numerical solutions. 相似文献
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An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials
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In this paper, a new adaption of homotopy analysis method is presented to handle nonlinear problems. The proposed approach is capable of reducing the size of calculations and easily overcome the difficulty arising in calculating complicated integrals. Furthermore, the homotopy polynomials that decompose the nonlinear term of the problem as a series of polynomials are introduced. Then, an algorithm of calculating such polynomials, which makes the solution procedure more straightforward and more effective, is constructed. Numerical examples are examined to highlight the significant features of the developed techniques. The algorithms described in this paper are expected to be further employed to solve nonlinear problems in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
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Qi Wang 《Applied mathematics and computation》2010,217(8):4122-4128
In this paper, the homotopy analysis method is successfully applied to solve approximate solutions of the differential-difference system. The solution of another relativistic Toda lattice system is considered. Comparisons made between the results of the proposed method and exact solutions reveal that the homotopy analysis method is very effective in solving differential-difference system. 相似文献
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Songxin Liang David J. Jeffrey 《Communications in Nonlinear Science & Numerical Simulation》2009,14(12):4057-4064
In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by Ganji et al. [D.D. Ganji, H. Tari, M.B. Jooybari, Variational iteration method and homotopy perturbation method for nonlinear evolution equations. Comput. Math. Appl. 54 (2007) 1018–1027]. It is found that the HPM is a special case of the HAM when =-1. However, the HPM solution is divergent for all x and t except t=0. It is also found that the solution given by the variational iteration method (VIM) is divergent too. On the other hand, using the HAM, one obtains convergent series solutions which agree well with the exact solution. This example illustrates that it is very important to investigate the convergence of approximation series. Otherwise, one might get useless results. 相似文献
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Fanwei Meng Jizhong Wang Zhaowen Zheng 《Proceedings of the American Mathematical Society》1998,126(2):391-395
Some oscillation criteria are given for the second order matrix differential system , where and are real continuous matrix functions with symmetric, . These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation , where is the identity matrix, is a given function on and .
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S. Mohammad Hosseini 《Journal of Computational and Applied Mathematics》2009,231(1):24-38
As the Tau method, like many other numerical methods, has the limitation of using a fixed step size with some high degree (order) of approximation for solving initial value problems over long intervals, we introduce here the adaptive operational Tau method. This limitation is very much problem dependent and in such case the fixed step size application of the Tau method loses the true track of the solution. But when we apply this new adaptive method the true solution is recovered with a reasonable number of steps. To illustrate the effectiveness of this method we apply it to some stiff systems of ordinary differential equations (ODEs). The numerical results confirm the efficiency of the method. 相似文献
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Zaid M. Odibat 《Applied mathematics and computation》2010,217(2):782-789
In this study, a reliable approach for convergence of the homotopy analysis method when applied to nonlinear problems is discussed. First, we present an alternative framework of the method which can be used simply and effectively to handle nonlinear problems. Then, mainly, we address the sufficient condition for convergence of the method. The convergence analysis is reliable enough to estimate the maximum absolute truncated error of the homotopy series solution. The analysis is illustrated by investigating the convergence results for some nonlinear differential equations. The study highlights the power of the method. 相似文献
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M.M. Rashidi G. Domairry S. Dinarvand 《Numerical Methods for Partial Differential Equations》2009,25(2):430-439
In this work, the homotopy analysis method (HAM) is applied to obtain the explicit analytical solutions for system of the Jaulent–Miodek equations. The validity of the method is verified by comparing the approximation series solutions with the exact solutions. Unlike perturbation methods, the HAM does not depend on any small physical parameters at all. Thus, it is valid for both weakly and strongly nonlinear problems. Besides, different from all other analytic techniques, the HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter ?. Briefly speaking, this work verifies the validity and the potential of the HAM for the study of nonlinear systems. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
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We present here a ‘similar’ solution for the nano boundary layer with nonlinear Navier boundary condition. Three types of flows are considered: (i) the flow past a wedge; (ii) the flow in a convergent channel; (iii) the flow driven by an exponentially-varying outer flows. The resulting differential equations are solved by the homotopy analysis method. Different from the perturbation methods, the present method is independent of small physical parameters so that it is applicable for not only weak but also strong nonlinear flow phenomena. Numerical results are compared with the available exact results to demonstrate the validity of the present solution. The effects of the slip length ?, the index parameters n and m on the velocity profile and the tangential stress are investigated and discussed. 相似文献
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In this paper, the homotopy analysis method (HAM) is employed to solve the linear optimal control problems (OCPs), which have a quadratic performance index. The study examines the application of the homotopy analysis method in obtaining the solution of equations that have previously been obtained using the Pontryagin’s maximum principle (PMP). The HAM approach is also applied in obtaining the solution of the matrix Riccati equation. Numerical results are presented for several test examples involving scalar and 2nd-order systems to demonstrate the applicability and efficiency of the method. 相似文献
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Abdul-Majid Wazwaz 《Applied mathematics and computation》2009,212(1):120-134
In this work, the variational iteration method (VIM) is used for analytic treatment of the linear and nonlinear ordinary differential equations, homogeneous or inhomogeneous. The method is capable of reducing the size of calculations and handles both linear and nonlinear equations, homogeneous or inhomogeneous, in a direct manner. However, for concrete problems, a huge number of iterations are needed for a reasonable level of accuracy. 相似文献
18.
An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the Fitzhugh–Nagumo equation. The homotopy analysis method (HAM) is one of the most effective method to obtain the exact solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of series solution. 相似文献
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In this paper, an algorithm is proposed for the solution of second-order boundary value problems with two-point boundary conditions. The Green’s function method is applied first to transform the ordinary differential equation into an equivalent integral one, which has already satisfied the boundary conditions. And then, the homotopy perturbation method is used to the resulting equation to construct the numerical solution for such problems. Numerical examples demonstrate the efficiency and reliability of the algorithm developed, it is quite accurate and readily implemented for both linear and nonlinear differential equations with homogeneous and nonhomogeneous boundary conditions. Furthermore, the lower order approximation is of higher accuracy for most cases. Some other extended applications of this algorithm are also exhibited. 相似文献
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M. Ghoreishi A. I. B. MD. Ismail A. K. Alomari 《Mathematical Methods in the Applied Sciences》2011,34(15):1833-1842
This paper presents general framework for solving the nth‐order integro‐differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth‐order integro‐differential equation. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献