Travelling wave solutions for some time-delayed equations through factorizations

Fahmy, E. S.

doi:10.1016/j.chaos.2007.02.007

Cited by 39 publications

(22 citation statements)

References 19 publications

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“…By applying the first-integral method, which is based on the ring theory of commutative algebra, a class of travelling solitary wave solutions for the generalized Burger's-Huxley equation, are obtained by Deng [14]. Fahmy [15] obtained travelling wave solutions for some time-delayed equations through factorizations. Wazwaz [16] derived travelling wave solutions for the Burger's, Fisher, Huxley equations and combined forms of these equations using the tanh-coth method.…”

Section: Introductionmentioning

confidence: 99%

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

Mohammadi

2012

Numerical Methods Partial

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The cubic B‐spline collocation scheme is implemented to find numerical solution of the generalized Burger's–Huxley equation. The scheme is based on the finite‐difference formulation for time integration and cubic B‐spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second‐order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

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Section: Introductionmentioning

confidence: 99%

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

Mohammadi

2012

Numerical Methods Partial

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“…In the past few years, a great deal of effort has been spent to compute the solution of the Burgers-Huxley equation. Recently various powerful mathematical methods such as spectral methods [3][4][5], Adomian decomposition method [6][7][8], homotopy analysis method [9,10], the tanh-coth method [11], variational iteration method [12,13], Hopf-Cole transformation [14], differential quadrature method [15], meshless method [16] and factorization method [17] have been used in solving the equation.…”

Section: Introductionmentioning

confidence: 99%

High-order finite difference schemes for numerical solutions of the generalized Burgers-Huxley equation

Sarı

Gürarslan

Zeytinoğlu

2010

Numer. Methods Partial Differential Eq.

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In this article, up to tenth-order finite difference schemes are proposed to solve the generalized BurgersHuxley equation. The schemes based on high-order differences are presented using Taylor series expansion. To establish the numerical solutions of the corresponding equation, the high-order schemes in space and a fourth-order Runge-Kutta scheme in time have been combined. Numerical experiments have been conducted to demonstrate the high-order accuracy of the current algorithms with relatively minimal computational effort. The results showed that use of the present approaches in the simulation is very applicable for the solution of the generalized Burgers-Huxley equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithms are seen to be very good alternatives to existing approaches for such physical applications.

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“…Recently, several new methods are used to find exact solutions of such nonlinear reaction diffusion equations such as, tanh-function method [16], extended tanh-function method [8], Jacobi elliptic function method [9], Lie symmetries [17], Adomian's Decomposition (ADM) [4], and factorization method [7].…”

Section: Introductionmentioning

confidence: 99%

Approximate Solution for Time-Delayed Convective Fisher Equation by Adm-Padé Technique

Alharbi

Fahmy

2010

Asian-European J. Math.

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We present an approximate solution to the time-delayed convective Fisher equation using ADM-Padé technique which is a combination of Adomian decomposition method and Padé approximation. This technique gives the approximate solution with faster convergence and higher accuracy than using ADM alone. 221 Asian-European J. Math. 2010.03:221-233. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/22/15. For personal use only. 222 A. Alharbi & E. S. Fahmy is clear that when τ = 0 equation (1.1) reduces to the convective Fisher equation.Here we will find an approximate solution for the modified model for time delayed convective Fisher equation using Adomian's Decomposition method and then apply the Padé approximants to the series solution derived from the ADM to improve the accuracy and speed up the convergence.Since the beginning of the 1980s Adomian [6] has presented and developed a so-called decomposition method for solving linear or nonlinear problems such as ordinary differential equations. Adomians decomposition method consists of splitting the given equation into linear and nonlinear parts. Then, ADM inverts a derivative operator contained in the linear operator on both sides, and identifies the initial and/or boundary conditions and the terms involving the independent variable alone as initial approximation. Next, ADM decomposes the unknown function into a series whose components are to be determined, and decomposes the nonlinear function in terms of special polynomials called Adomians polynomials. Finally, ADM finds the successive terms of the series solution by recurrent relation using these Adomians polynomials. ADM is quantitative rather than qualitative, analytic, requiring neither linearization nor perturbation, and continuous with no resort to discretization and consequent computer-intensive calculations. Some applications [3] of the method show its advantages. However, ADM has some drawbacks. By using ADM, we get a series solution, in practice a truncated series solution. Although the series can be rapidly convergent in a very small region, it has very slow convergence rate in the wider region we examine, and the truncated series solution is an inaccurate solution in that region, which will greatly restrict the application area of the method. Many examples given in [13] can be used to support this assertion.We have used an extension of ADM [10], which can improve the convergence rate of the series solution. Padé approximant [5] approximates a function by the ratio of two polynomials. The coefficients of the powers occurring in the polynomials are determined by the coefficients in the power series expansion of the function. Generally, the Padé approximant can enlarge the convergence domain of the truncated power series, and can improve greatly the convergence rate of the truncated power series. In order to improve the accuracy of ADM, we propose to modify Adomians series solution for the time delayed convective Fisher equation by using the Padé approximant.Since the ADM serie...

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Travelling wave solutions for some time-delayed equations through factorizations

Cited by 39 publications

References 19 publications

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

High-order finite difference schemes for numerical solutions of the generalized Burgers-Huxley equation

Approximate Solution for Time-Delayed Convective Fisher Equation by Adm-Padé Technique

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