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Article
Publication date: 28 October 2014

Talaat El-Sayed El-Danaf, Mfida Ali Zaki and Wedad Moenaaem

The purpose of this paper is to investigate the possibility of extension to the variational iteration and the Adomian decomposition methods for solving nonlinear Huxley equation…

289

Abstract

Purpose

The purpose of this paper is to investigate the possibility of extension to the variational iteration and the Adomian decomposition methods for solving nonlinear Huxley equation with time-fractional derivative.

Design/methodology/approach

Objectives achieved the main methods: the fractional derivative of f (x) in the Caputo sense is first stated. Second, the time-fractional Huxley equation is written in a differential operator form where the differential operator is in Caputo sense. After acting on both sides by the inverse operator of the fractional differential operator in Caputo sense, the Adomian's decomposition is then used to get the power series solution of the resulted time-fractional Huxley equation. Also, a second objective is achieved by applying the variational iteration method to get approximate solutions for the time-fractional Huxley equation.

Findings

There are some important findings to state and summarize here. First, the variational iteration method and the decomposition method provide the solutions in terms of convergent series with easily computable components for this considered problem. Second, it seems that the approximate solution of time-fractional Huxley equation using the decomposition method converges faster than the approximate solution using the variational iteration method. Third, the variational iteration method handles nonlinear equations without any need for the so-called Adomian polynomials. However, Adomian decomposition method provides the components of the exact solution, where these components should follow the summation given in Equation (21).

Originality/value

This paper presents new materials in terms of employing the variational iteration and the Adomian decomposition methods to solve the problem under consideration. It is expected that the results will give some insightful conclusions for the used techniques to handle similar fractional differential equations. This emphasizes the fact that the two methods are applicable to a broad class of nonlinear problems in fractional differential equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 18 September 2009

Talaat S. El Danaf and Faisal E.I. Abdel Alaal

The purpose of this paper is to propose a non‐polynomial spline‐based method to obtain numerical solutions of a dissipative wave equation. Applying the Von Neumann stability…

339

Abstract

Purpose

The purpose of this paper is to propose a non‐polynomial spline‐based method to obtain numerical solutions of a dissipative wave equation. Applying the Von Neumann stability analysis, the developed method is shown to be conditionally stable for given values of specified parameters. A numerical example is given to illustrate the applicability and the accuracy of the proposed method. The obtained numerical results reveal that our proposed method maintains good accuracy.

Design/methodology/approach

A non‐polynomial spline is proposed based on the dissipative wave equation, which gives nonlinear system of algebraic equations; by solving these equations, the numerical solution is found.

Findings

It is found that the method gives more accurate numerical results for such nonlinear partial differential equations. The stability is good.

Research limitations/implications

Any nonlinear or linear partial differential equation can be solved by such method.

Practical implications

We compare between the numerical and analytic solutions of the dissipative wave equation, also the error norms which were small.

Originality/value

This paper presents a new method to solve such problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 19 no. 8
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 19 April 2011

Talaat S. El Danaf

The purpose of this paper is to demonstrate how numerical solutions of the nonlinear Huxley equation are obtained by collocation‐based method using cubic B‐spline over finite…

220

Abstract

Purpose

The purpose of this paper is to demonstrate how numerical solutions of the nonlinear Huxley equation are obtained by collocation‐based method using cubic B‐spline over finite elements.

Design/methodology/approach

For the numerical procedure, time derivative is discretized using usual finite difference scheme. Solution and its principal derivatives over the subintervals are approximated by the combination of the cubic B‐spline and unknown element parameters.

Findings

The numerical results are found to be in good agreement with the exact solution. Also the method is very accurate and conditionally stable; the results are very accurate at a small h (discretization) of x so this method can be applied for any nonlinear partial differential equations.

Originality/value

The paper demonstrates how numerical solutions of the nonlinear Huxley equation are obtained by collocation‐based method using cubic B‐spline over finite elements.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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