Analytic and approximate solutions for Nagumo telegraph reaction diffusion equation

Abdusalam, H. A.

doi:10.1016/j.amc.2003.08.050

Cited by 31 publications

(25 citation statements)

References 11 publications

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“…Abbasbandy [17] solved the FitzHugh-Nagumo (FN) equation by the homotopy analysis method. Abdusalam [18] proposed an analytic and approximate method for Nagumo telegraph reaction-diffusion equation. Ablowitz and Zepetella [19] established the explicit solutions of Fisher's equation for a special wave speed.…”

Section: Reaction-diffusion Modelingmentioning

confidence: 99%

Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

Hariharan

Kannan

2014

Applied Mathematical Modelling

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a b s t r a c tWavelet method is a recently developed tool in applied mathematics. Investigation of various wavelet methods, for its capability of analyzing various dynamic phenomena through waves gained more and more attention in engineering research. Starting from 'offering good solution to differential equations' to capturing the nonlinearity in the data distribution, wavelets are used as appropriate tools at various places to provide good mathematical model for scientific phenomena, which are usually modeled through linear or nonlinear differential equations. Review shows that the wavelet method is efficient and powerful in solving wide class of linear and nonlinear reaction-diffusion equations. This review intends to provide the great utility of wavelets to science and engineering problems which owes its origin to 1919. Also, future scope and directions involved in developing wavelet algorithm for solving reaction-diffusion equations are addressed.Crown

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Section: Reaction-diffusion Modelingmentioning

confidence: 99%

Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

Hariharan

Kannan

2014

Applied Mathematical Modelling

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“…Now, we will approximate the wave shapes by linearizing about both I = 0 and I = N , as Jones and Sleeman [10] have done for Fisher's equation (see also [1,5,14]). …”

Section: Application Of Telegraph Reaction Diffusion Equation To Aviamentioning

confidence: 99%

Approximate Traveling Wave Solution of Avian Flu Telegraph Reaction Diffusion Equation

Ahmed

2010

Int. J. Biomath.

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It is known that, the telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion in several branches of sciences. In this paper we generalize the governing equation of distributed-infective model which represents the spread of avian flu to the telegraph reaction diffusion equation and presents its approximate traveling wave solution by using linear piecewise approximation.

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“…models an infinitesimal piece of a telegraph wire as an electrical circuit, and it describes the voltage and current in a double conductor with distance

x

and time

t

. The telegraph equation is in particular important as it is commonly used in the study and modeling of signal analysis for transmission and propagation of electrical signals in a cable transmission line , and in reaction diffusion occurring in many branches of sciences .…”

Section: Introductionmentioning

confidence: 99%

“…(1.1a) models an infinitesimal piece of a telegraph wire as an electrical circuit, and it describes the voltage and current in a double conductor with distance x and time t [6]. The telegraph equation is in particular important as it is commonly used in the study and modeling of signal analysis for transmission and propagation of electrical signals in a cable transmission line [7,8], and in reaction diffusion occurring in many branches of sciences [9,10]. The numerical solution of second order hyperbolic PDEs has been studied extensively by a variety of techniques such as the finite element methods [11,12], finite-difference schemes [3,[13][14][15], combined finite difference scheme and Haar wavelets [6], discrete eigenfunctions method [7], Legendre multiwavelet approximations [16], the singular dynamic method [17], interpolating scaling functions [18], cubic and quartic B-spline collocation methods [19,20], nonpolynomial spline methods [21], the reduced differential transform method [22], and so forth.…”

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confidence: 99%

High‐order numerical solution of second‐order one‐dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method

Elgindy

2015

Numerical Methods Partial

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SUMMARYWe present a high-order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second-order onedimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well-conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out in order to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications. This is a preprint copy that has been accepted for publication in Numerical Methods for Partial Differential Equations.

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Analytic and approximate solutions for Nagumo telegraph reaction diffusion equation

Cited by 31 publications

References 11 publications

Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

Approximate Traveling Wave Solution of Avian Flu Telegraph Reaction Diffusion Equation

High‐order numerical solution of second‐order one‐dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method

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