A Coupled Method of Laplace Transform and Legendre Wavelets for Lane‐Emden‐Type Differential Equations

Yin, Fukang; Song, Junqiang; Lu, Fengshun; Leng, Hongze

doi:10.1155/2012/163821

Cited by 19 publications

(15 citation statements)

References 26 publications

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“…G. Hariharan, R. Rajaraman [137] had solved a few reaction-diffusion problems by the Laplace Legendre wavelets method. Yin et al [138,139] introduced the Laplace Legendre wavelets method for solving Klein-Gordon and Lane-Emden-Type Differential Equations. Recently, Hariharan and Kannan [140] reviewed the Haar wavelets for solving differential and integral equations arising in science and engineering.…”

Section: Function Approximationmentioning

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: Function Approximationmentioning

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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“…The solution of the Lane-Emden equations is numerically challenging due to the singularity behavior at the origin and nonlinearities. There are a lot of literatures for approximate solutions to (1)(2), which include Adomian decomposition method [4][5][6], homotopy perturbation method [7][8], variational iteration method [9][10], differential transformation method [11][12], Legendre wavelets method [13][14][15], Legendre tau method [16], Sinc-collocation method [17], Chebyshev spectral method [18], nonclassical Radau collocation method [19], etc.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution to Singular Ordinary Differential Equations of Lane-Emden type by Legendre collocation method

Zhao¹,

Wu²

2016

Proceedings of the 3rd International Conference on Material Engineering and Application (ICMEA 2016)

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The purpose of this paper is to propose an efficient numerical method for solving singular ordinary differential equations of Lane-Emden type. The singularity at initial point of the problem leads to failure of many methods such as Euler method, Runge-Kutta method, etc. The proposed method is based on shifted Legendre-Gauss-Radau collocation points and the corresponding interpolation. The first step is to convert second-order ordinary differential equation to an equivalent first order ordinary differential system, and the shifted Legendre-Gauss-Radau points are utilized to collocate the first order ordinary differential system except the initial point. Then computation of the nonlinear initial value problem is reduced to a nonlinear algebraic system. Since evaluation includes no singular point, the difficulty of singularity is overcome. Due to the stability of Gauss-type interpolation, the proposed method possesses high accuracy which is observed by several numerical tests. Also, the method can be done by proceeding in time step by step. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.

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“…Wavelet Analysis, as a relatively new and emerging area in Applied Mathematical Research, has received considerable attention in dealing with PDEs [27][28][29][30][31][32][33][34]. In the last decades, fractional calculus found many applications in various fields of physical sciences such as viscoelasticity, diffusion, control, relaxation processes, signal processing, electromagnetism, biosciences, fluid mechanics, electrochemistry, fluid mechanics and so on.…”

Section: Introductionmentioning

confidence: 99%

“…Yousefi [32] introduced the Legendre wavelets for solving Lane-Emden type differential equations. Recently, Fukang Yin et al [34] introduced a coupled method of Laplace Transform and Legendre wavelets for Lane-Emden type differential Equations.…”

Section: Introductionmentioning

confidence: 99%

An efficient wavelet based approximation method to time fractional Black-Scholes European option pricing problem arising in financial market

Hariharan

Padma²,

Pirabaharan³

2013

ams

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In this paper, a wavelet based hybrid method is employed to provide the quick and accurate solutions of fractional Black-Scholes equation with boundary condition for a European option pricing (EOP) problem. The fractional Black-Scholes is used as a model for valuing European or American call and put options on a non-dividend paying stock. To the best of our knowledge, until now there is no rigorous Legendre wavelet solutions have been reported for the fractional Black-Scholes 3446 G. Hariharan, S. Padma and P. Pirabaharan equations (BSE). The fundamental idea of wavelet method is to convert the fractional Block-Scholes equations into a group of algebraic equations, which involves a finite number of variables. The illustrative examples are given to demonstrate the applicability and validity of the method. Moreover the use of wavelets is found to be simple, flexible, accurate and small computation costs.

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A Coupled Method of Laplace Transform and Legendre Wavelets for Lane‐Emden‐Type Differential Equations

Cited by 19 publications

References 26 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

Numerical Solution to Singular Ordinary Differential Equations of Lane-Emden type by Legendre collocation method

An efficient wavelet based approximation method to time fractional Black-Scholes European option pricing problem arising in financial market

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