A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations

Mohammadi, Fakhrodin; Hosseini, Mohammad Mehdi

doi:10.1016/j.jfranklin.2011.04.017

Cited by 111 publications

(62 citation statements)

References 12 publications

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“…The product operation of Legendre wavelet vector bases is approximated as 16) where the element of the matrix of the product operation is composed of linear combinations of elements of the vector 0 Q . In this paper, we address the general computational procedure for this product operation by using the DLWG method, which is similar to that of the literature [24].…”

Section: Computation Of Integer Power Operatormentioning

confidence: 99%

“…For example, these equations can describe the equilibrium density distribution in self-gravitating sphere of polytrophic isothermal gas, and characterize the principle of thermionic currents and the theory of stellar structure, etc. In general, the equations mentioned above have a form as follows Legendre wavelet operational matrix methods, which can convert this equation to a system of equations by computing operational matrix of integration or derivative [12][13][14][15][16][17][18][19]. For example, Razzaghi Yousefi [14] derived the Legendre wavelet operational matrix of integration to solve Eq.…”

Section: Introductionmentioning

confidence: 99%

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Discontinuous Legendre Wavelet Galerkin Method for Solving Lane-Emden Type Equation

Zheng¹,

Wei²,

He³

2016

JAAM

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Abstract. This paper presents a new numerical approach, i.e., discontinuous Legendre wavelet Galerkin (DLWG) technique, to solve the Lane-Emden equations. This scheme incorporates Legendre wavelet into discontinuous Galerkin (DG) method, thus it has the advantages of both wavelet Galerkin (WG) method and DG technique. Specifically, the variational formulation of the equation and numerical fluxes are first derived and clearly calculated, then the Lane-Emden equations are converted into solutions to systems of equations. It is pointed that the proposed approach needs less storage space and execution time than the other methods because of the use of discontinuous elements producing lower dimensional, block-diagonal and sparse mass matrices. Furthermore, numerical experiments demonstrate the efficiency and applicability of this technique.

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Section: Computation Of Integer Power Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Discontinuous Legendre Wavelet Galerkin Method for Solving Lane-Emden Type Equation

Zheng¹,

Wei²,

He³

2016

JAAM

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“…Shifted Legendre polynomials and their properties are described in [8,9]. In this article, we only pay close attention to two important properties of shifted Legendre polynomials.…”

Section: A the Properties Of Shifted Legendre Polynomialsmentioning

confidence: 99%

Solving linear PDEs with the aid of two-dimensional Legendre wavelets

Yin¹,

Song²

2012

Proceedings of 2012 National Conference on Information Technology and Computer Science

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Abstract-In this paper, we develop a method, which using twodimensional Legendre wavelets, to solve linear PDEs. Based on the properties of shifted Legendre polynomials, we give a brief proof about the general procedure of two-dimensional operational matrices of integration, and then employ aforementioned matrices to find the solution of the PDEs. The proposed method is mathematically simple and fast. To demonstrate the efficiency of the method, two test problems (solution of the diffusion, Poisson) are discussed. The experimental results showed that the accuracy of the method is very high and only need a small number of collocation points.

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“…Wavelets permit the accurate representation of a variety of functions and operators [14,15,17,18,19,20,21,16]. In this paper, an stochastic operational matrix for the Chebyshev wavelets is derived.…”

Section: Introductionmentioning

confidence: 99%

A Chebyshev wavelet operational method for solving stochastic Volterra-Fredholm integral equations

Mohammadi

2015

IJAMR

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In this paper, the stochastic operational matrix of Itô-integration for the Chebyshev wavelets is applied for solving stochastic Volterra-Fredholm integral equations. The main characteristic of the presented method is that it reduces stochastic Volterra-Fredholm integral equations into a linear system of equations. Convergence and error analysis of the Chebyshev wavelets basis is considered. The efficiency and accuracy of the proposed method was demonstrated by some non-trivial examples and comparison with the other existing methods.

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A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations

Cited by 111 publications

References 12 publications

Discontinuous Legendre Wavelet Galerkin Method for Solving Lane-Emden Type Equation

Discontinuous Legendre Wavelet Galerkin Method for Solving Lane-Emden Type Equation

Solving linear PDEs with the aid of two-dimensional Legendre wavelets

A Chebyshev wavelet operational method for solving stochastic Volterra-Fredholm integral equations

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