The linear Legendre mother wavelets operational matrix of integration and its application

Khellat, Farhad; Yousefi, S. A.

doi:10.1016/j.jfranklin.2005.11.002

Cited by 63 publications

(26 citation statements)

References 9 publications

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“…In view of successful application of wavelet operational matrix in system analysis [14,15], system identification [16,17], optimal control [18][19][20] and numerical solution of integral and differential equations [21][22][23][24][25][26], together with the characteristic of wavelet functions, we hold that they should be applicable to solve the fractional order systems.…”

Section: Introductionmentioning

confidence: 99%

Solving a nonlinear fractional differential equation using Chebyshev wavelets

2010

Communications in Nonlinear Science and Numerical Simulation

227

111

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Section: Introductionmentioning

confidence: 99%

Solving a nonlinear fractional differential equation using Chebyshev wavelets

2010

Communications in Nonlinear Science and Numerical Simulation

227

111

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“…The differential and integral expressions, the performance index and the boundary conditions are converted into some algebraic equations, which can be solved for the unknown coefficients. Recently, Yousefi et al [24,25,26,12,20,19], have presented Legendre wavelets and Bernstein operational matrix for solving the variational problems and differential equations. In [27,28,29], the operational matrix form in Hybrid of block-pulse functions and another set of functions like Taylor series, Legendre or Chebyshev has been used to find the solution of the various classes of dynamical systems.…”

Section: Research Literaturementioning

confidence: 99%

Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model

Parand

Hossayni

Rad

2016

Applied Mathematical Modelling

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Please cite this article as: K. Parand, Sayyed A. Hossayni, J.A. Rad, An operation matrix method based on Bernstein polynomials for Riccati differential equation and Volterra population model, Applied Mathematical Modelling (2015), Highlight• We present an exact formulation for the operational matrix.• This method transforms the original problem into a algebraic equations system.• Riccati equation and Volterra population model are solved by new method. AbstractIn this paper, we present a modified configuration, including exact formulation for the operational matrix form of the integration, differentiation and product operators to be applied in the Galerkin method. Up to now, so many studies have been conducted on the methods of obtaining operational matrices (derivative, integral and product) for Fourier, Chebyshev, Legendre and Jacobi polynomials and, sometimes, the non-orthogonal bases, which almost all of them operate approximately; but, in this research, we aim to obtain the exact operational matrices (EOMs), which can be used for many classes of orthogonal and non-orthogonal polynomials. Similar to the previous works, this method transforms the original problem into a system of nonlinear, algebraic equations. To keep the simplicity of the procedure, samples have been considered in one dimensional contexts; however, the proposed technique can be employed for two and three-dimensional problems, as well. For confirming the accuracy of the new approach and for showing the performance of EOMs over ordinary operational matrices (OOMs), two examples are presented. The corresponding results demonstrated the increased accuracy of the new method. Also the convergence of the EOM method is studied numerically and analytically, to prove the method efficiency.

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“…Operational matrices are used in several areasof numerical analysis and they hold particular importance in various subjects such as integral equations [20], differential and partial differential equations [21] and etc. Also many textbooks and papers have employed the operational matrices for spectral methods [9].…”

Section: The Operational Matrices With Respect To Rational Laguerrefumentioning

confidence: 99%

Rational Laguerre Functions And Their Applications

Aminataei¹,

Asl²,

Kalatehbojdi³

2015

J. Math. Computer Sci.

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In this work, we introduce a new class of rational basis functions defined on [ , ) ab and based on mapping the Laguerre polynomials on the bounded domain [ , ) ab . By using these rational functions as basic functions, we implement spectral methods for numerical solutions of operator equations. Also the quadrature formulae and operational matrices (derivative, integral and product) with respect to these basis functions are obtained. We show that using quadrature formulae based on rational Laguerre functions give us very good results for numerical integration of rational functions and also implementing spectral methods based on these basis functions for solving stiff systems of ordinary differential equationsgive us suitable results. The details of the convergence rates of these basis functions for the solutions of operator equations are carried out, both theoretically and computationally and the error analysis is presented in   2 [ , ) L a b space norm.

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The linear Legendre mother wavelets operational matrix of integration and its application

Cited by 63 publications

References 9 publications

Solving a nonlinear fractional differential equation using Chebyshev wavelets

Solving a nonlinear fractional differential equation using Chebyshev wavelets

Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model

Rational Laguerre Functions And Their Applications

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