A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order

Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S.

doi:10.1016/j.camwa.2011.07.024

Cited by 273 publications

(136 citation statements)

References 23 publications

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“…The operational differential matrix based on the Jacobi-Gauss-Lobatto points is also obtained, which can be seen as a generalization of the classical differential matrix. The computational cost for deriving the operational differential matrix is O(N 2 ), which is much less than O(N 3 ) in [5,6].…”

Section: Resultsmentioning

confidence: 99%

“…The similar work can be found in [21], where the operational matrix D (α) based on the explicit form of the Legendre polynomials was obtained, which takes O(N 3 ) arithmetic operations. The similar matrix D (α) based on the Chebyshev polynomials can be found in [5,6], where the computational complexity is O(N 3 ).…”

Section: Approximation To the Caputo Derivativementioning

confidence: 99%

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Spectral approximations to the fractional integral and derivative

Zeng

Liu

2012

fcaa

153

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AbstractIn this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

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Section: Resultsmentioning

confidence: 99%

Section: Approximation To the Caputo Derivativementioning

confidence: 99%

Spectral approximations to the fractional integral and derivative

Zeng

Liu

2012

fcaa

153

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“…Some of Chebyshev polynomials properties are: orthogonality, recursive, real zeros, complete for the space of polynomials, etc. For these reasons, many researchers have employed these polynomials in their research [17,18,19,20,21,22,23].…”

Section: The Chebyshev Functionsmentioning

confidence: 99%

Systems of nonlinear Volterra integro-differential equations of arbitrary order

Parand

Delkhosh

2018

B Soc Paran Mat

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In this paper, a new approximate method for solving the system of nonlinear Volterra integro-differential equations of arbitrary (integer and fractional) order is introduced. For this purpose, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) based on the classical Chebyshev polynomials of the first kind has been introduced that can be used to obtain the solution of the integro-differential equations (IDEs). Also, we construct the fractional derivative operational matrix of order α in the Caputo's definition for GFCFs. This method reduces a system of IDEs by collocation method into a system of algebraic equations. Some examples to illustrate the simplicity and the effectiveness of the propose method have been presented.

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“…Many researchers [1,5,6,10,15] are stated that the various result in different kind of wavelet models ( Haar, Legendre, Galerkin, chebyshev etc.,). As particularly researcher has discussed chebyshev polynomial because of this polynomial is most suitable in numerical analysis including polynomial approximation, integral and differential equation [2,7,8,11]. The operational matrices of the derivatives have been determined for Chebyshev polynomials [7] and Legendre polynomials [4], and applied together with tau and pseudo spectral methods to solve some types of PDEs.…”

mentioning

confidence: 99%

“…In this paper we derive new explicit formula for the matrix derivatives of chebyshev polynomial of third degree because of this kind of polynomial is an important application of the numerical analysis of optimal controlling system of the matrix derivatives. Integral and differential equation [2,7,8,11] The operational matrices of the derivatives have been determined for Chebyshev polynomials [7] and Legendre polynomials [4], and applied together with tau and pseudo spectral methods to solve some types of PDEs. The operational matrix of integration has been determined for several types of orthogonal polynomials, such as Chebyshev polynomials of the first kind [12], Legendre polynomials [13].…”

mentioning

confidence: 99%