Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets

Chen, Yiming; Wei, Yan-Qiao; Liu, Da‐Yan; Yu, Hao

doi:10.1016/j.aml.2015.02.010

Cited by 108 publications

(59 citation statements)

References 22 publications

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“…The fractional differential equations model areal problem in life that needs a solution. Therefore, there are many different numerical methods that solve these equations, such as the predictor‐corrector method, Legendre wavelets, Legendre spectral method, Legendre collocation method, pseudo‐spectral scheme, Haar wavelet collocation method, Chebyshev spectral methods,() other techniques,() and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

El‐Sayed

Agarwal

2019

Math Methods in App Sciences

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In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.

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Section: Introductionmentioning

confidence: 99%

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

El‐Sayed

Agarwal

2019

Math Methods in App Sciences

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“…So, we just discuss the numerical precision of these three methods in this section. Table 4 illustrates that Shannon-Gabor wavelet 6 and the Shannon-Cosine wavelet numerical methods have better numerical stability compared to the Shannon wavelet. As N = 11.261199765911442, the numerical error obtained by the Shannon-Cosine spectral method is smaller than Shannon-Gabor wavelet method with the same parameters j and α.…”

Section: Shannon-cosine Spectral Methods On Fractional Fokker-planck Ementioning

confidence: 99%

“…Many typical orthogonal polynomials such as Legendre, 4,6,15,18 Chebyshev 19,20 polynomial expansions have been employed to construct the orthogonal wavelets to solve various integro-differential equations, which improve the numerical accuracy and efficiency of the corresponding spectral methods. It is well known that Daubechies wavelet is also an orthogonal one, which is complicated to be used in the spectral method as it has no analytical expression.…”

Section: Introductionmentioning

confidence: 99%

Shannon–Cosine wavelet spectral method for solving fractional Fokker–Planck equations

Mei

Gao

2018

Int. J. Wavelets Multiresolut Inf. Process.

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The windowed-Shannon wavelet is not recommended generally as the window function will destroy the partition of unity of Shannon mother wavelet. A novel windowing scheme is proposed to overcome the shortcoming of the general windowed-Shannon function, and then, a novel and efficient Shannon-Cosine wavelet spectral method is provided for solving the fractional PDEs. Taking full advantage of the waveform of sinc function to hold the partition of unity, Shannon-Cosine wavelet is constructed, which is composed of Shannon wavelet and the trigonometric polynomials. It was proved that the proposed wavelet function meets the requirements of being a trial function and possesses many other excellent properties such as normalization, interpolation, two-scale relations, compact support domain, and so on. Therefore, it is a real wavelet function instead of a general Shannon-Gabor wavelet which is a kind of quasi-wavelet. Next, by means of the Shannon-Cosine wavelet collocation method, the corresponding algebraic equation system of the fractional Fokker-Planck equation can be obtained. Approximate solutions of the fractional Fokker-Plank equations are compared with the exact solutions. These calculations illustrate that the accuracy of the Shannon-Cosine wavelet collocation solutions is quite high even using a small number of grid points.

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“…Recently, spectral methods using continuous orthogonal polynomials, such as Jacobi, Chebyshev and Legendre polynomials, have been developed for solving different kinds of VO fractional differential equations. Chen et al proposed Legendre wavelets functions to solve a class of nonlinear VO fractional differential equations [8]. Bernstein polynomials are used to numerically solve the VO fractional partial differential equations by Wang et al [35].…”

Section: Introductionmentioning

confidence: 99%

A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation

2018

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In this paper, we consider the discrete Hahn polynomials fH n g and investigate their application for numerical solutions of the time fractional variable order mobile-immobile advection-dispersion model which is advantageous for modeling dynamical systems. This paper presented the operational matrix of derivative of discrete Hahn polynomials. The main advantage of approximating a continuous function by Hahn polynomials is that they have a spectral accuracy in the interval [0, N]. Furthermore, for computing the coefficients of the expansion uðxÞ ¼ P 1 n¼0 c n H n ðxÞ, we have to only compute a summation and the calculation of coefficients is exact. Also an upper bound for the error of the presented method, with equidistant nodes, is investigated. Illustrative examples are provided to show the accuracy and efficiency of the presented method. Using a small number of Hahn polynomials, significant results are achieved which are compared to other methods.

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Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets

Cited by 108 publications

References 22 publications

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

Shannon–Cosine wavelet spectral method for solving fractional Fokker–Planck equations

A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation

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