Some new results on products of Apostol–Bernoulli and Apostol–Euler polynomials

He, Yuan

doi:10.1016/j.jmaa.2015.05.055

Cited by 16 publications

(10 citation statements)

References 22 publications

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“…is for normality, the dilation parameter is a = 2 −(l−1) and the translation parameter b = (i − 1)2 −(l−1) Here P j (t) are the well-known Euler polynomials of order j which can be defined by means of the following generating functions [5],…”

Section: Euler Waveletsmentioning

confidence: 99%

“…Finite difference and Finite element based numerical methods need a large amount of computation and usually the effect of round-off error causes the loss of accuracy [1,2]. In recent days, numerical methods via wavelets are tremendously growing much faster [4][5][6][7][8][9][10]. Hence, we are able to develop new wavelet method for the solutions of parabolic partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

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Euler wavelet based numerical scheme for the solutions of parabolic partial differential equations

Shiralashetti¹,

Hanaji²

2020

Malaya J. Mat.

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Many physical problems when analysed assumes the form of a partial differential equations. Recently wavelet transforms serves as a very useful tool in solving partial differential equations. In this paper, we proposed an efficient numerical scheme based on Euler wavelets for the solutions of parabolic partial differential equations. Some Illustrative examples are included to demonstrate the validity and applicability of the proposed scheme. Numerical findings of the problems with both initial and boundary conditions shows the efficiency and accuracy of the present scheme.

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Section: Euler Waveletsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Euler wavelet based numerical scheme for the solutions of parabolic partial differential equations

Shiralashetti¹,

Hanaji²

2020

Malaya J. Mat.

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“…is for normality, the dilation parameter is a = 2 −(k−1) and the translation parameter b = (n − 1)2 −(k−1) . Here, E m (t) are the well-known Euler polynomials of order m which can be defined by means of the following generating functions [He (2015)]…”

Section: Euler Waveletsmentioning

confidence: 99%

Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets

YanxinWang¹,

Zhu²,

Wang³

2019

Computer Modeling in Engineering &Amp; Sciences

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An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper. The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented. Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations. And the convergence of the Euler wavelets basis is given. The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.Many phenomena in fluid mechanics, chemistry, physics, finance and other sciences can be described successfully by models using mathematical tools from fractional calculus, i.e. the theory of derivatives and integrals of fractional order [Miller and Ross (1993)]. Due to the fractional order exponents in differential operators, analytical solutions of fractional equations are usually difficult to obtain. Consequently, different methods have been developed to give numerical solutions for fractional equations, including fractional differential transform method [Wei and Chen (2014)], Adomian decomposition method [Song and Wang (2013)], Chebyshev pseudo-spectral method [Khader and Sweilam (2013)], Homotopy perturbation method [Abdulaziz, Hashim and Momani (2008)], Homotopy analysis method [Dehghan, Manafian and Saadatmandi (2010)], and wavelet method [Li and Zhao (2010); Wang and Fan (2012); Wang and Zhu (2016)]. Recently, the concepts of fractional derivatives of variable order have been introduced and some research works of the relative practical applications have arisen [Sun, Chen and Chen (2009); Samko (2013)]. Several numerical approximation methods are proposed to solve the variable order fractional differential equation [Chen, Liu, Li et al.

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“…Note that the corresponding two higher-order convolution identities for the Apostol-Euler polynomials stated in [18,Theorem 3.2] are only complete on condition that λ 1 = · · · = λ k .…”

Section: Convolution Identities For Apostol-euler Polynomialsmentioning

confidence: 99%

General convolution identities for Apostol-Bernoulli, Euler and Genocchi polynomials

He¹,

Kim²

2016

J. Nonlinear Sci. Appl.

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We perform a further investigation for the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. By making use of the generating function methods and summation transform techniques, we establish some general convolution identities for the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. These results are the corresponding extensions of some known formulas including the general convolution identities discovered by Dilcher and Vignat [K.

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Some new results on products of Apostol–Bernoulli and Apostol–Euler polynomials

Cited by 16 publications

References 22 publications

Euler wavelet based numerical scheme for the solutions of parabolic partial differential equations

Euler wavelet based numerical scheme for the solutions of parabolic partial differential equations

Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets

General convolution identities for Apostol-Bernoulli, Euler and Genocchi polynomials

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