New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials

Summary A novel collocation method based on Genocchi wavelet is presented for the numerical solution of fractional differential equations and time‐fractional partial differential equations with delay. In this work, to achieve the approximate solution with height accuracy, we employed the operational matrix of integer derivative and the pseudo‐operational matrix of fractional derivative in Caputo sense. Also, based on Genocchi function properties, we presented delay and pantograph operational matrices of Genocchi wavelet functions (GWFs). Due to operational and pseudo‐operational matrices, the equations under this study can be turned into nonlinear algebraic equations with the unknown GWF coefficients. For illustrating the upper bound of error for the proposed method, we estimate the error in the sense of Sobolev space. In addition, to demonstrate the efficacy of the pseudo‐operational matrix of fractional derivative, we investigate the upper bound of error for the mentioned matrix. Finally, the algorithm based on the proposed approach is implemented for some numerical experiments to confirm accuracy and applicability.

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“…The classical Genocchi polynomial G m ( ξ ) is defined by means of the exponential generating functions

\frac{2 t e^{ξ η}}{e^{η} + 1} = \sum_{m = 0}^{\infty} G_{m} (ξ) \frac{η^{m}}{m!}, (| η | < π),

where Genocchi polynomials of order m are defined on interval [0,1] as

G_{m} false(ξ false) = true \sum_{k = 0}^{m} ((), \begin{matrix} center \\ array m \\ array k \end{matrix}) g_{m - k} ξ^{k},

where

g_{k} = 2 B_{k} - 2^{k + 1} B_{k}

is the Genocchi number and B k is the well‐known Bernoulli number . The first few Genocchi numbers are

g_{0} = 0, g_{1} = 1, g_{2} = - 1, g_{4} = 1, g_{6} = - 3,

where g 2 k +1 =0, k =1,2,….…”

Section: Gwfs and Their Propertiesmentioning

confidence: 99%

“…The classical Genocchi polynomial G m ( ) is defined by means of the exponential generating functions [46][47][48]…”

Section: Genocchi Functionsmentioning

confidence: 99%

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

Dehestani

Ordokhani

Razzaghi

2019

Numerical Linear Algebra App

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“…The figures confirm that when q approaches 1 our results approach the exact solution. We also compare the absolute error obtained by our method and those obtained in [37] and [17] at t = 1 in Table 6. t y(t) Error in [37] with h = 0.002Error in [17] with N = 10Our Error with N = 10 1y 1 (t)2.5606 × 10 −7…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…Another very effective approach for solving FDEs is based on using operational matrix of fractional derivatives, to eliminate the differential operators, in order to reducing the underlying problem into solving a system of algebraic equations. In this respect, several authors employed these operational matrices for obtaining numerical solutions of FDEs, such as, Legendre polynomials [35], Legendre wavelets [16], Bernstein polynomials [33] and Genocchi polynomials [17].…”

Section: Introductionmentioning

confidence: 99%

Bernoulli Operational Matrix of Fractional Derivative for Solution of Fractional Diﬀerential Equations

Belgacem¹,

Bokhari²,

Amir³

2018

(GLM)

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The aim of this paper is to present a numerical method based on Bernoulli polynomials for numerical solutions of fractional differential equations(FDEs). The Bernoulli operational matrix of fractional derivatives[31] is derived and used together with tau and collocation methods to reduce the FDEs to a system of algebraic equations. Hence, the solutions obtained using this method give good approximations. Illustrative examples are included to demonstrate the validity and applicability of the proposed method.

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“…Since the results were encouraging, we hope to apply this method to Genocchi polynomials. Some of the applications using Genocchi polynomials to solve differential equation problems are shown in [16], and also [17][18][19]. This proposed method is able to reduce the FDWE and FGKE to only solve the linear system of algebraic equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

2018

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Abstract:In this paper, two-dimensional Genocchi polynomials and the Ritz-Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein-Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz-Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples.

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New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials

Cited by 39 publications

References 19 publications

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

Bernoulli Operational Matrix of Fractional Derivative for Solution of Fractional Diﬀerential Equations

Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

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