Operational matrix based on Genocchi polynomials for solution of delay differential equations

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.

show abstract

“…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: 65%

“…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

View full text Add to dashboard Cite

show abstract

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…It is very important to note that this polynomial shares some great advantages with Bernoulli and Euler polynomials for approximating an arbitrary function over some classical orthogonal polynomials; we refer the reader to [6] for these advantages. On top of that, we 2 International Journal of Differential Equations had successfully applied the operational matrix via Genocchi polynomials for solving integer-order delay differential equations [12] and fractional optimal control problems [13], and the numerical solutions obtained are comparable or even more accurate compared to some existing well-known methods. Motivated by these advantages, in this paper, we intend to extend the result for integer-order delay differential equations in [12] to fractional delay differential equations or so-called generalized fractional pantograph equations.…”

Section: Introductionmentioning

confidence: 99%

“…On top of that, we 2 International Journal of Differential Equations had successfully applied the operational matrix via Genocchi polynomials for solving integer-order delay differential equations [12] and fractional optimal control problems [13], and the numerical solutions obtained are comparable or even more accurate compared to some existing well-known methods. Motivated by these advantages, in this paper, we intend to extend the result for integer-order delay differential equations in [12] to fractional delay differential equations or so-called generalized fractional pantograph equations. To the best of our knowledge, this is the first time that the operational matrix based on Genocchi polynomials is applied to solve the fractional pantograph equations.…”

Section: Introductionmentioning

confidence: 99%

Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

Isah

Phang

2017

International Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.

show abstract

Operational matrix based on Genocchi polynomials for solution of delay differential equations

Cited by 24 publications

References 17 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

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

Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

Contact Info

Product

Resources

About