Modified Laguerre Wavelets Method for delay differential equations of fractional-order

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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“…Consider the fractional multipantograph differential equation with variable coefficients

\{\begin{matrix} D^{ν} u false(t false) = - u false(t false) - \exp ({, \frac{- t}{2}) \sin ({, \frac{t}{2}) u ({, \frac{t}{2}) - 2 \exp ({, \frac{- 3 t}{4}) \cos ({, \frac{t}{2}) \sin ({, \frac{t}{4}) u ({, \frac{t}{4}), \\ u false(0 false) = 1 . \end{matrix} 0 < ν \leq 1, 0 \leq t \leq 1,

For a special value of ν =1, the exact solution of this problem is

u false(t false) = \exp false(- t false) \cos false(t false)

. We implemented the present method for various M , k and demonstrated the results in Table and Figure .…”

Section: Numerical Approach Implementationmentioning

confidence: 99%

“…41 Haar wavelet method has been utilized for dealing with fractional-order differential equations. 42 More information about this topic can be found in other works 33,34,43 and the references therein.…”

Section: Introductionmentioning

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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“…Wavelets form a family of functions which formulated from dilation and translation of a single function ( ) t ψ called the mother wavelet. When the dilation parameter p and the translation parameter q vary continuously, we have the following family of continuous wavelets [12]…”

Section: Laguerre Waveletsmentioning

confidence: 99%

“…To obtain numerical solutions of fractional and fractional-order delay differential equations, there are many numerical methods such as, Galerkin method, collocation method, operation matrix of integration method and operational matrix of differentiation, etc and different wavelets basis have been used. Modified Laguerre wavelets method are applied to delay differential equations of fractional-order [12]. The Legendre wavelet method is presented by Rehman and Khan [13].…”

Section: Introductionmentioning

confidence: 99%

Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations

Seçer

Özdemir

2019

Therm sci

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The application of modified Laguerre wavelet with respect to the given conditions by Galerkin method to an approximate solution of fractional and fractional-order delay differential equations is studied in this paper. For the concept of fractional derivative is used Caputo sense by using Riemann-Liouville fractional integral operator. The presented method here is tested on several problems. The approximate solutions obtained by presented method are compared with the exact solutions and is shown to be a very efficient and powerful tool for obtaining approximate solutions of fractional and fractional-order delay differential equations. Some tables and figures are presented to reveal the performance of the presented method.

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“…It is a functional differential equation with proportional delays. Due to its vast applications in science and engineering it took the attention of many researchers to propose numerous schemes for the simulation of Pantograph equation (Rahimkhani, et al, 2018;Iqbal, Saeed & Mohyud-Din, 2015;Raja, 2014;Bharawy, Al-Zahrani, Alhamed & Baleanu, 2014;Saadatmandi & Dehghan, 2009;Isah, Phang & Phang, 2017;Syam & Jaradat, 2017).…”

Section: Introductionmentioning

confidence: 99%

Series solution of fractional Pantograph equations via Taylor series

Shaikh¹,

Hanif²,

Khan³

et al. 2019

3C Tecnología

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This article is devoted to develop a numerical approximation called Taylor minimization method for initial and boundary value fractional Pantograph equations, which governs the modelling of the train system, with neutral and multi-term delays. Taylor optimization technique is basically composed of truncated Taylor series approximation of unknown function while employment of procedure is accompanied by an optimization strategy that is simulated annealing for carrying out the learning phase of unknown Taylor series coefficients. The proposed technique is implemented on various models of Pantograph equations to study the applicability and effectiveness of the planned scheme while error analysis and comparison with previous methods are performed to validate the results. To measure the capability of convergence the data for 100 numbers of independent runs is demonstrated in the form of pictorial presentation.

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Modified Laguerre Wavelets Method for delay differential equations of fractional-order

Cited by 46 publications

References 9 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

Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations

Series solution of fractional Pantograph equations via Taylor series

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