Legendre wavelet method for fractional delay differential equations

Yuttanan, Boonrod; Razzaghi, Mohsen; Vo, Thieu N.

doi:10.1016/j.apnum.2021.05.024

Cited by 28 publications

(18 citation statements)

References 61 publications

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“…In Table 8, we give comparison results for the acquired absolute errors of the approximate solution between the proposed method, the Bernoulli wavelet method [44], the spectral method based on a modification of hat functions [49], and the Legendre wavelet method [42] for c = 1/2. From Table 8, we can see that the present method provides higher efficiency even for a small number of basis elements from the fractional Taylor vector.…”

Section: Examplementioning

confidence: 99%

“…For most FDDEs, exact solutions are not known. Therefore, various numerical techniques such as the Legendre wavelet method [42], the Chebyshev operational matrix method [43], the Bernoulli wavelets method [44], the shifted Gegenbauer-Gauss collocation method [45], the Haar wavelet collocation method [46], the modified operational matrix method [47], the discontinuous Galerkin method [48], spectral methods [49], etc., have been developed and applied to provide approximate solutions.…”

Section: Introductionmentioning

confidence: 99%

“…The results for the absolute errors of the approximate solutions provided by the present method and the methods of[42,44,49].…”

mentioning

confidence: 96%

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Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

Avcı

2021

Fractal Fract

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In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The main characteristic of this approach is, by utilizing the operational matrix of fractional integration, to reduce the given differential equation to a set of algebraic equations with unknown coefficients. This equation system can be solved efficiently using a computer algorithm. A bound on the error for the best approximation and fractional integration are also given. Several examples are given to illustrate the validity and applicability of the technique. The efficiency of the presented method is revealed by comparing results with some existing solutions, the findings of some other approaches from the literature and by plotting absolute error figures.

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Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

Avcı

2021

Fractal Fract

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“…Numerical methods offered a powerful substitute means for solving the DEs under the given initial conditions. Numerous methods have been developed in recent years to solve fractional-order differential equations (FODEs), including the homotopy perturbation method [1], the differential transform method [2], the operational matrix method [3], the conformable Shehu transform decomposition method [4], the variational iteration method [5], the Jacobi collocation method [6], the conformable Shehu transform iterative method [7], the spectral tau method [8], the Legendre wavelet method [9], the fractional natural decomposition method [10], the power series method with the conformable operator [11], and the Chebyshev polynomial method [12].…”

Section: Introductionmentioning

confidence: 99%

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Liaqat

Khan

Akgül

et al. 2022

Journal of Function Spaces

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Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor’s series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.

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“…Stabilization problem of neutral FDDEs is given in [31]. Various numerical schemes [32,33,34,35] are designed by the researchers. Some issues related with the initialization of FDDEs are examined in [36].…”

Section: Introductionmentioning

confidence: 99%

Stability and Bifurcation Analysis of a Fractional Order Delay Differential Equation Involving Cubic Nonlinearity

Bhalekar,

Gupta

2022

Preprint

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Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equationWe provide linearization of this system in a neighbourhood of equilibrium points and propose linearized stability conditions. To discuss the stability of equilibrium points, we propose various conditions on the parameters δ, , p, q and τ . Even though there are five parameters involved in the system, we are able to provide the stable region sketch in the qδ−plane for any positive and p. This provides the complete analysis of stability of the system. Further, we investigate chaos in the proposed model. This system exhibits chaos for a wide range of delay parameter.

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Legendre wavelet method for fractional delay differential equations

Cited by 28 publications

References 61 publications

Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Stability and Bifurcation Analysis of a Fractional Order Delay Differential Equation Involving Cubic Nonlinearity

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