Fractional-order Fibonacci-hybrid functions approach for solving fractional delay differential equations

Sabermahani, Sedigheh; Ordokhani, Yadollah; Yousefi, S. A.

doi:10.1007/s00366-019-00730-3

Cited by 21 publications

(12 citation statements)

References 35 publications

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“…Similar works [9][10][11] were also carried out by using Chebyshev polynomials approximation for pantograph-type differential and integro-differential equations. Rabiei and Ordokhani 12 and Sabermahani et al 12,13 applied Boubaker and Fibonacci polynomials as basic functions for solving such problem. More recently, in Saeed et al 14 and Rahimkhani et al 15 employed wavelet as the basis functions for dealing with the fractional pantograph differential equations.…”

Section: Introductionmentioning

confidence: 99%

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

Yang¹,

Lv²

2020

Math Methods in App Sciences

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This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.

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Section: Introductionmentioning

confidence: 99%

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

Yang¹,

Lv²

2020

Math Methods in App Sciences

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“…Sabermahani et al (2020a) introduced a formulation for fractional-order general Lagrange scaling functions and employed these functions for solving fractional differential equations. Sabermahani et al (2020b) used hybrid functions of block-pulse and fractional-order Fibonacci polynomials for obtaining the approximate solution of fractional delay differential equations. Sabermahani et al (2020c) presented a numerical technique based on two-dimensional Müntz-Legendre hybrid functions to solve fractional-order partial differential equations in the sense of the Caputo derivative.…”

Section: Introductionmentioning

confidence: 99%

Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials

et al. 2020

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This paper addresses the application of generalized polynomials for solving nonlinear systems of fractional-order partial differential equations with initial conditions. First, the solutions are expanded by means of generalized polynomials through an operational matrix. The unknown free coefficients and control parameters of the expansion with generalized polynomials are evaluated by means of an optimization process relating the nonlinear systems of fractionalorder partial differential equations with the initial conditions. Then, the Lagrange multipliers are adopted for converting the problem into a system of algebraic equations. The convergence of the proposed method is analyzed. Several prototype problems show the applicability of the algorithm. The approximations obtained by other techniques are also tested confirming the high accuracy and computational efficiency of the proposed approach.

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“…Several numerical methods to solve the delay differential equations has been presented such as hybrid-Chebyshev polynomials, 14 fractional-order general Lagrange scaling functions, 15 hybrid-Legendre polynomials, hybrid-Taylor polynomials, 16 and fractional-order hybrid functions. 17 Another important class of delay problems is optimal control problems that are used to model many of the phenomena, which we mention here.…”

Section: Introductionmentioning

confidence: 99%

“…Several numerical methods to solve the delay differential equations has been presented such as hybrid‐Chebyshev polynomials, fractional‐order general Lagrange scaling functions, hybrid‐Legendre polynomials, hybrid‐Taylor polynomials, and fractional‐order hybrid functions …”

Section: Introductionmentioning

confidence: 99%

Fibonacci wavelets and their applications for solving two classes of time‐varying delay problems

Sabermahani

Ordokhani

Yousefi

2019

Optim Control Appl Methods

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Summary In this paper, a numerical method for solving time‐varying delay equations and optimal control problems with time‐varying delay systems is discussed. This method is based upon Fibonacci wavelets and Petrov‐Galerkin method. To solve these problems, first, the Fibonacci wavelets are presented. With the aid of operational matrices of integration and delay for Fibonacci wavelets and using Petrov‐Galerkin method and Newton's iterative method, we solve two classes of time‐varying delay problems, numerically. The approximate solutions achieved by this method satisfy all the initial conditions. In addition, an estimation of the error is given. Numerical results are included to demonstrate the accuracy and applicability of the present technique.

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Fractional-order Fibonacci-hybrid functions approach for solving fractional delay differential equations

Cited by 21 publications

References 35 publications

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials

Fibonacci wavelets and their applications for solving two classes of time‐varying delay problems

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