Fractional-order shifted Legendre collocation method for solving non-linear variable-order fractional Fredholm integro-differential equations

Abdelkawy, Mohamed; Amin, A. Z. M.; Lopes, António M.

doi:10.1007/s40314-021-01702-4

Cited by 13 publications

(8 citation statements)

References 60 publications

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“…As stated previously by the choice of parameter θ ̸ = 1 2 in many cases we can not solve the problem with the VSM. Hence, the proposed method can be an appropriate tool to solve the problem for different choices of θ ∈ (0, 1).…”

Section: The Choice Of Parameter αmentioning

confidence: 99%

“…It should be noted that even for θ = 1 2 , the ODE systems ( 21) and ( 22) are nonlinear. Also, the solutions…”

Section: The Generalized Model For An Oligopoly Marketmentioning

confidence: 99%

“…The authors employed the FLFs to obtain the numerical solution of the fractional-order differential equations. Abdelkawy et al [1] used FLFs to solve the nonlinear systems of variable order fractional Fredholm integro-differential equations. Similar to FLFs, the fractional-order shifted Jacobi functions have been constructed in Abdelkawy et al [2] employed to find the approximate solution of variable-order fractional integro-differential equations.…”

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confidence: 99%

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Generalized model of stochastic common property fishery differential game: Numerical study

Nikooeinejad,

Heydari

2024

Preprint

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In a generalized model of stochastic common property differential game problem, we have a more flexible framework that extends the model to handle the natural growth, price and cost functions for a wide range of parameters. Unlike previous studies that appeared in the literature, instead of keeping the model as parsimonious as possible, we consider the nonlinear resource stock dynamics and nonquadratic payoff functions, focusing on monopoly, duopoly and oligopoly games with a feedback information structure. It is observed that the choice of model parameters has an important influence on the nature and analytical behavior of solution of the nonlinear HJB PDEs (ODEs) and consequently on Nash equilibrium strategies for finite- (infinite-) horizon planning. We can not derive a closed-form solution for this class of generalized models. Hence, this paper presents a collocation method based on the fractional-order shifted Chelyshkov polynomials to solve the nonlinear Hamilton-Jacobi-Bellman PDEs (ODEs) in the generalized stochastic common property differential game problem with finite- (infinite-) horizon. Numerical results show that the choice of different parameters in the approximate solutions based on the fractional-order shifted Chelyshkov polynomials greatly affect the performance of the collocation method. Also, the results show that the firms' optimal extraction plans are strongly influenced by the natural growth, price and cost functions with respect to different parameters.

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Section: The Choice Of Parameter αmentioning

confidence: 99%

“…It should be noted that even for θ = 1 2 , the ODE systems ( 21) and ( 22) are nonlinear. Also, the solutions…”

Section: The Generalized Model For An Oligopoly Marketmentioning

confidence: 99%

mentioning

confidence: 99%

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Generalized model of stochastic common property fishery differential game: Numerical study

Nikooeinejad,

Heydari

2024

Preprint

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“…The author in [21] applied two schemes based on the Fibonacci operational matrix to treat the nonlinear fractional Klein-Gordon equation. The author in [22] employed the fractional-order shifted Legendre collocation method for a type of fractional Fredholm integro-differential equations. Another type of FDEs is treated using the implicit wavelet collocation method in [23]).…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

Abd-Elhameed

2023

Fractal Fract

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The main aim of this paper is to introduce a new class of orthogonal polynomials that generalizes the class of Chebyshev polynomials of the first kind. Some basic properties of the generalized Chebyshev polynomials and their shifted ones are established. Additionally, some new formulas concerned with these generalized polynomials are established. These generalized orthogonal polynomials are employed to treat the multi-term linear fractional differential equations (FDEs) that include some specific problems that arise in many applications. The basic idea behind the derivation of our proposed algorithm is built on utilizing a new power form representation of the shifted generalized Chebyshev polynomials along with the application of the spectral Galerkin method to transform the FDE governed by its initial conditions into a system of linear equations that can be efficiently solved via a suitable numerical solver. Some illustrative examples accompanied by comparisons with some other methods are presented to show that the presented algorithm is useful and effective.

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“…Spectral methods [19][20][21] have been widely used in various fields for forty years. Firstly, Fourier-expanded spectral techniques have been used in a few contexts, like periodic boundary conditions and simple geometric areas.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

Amin,

Abdelkawy,

Solouma

et al. 2023

Fractal Fract

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One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings.

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Fractional-order shifted Legendre collocation method for solving non-linear variable-order fractional Fredholm integro-differential equations

Cited by 13 publications

References 60 publications

Generalized model of stochastic common property fishery differential game: Numerical study

Generalized model of stochastic common property fishery differential game: Numerical study

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

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