An investigation of fractional Bagley-Torvik equation

Fazli, Hossein; Nieto, Juan J.

doi:10.1515/math-2019-0040

Cited by 19 publications

(17 citation statements)

References 27 publications

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“…Comparison of the profiles of u(t) versus t represented by Equation(9) and Equation(18) of[31].FromFigure 3, Equation (10) is equivalent to the results shown in Exp. (3.1) by Fazli and Nieto[33] by different technique.Again, for λ 2 = λ 1 = λ 0 = 1, u 0 = u 1 = 1, α = 0.5, β = 2 and f (t) = t 3 + + 7t + 1, Equation(7)reduces to…”

mentioning

confidence: 95%

“…More recently, Bansal and Jain [31] discussed analytical solution of BTE by a generalized differential transform method, Anjara and Solofoniaina [32] solved the equation by Adomian's method, Fazli and Nieto [33] proved the results for the existence and approximations of the solutions of BTE. Gamel et al [34] used the Chelyshkov-Tau approach for solving BTE.…”

Section: Introductionmentioning

confidence: 99%

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An Investigation of Fractional Bagley–Torvik Equation

Zafar

Kudra

Awrejcewicz

2019

Entropy

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In this article, we will solve the Bagley–Torvik equation by employing integral transform method. Caputo fractional derivative operator is used in the modeling of the equation. The obtained solution is expressed in terms of generalized G function. Further, we will compare the obtained results with other available results in the literature to validate their usefulness. Furthermore, examples are included to highlight the control of the fractional parameters on he dynamics of the model. Moreover, we use this equation in modelling of real free oscillations of a one-degree-of-freedom mechanical system composed of a cart connected with the springs to the support and moving via linear rolling bearing block along a rail.

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mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

An Investigation of Fractional Bagley–Torvik Equation

Zafar

Kudra

Awrejcewicz

2019

Entropy

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“…The existence and uniqueness results of fractional-order differential equations (FDEs) have been investigated extensively in the literature. Some of them are presented as follows: Fazli and Nieto [11] investigated the existence and uniqueness of the solution of FDEs of Bagley-Torvik type by considering the existence of coupled lower and upper solutions. Pang et al [12] investigated the existence and uniqueness of the solution of the generalized FDEs with initial conditions by proposing a novel max-metric containing a Caputo derivative.…”

Section: Introductionmentioning

confidence: 99%

Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials

et al. 2022

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In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational matrix of the fractional-order derivatives in the Caputo sense, which is then used with spectral tau and spectral collocation methods to reduce the FDEs to a system of algebraic equations. Several numerical examples are given to show the accuracy of this method. These examples show that the obtained results have good agreement with the analytical solutions in both linear and non-linear FDEs. In addition to this, the numerical results obtained by using our method are compared with the numerical results obtained otherwise in the literature.

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“…where λ i , i = 0, 1, 2 are constants, λ 2 = 0, f is a real-valued function and D ν is the fractional derivative ν = 1 2 or ν = 3 2 . Because of the applications of the Bagley-Torvik equation, many researches tackled the problem of finding the analytic and numerical solution of this equation [3,[10][11][12][13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Analytic and numerical solutions of discrete Bagley–Torvik equation

et al. 2021

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In this research article, a discrete version of the fractional Bagley–Torvik equation is proposed: $$ \nabla _{h}^{2} u(t)+A{}^{C} \nabla _{h}^{\nu }u(t)+Bu(t)=f(t),\quad t>0, $$ ∇ h 2 u ( t ) + A C ∇ h ν u ( t ) + B u ( t ) = f ( t ) , t > 0 , where $0<\nu <1$ 0 < ν < 1 or $1<\nu <2$ 1 < ν < 2 , subject to $u(0)=a$ u ( 0 ) = a and $\nabla _{h} u(0)=b$ ∇ h u ( 0 ) = b , with a and b being real numbers. The solutions are obtained by employing the nabla discrete Laplace transform. These solutions are expressed in terms of Mittag-Leffler functions with three parameters. These solutions are handled numerically for some examples with specific values of some parameters.

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An investigation of fractional Bagley-Torvik equation

Cited by 19 publications

References 27 publications

An Investigation of Fractional Bagley–Torvik Equation

An Investigation of Fractional Bagley–Torvik Equation

Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials

Analytic and numerical solutions of discrete Bagley–Torvik equation

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