Analytical solution of the generalized Bagley–Torvik equation

Pang, Denghao; Wei, Jiang; Du, Jiaxing; Niazi, Azmat Ullah Khan

doi:10.1186/s13662-019-2082-8

Cited by 8 publications

(5 citation statements)

References 32 publications

(41 reference statements)

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“…In the study of Alshammari et al [21], residual power series are used to obtain the numerical solution of a class of Bagley-Torvik problems in Newtonian fluid, and in the study of Karaaslan et al [22], using the discontinuous Galerkin method that can be combined in the equation of motion of a plate immersed in a Newtonian fluid, the numerical solution of Bagley-Torvik equation has been discussed. Analytical solutions of the generalized Bagley-Torvik equation [23], Sumudu transformation method [24], generalized differential transform [25], Sine-Gordon expansion method, and Bernoulli equation method [26] are analytical solutions for solving the Bagley-Torvik equation in this work.…”

Section: Introductionmentioning

confidence: 99%

Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

Akhlaghi,

Tavassoli Kajani,

Allame

2023

Advances in Mathematical Physics

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This paper uses Müntz orthogonal functions to numerically solve the fractional Bagley–Torvik equation with initial and boundary conditions. Müntz orthogonal functions are defined on the interval 0 , 1 and have simple and distinct real roots on this interval. For the function f ∈ L 2 0 , 1 , we obtain the best unique approximation using Müntz orthogonal functions. We obtain the Riemann–Liouville fractional integral operator for Müntz orthogonal functions so that we can reduce the complexity of calculations and increase the speed of solving the problem, which can be seen in the process of running the Maple program. To solve the fractional Bagley–Torvik equation with initial and boundary conditions, we use Müntz orthogonal functions and consider simple and distinct real roots of Müntz orthogonal functions as collocation points. By using the Riemann–Liouville fractional integral operator that we define for the Müntz orthogonal functions, the process of numerically solving the fractional Bagley–Torvik equation that is solved using Müntz orthogonal functions is reduced, and finally, we reach a system of algebraic equations. By solving algebraic equations and obtaining the vector of unknowns, the fractional Bagley–Torvik equation is solved using Müntz orthogonal functions, and the error value of the method can be calculated. The low error value of this numerical solution method shows the high accuracy of this method. With the help of the Müntz functions, we obtain the error bound for the approximation of the function. We have obtained the error bounds for the numerical method using which we solved the fractional Bagley–Torvik equation with initial and boundary conditions. Finally, we have given a numerical example to show the accuracy of the solution of the method presented in this paper. The results of solving this example using Müntz orthogonal functions and comparing the results with other methods that have been used the solve this example show the higher accuracy of the method proposed in this paper.

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Section: Introductionmentioning

confidence: 99%

Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

Akhlaghi,

Tavassoli Kajani,

Allame

2023

Advances in Mathematical Physics

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“…The Bagley-Torvik equation successfully simulated frequency-dependent damping substances utilising fractional derivatives of order (1/2 or 3/2). Several authors have been investigated the analytical and numerical solutions for Bagley-Torvik equations, see the papers [35][36][37][38][39][40][41] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Solvability of fractional differential equations with applications of Morgan Voyce polynomials

Kumar,

Singh,

Kumar

2024

Phys. Scr.

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In this research, a new computational approach is presented to address multi-order fractional differential equations, including the renowned Bagley–Torvik and Painlevé equations. These equations are pivotal in scientific and engineering realms, like modelling the movement of a submerged plate restricted in a Newtonian fluid and gas in a fluid, along with simulating the coupled oscillations. We utilise the collocation approach employing a novel operational matrix derived for Morgan-Voyce polynomials via the Atangana-Baleanu fractional derivative. Initially, we introduce the fractional differential matrix to convert the problem and its constraints into a system of algebraic equations with unknown coefficients. These coefficients aid in finding numerical solutions for the given equations. To assess the efficiency of proposed method, various examples are simulated utilising the proposed approach and the outcomes are compared with existing results for different derivatives.

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“…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. Abbas [13] studied the existence and uniqueness of the solution of FDEs by using Banach's contraction principle together with Krasnoselskii fixed point theorem.…”

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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Analytical solution of the generalized Bagley–Torvik equation

Cited by 8 publications

References 32 publications

Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

Solvability of fractional differential equations with applications of Morgan Voyce polynomials

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

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