A fast multiscale Galerkin algorithm for solving boundary value problem of the fractional Bagley–Torvik equation

Chen, Jian

doi:10.1186/s13661-020-01391-8

Cited by 2 publications

(1 citation statement)

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“…To solve the fractional Bagley-Torvik equation, several numerical solutions and analytical solutions have been used. Hybrid functions approximation [1] fractional-order Legendre collocation method [2], Haar wavelet [3], Laplacetransform [4], Laguerre polynomials [5], shifted Chebyshev operational matrix [6], Legendre artificial neural network method [7], Chebyshev collocation method [8], the fractional Taylor method [9], exponential integrators [10], Gegenbauer wavelet method [11], Müntz-Legendre polynomials [12], discrete spline methods [13], Hermit solution [14], local discontinuous Galerkin approximations [15], numerical inverse Laplace transform [16], generalized Fibonacci operational tau algorithm [17], Jacobi collocation methods [18], polynomial least squares method [19], and fast multiscale Galerkin algorithm [20] are methods by which Bagley-Torvik equation solved numerically. 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.…”

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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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