The Numerical Solution of the Bagley-Torvik Equation by Exponential Integrators

Esmaeili, Shahrokh

doi:10.24200/sci.2017.4503

Cited by 6 publications

(14 citation statements)

References 24 publications

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“…We must mention that several other approaches have been however discussed in the literature: see, for instance, the generalized Adams methods [10], extensions of the Runge-Kutta methods [11], generalized exponential integrators [12,13], spectral methods [14,15], spectral collocation methods [16], methods based on matrix functions [17][18][19][20], and so on. In this paper, for brevity, we focus only on PI rules and FLMMs, and we refer the reader to the existing literature for alternative approaches.…”

Section: Multi-step Methods For Fdesmentioning

confidence: 99%

“…To avoid the solution of the nonlinear equations in Equation (12) for the evaluation of y n , a predictor-corrector (PC) approach is sometimes preferred, in which a first approximation of y n is predicted by means of the explicit PI rectangular rule (10) and hence corrected by the implicit PI trapezoidal rule (12) according to:…”

Section: Product-integration Rulesmentioning

confidence: 99%

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Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

2018

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Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines.

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Section: Multi-step Methods For Fdesmentioning

confidence: 99%

Section: Product-integration Rulesmentioning

confidence: 99%

Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

2018

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“…However, even if many numerical methods for FDEs can be extended to MTFDEs, delicate issues such as numerical stability, convergence or accuracy cannot be easily predicted in this case. Many authors have worked thoroughly on their numerical solution [15][16][17][18][19][20][21][22]. We restrict our attention to the linear case that includes important models, such as the Bagley-Torvik equation [23], the fractional oscillation equation [24] and many others.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

Popolizio

2018

Mathematics

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Abstract:Multiterm fractional differential equations (MTFDEs) nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods for fractional differential equations (FDEs) to this case. In this paper, we first transform the MTFDEs into equivalent systems of FDEs, as done by Diethelm and Ford; in this way, the solution can be expressed in terms of Mittag-Leffler (ML) functions evaluated at matrix arguments. We then propose to compute it by resorting to the matrix approach proposed by Garrappa and Popolizio. Several numerical tests are presented that clearly show that this matrix approach is very accurate and fast, also in comparison with other numerical methods.

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“…In particular, for α = 1/2 and β = 0, we have ω 5 (t) = 1 − 1 2 t 2 . Absolute errors of the 5th approximate solution for FPTE (17) and (18) are computed for α = 1/2, with selected nods of t with step size 0.16 and summarized in Table 3, while Table 4 shows the numerical results of the RPS algorithm and exponential integrators method (EIM) [36] for the parameter value β = 0.2 and different values of t in [0, 5]. Example 4.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Thus, according to the RPS algorithm, the 5th approximate solution of FBTEs (19) and (20) is given as ω 5 (t) = 1 − ϕ(0) − 1 2 t 4α Γ(4α+1) + 1 4 − 1 2 ϕ(0) t 5α Γ(5α+1) . The resulting values of the RPS algorithm and some numerical methods, including the Fermat Tau method (FTM) [38], the generalized Taylor method (GTM) [36], and the fractional Taylor method (FrTM) [37], for inputs t between 0 and 1 with a step of 0.1, are given in Table 5. From this table, it can be illustrated that the result obtained by our scheme is in good agreement with the state-of-the-art numerical solvers.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

et al. 2019

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Featured Application: Fractional differential equations play a significant role in modeling certain dynamical systems arising in many fields of applied sciences and engineering. In this paper, the authors develop an attractive analytic-numeric technique, residual power series (RPS) method, for solving fractional Bagley-Torvik equations with a source term involving Caputo fractional derivative. In regard to its simplicity, the method can be applicable to a wide class of fractional partial differential equations, fractional fuzzy differential equations, fractional oscillator equations, and so on.Abstract: Numerical simulation of physical issues is often performed by nonlinear modeling, which typically involves solving a set of concurrent fractional differential equations through effective approximate methods. In this paper, an analytic-numeric simulation technique, called residual power series (RPS), is proposed in obtaining the numerical solution a class of fractional Bagley-Torvik problems (FBTP) arising in a Newtonian fluid. This approach optimizes the solutions by minimizing the residual error functions that can be directly applied to generate fractional PS with a rapidly convergent rate. The RPS description is presented in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm in addressing some test problems. The results indicate that the RPS algorithm is reliable and suitable in solving a wide range of fractional differential equations applying in physics and engineering.

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The Numerical Solution of the Bagley-Torvik Equation by Exponential Integrators

Cited by 6 publications

References 24 publications

Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

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