Numerical Solution of the Bagley-Torvik Equation

By Kai Diethelm, Neville J. Ford. BIT Numerical Mathematics, 2002

295 citations recorded by the source corpus.

We consider the numerical solution of the Bagley-Torvik equation Ay″(t) + BD*3/2y(t) + Cy(t) = f(t), as a prototype fractional differential equation with two derivatives. Approximate solutions have recently been proposed in the book and papers of Podlubny in which the solution obtained with approximate methods is compared to the exact solution. In this paper we consider the reformulation of the Bagley-Torvik equation as a system of fractional differential equations of order 1/2. This allows us to propose numerical methods for its solution which are consistent and stable and have arbitrarily high order. In this context we specifically look at fractional linear multistep methods and a predictor-corrector method of Adams type.

Subfields: Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/e072e55678ca3927b5861689a34860db7a94ab6b

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Citations in the corpus

New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions (2014, background): “In particular, let y(x) = y1(x), the system that we shall consider is of the form [12] D 1 2 y1 (x) = y2 (x) , D 1 2 y2 (x) = y3 (x) , D 1 2 y3 (x) = y4 (x) , D 1 2 y4 (x) = −y1 (x)− y4 (x) + x+ 1, (6.”
A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations (2011, methodology): “The Bagley–Torvik equation is numerically considered by Podlubny [2] and Diethelm and Ford [28].”
Finite difference/spectral approximations for the time-fractional diffusion equation (2007, methodology): “Different models using FPDEs have been proposed [4,14,16], and there has been significant interest in developing numerical schemes for their solution.”
Numerical analysis for distributed-order differential equations (2009, methodology): “Step 2 finally will consist of solving the system we have constructed numerically by any of the available methods; one could for example use the Adams method of [16] or the backward differentiation scheme of [9] or a fractional multistep method (see [20, 21, 22, 14]) which is likely to be efficient ”
Analytical approximate solutions for nonlinear fractional differential equations (2002, methodology): “For l 1⁄4 2, m 1⁄4 3 2 , (39) is the Bagley–Tovik equation which describes the motion of a rigid plate immersed in a Newtonian fluid [18] and the initial value problem has been solved analytically using the Laplace transform [1], and numerically [19] using the approach developed in [20].”
Numerical solution of multi‐term fractional differential equations (2009, unknown): “However, much of the published work has been concerned with linear single term equations, with those of order less than unity mostly investigated [7] or with multi-term FDEs, whose order obeys to certain constraint [17], or of a concrete form [5].”
A new operational matrix for solving fractional-order differential equations (2010, background): “As the first example, we consider the following initial value problem in the case of the inhomogeneous Bagley–Torvik equation [44] D2y(x)+ D 3 2 y(x)+ y(x) = 1+ x, y(0) = 1, y(0) = 1.”