Approximate solution of the Bagley-Torvik equation by hybridizable discontinuous Galerkin methods

In this paper, we introduce a hybridizable discontinuous Galerkin method for numerically solving a boundary value problem associated with the Bagley-Torvik equation that arises in the study of the motion of a plate immersed in a Newtonian fluid. One of the main features of these methods is that they are efficiently implementable since it is possible to eliminate all internal degrees of freedom and obtain a global linear system that only involves unknowns at the element interfaces. We display the results of a series of numerical experiments to ascertain the performance of the method.

[1]  Svatoslav Staněk,et al.  Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation , 2012 .

[2]  Fatih Celiker,et al.  Hybridizable Discontinuous Galerkin Methods for Timoshenko Beams , 2010, J. Sci. Comput..

[3]  Fatih Celiker,et al.  A projection-based error analysis of HDG methods for Timoshenko beams , 2010, Math. Comput..

[4]  Santanu Saha Ray,et al.  Analytical solution of the Bagley Torvik equation by Adomian decomposition method , 2005, Appl. Math. Comput..

[5]  Fatih Celiker,et al.  Element-by-Element Post-Processing of Discontinuous Galerkin Methods for Timoshenko Beams , 2006, J. Sci. Comput..

[6]  Suayip Yüzbasi,et al.  Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials , 2013, Appl. Math. Comput..

[7]  Fatih Celiker,et al.  Locking-Free Optimal Discontinuous Galerkin Methods for a Naghdi-Type Arch Model , 2012, J. Sci. Comput..

[8]  M. Rehman,et al.  A numerical method for solving boundary value problems for fractional differential equations , 2012 .

[9]  Fatih Celiker,et al.  Superconvergence of the numerical traces of discontinuous Galerkin and Hybridized methods for convection-diffusion problems in one space dimension , 2007, Math. Comput..

[10]  Şuayip Yüzbaşı,et al.  A numerical approximation for Volterra’s population growth model with fractional order , 2013 .

[11]  Z. Hammouch,et al.  Approximate analytical solutions to the Bagley-Torvik equation by the Fractional Iteration Method , 2012 .

[12]  M. Anwar,et al.  A collocation-shooting method for solving fractional boundary value problems , 2010 .

[13]  Fatih Celiker,et al.  HDG Methods for Naghdi Arches , 2014, J. Sci. Comput..

[14]  Weiwei Zhao,et al.  Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations , 2010, Appl. Math. Comput..

[15]  Aydin Kurnaz,et al.  The solution of the Bagley-Torvik equation with the generalized Taylor collocation method , 2010, J. Frankl. Inst..

[16]  R. Bagley,et al.  On the Appearance of the Fractional Derivative in the Behavior of Real Materials , 1984 .

[17]  Şuayip Yüzbaşı,et al.  Numerical solution of the Bagley–Torvik equation by the Bessel collocation method , 2013 .

[18]  Fatih Celiker,et al.  Element-by-element post-processing of discontinuous galerkin methods for naghdi arches , 2011 .

[19]  Santanu Saha Ray,et al.  On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation , 2012, Appl. Math. Comput..

[20]  Z. Wang,et al.  General solution of the Bagley–Torvik equation with fractional-order derivative , 2010 .

[21]  Jan Cermák,et al.  Exact and discretized stability of the Bagley-Torvik equation , 2014, J. Comput. Appl. Math..

[22]  Raytcho D. Lazarov,et al.  Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

[23]  Kassem Mustapha,et al.  A hybridizable discontinuous Galerkin method for fractional diffusion problems , 2014, Numerische Mathematik.

[24]  I. L. El-Kalla,et al.  Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations , 2010 .

[25]  N. Ford,et al.  Numerical Solution of the Bagley-Torvik Equation , 2002, BIT Numerical Mathematics.

[26]  Huazhong Tang,et al.  High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations , 2012 .