Stability And Convergence Analysis Of Numerical Method For The Fractional-order Bagley-Torvik Equation

Fractional differential equations have been widely attention and application in mechanics of materials, fluid mechanics, the viscoelasticity system, biology, and many other scientific fields. The fractional order Bagley-Torvik equation, as a proto-type fractional differential equation with two derivatives, is more typical fractional order differential equation. It was first presented in1984by Torvik and Bagley in the modelling of the motion of a rigid plate immersed in a Newtonian viscous fluid. And then, many scholars have studied and expanded this kind of problem, the properties of fractional order Bagley-Torvik equation itself are obtained, and the stability and convergence properties of a few kinds of numerical methods for solving this problem are given.In this paper, we first research the stability and convergence of the numerical method (denoted by GL2), which is introduced by Shen Shujun and Liu Fawang in2004for solving the fractional Bagley-Torvik equation with homogeneous initial conditions as form where A,B,C are real constants with A≠0and function f(t)∈L(O,T), that is∫OT|f (t)|dt<∞.We proved that the method GL2is stable about initial values and the right hand side function and that this method is convergence with second order; Then we improved the above methods and constructed a new numerical scheme. It is proved that the order of truncation errors of the new scheme is more higher then GL2. Sev-eral numerical tests are carried out at the same time and the results further support the validity of the theoretical results.