Numerical Methods Of Multi-order Fractional Ordinary Differential Equations And Applications

The use of fractional differential and integral operator in mathematical has become increasingly widespread in recent years. Various field of science and engineering involved fractional calculation. For example, system biology [25], physics [16], Chemistry and biochemistry [26], hydrology application [14], fractional-order controllers [24], polymer rheology [20], and fractional derivative viscoelastic [18]. Nonlinear dynamical systems with fractional damping also play an important role in engineering, seismic wave attenuation and polymer rheology [20]. The most significant advantage of the fractional order models in comparison with integer-order models is based on important fundamental physical considerations. So viscoelastic models involving fractional derivatives instead of common derivatives are a good research issue. It is known that fractional model of viscous damping have long been attracted the attention of investigators. However, because of the absence of appropriate mathematical methods, numerical methods and theoretical analysis of fractional calculation are very difficult tasks.In this paper, we consider the numerical method of multi-term fractional ordinary differential equations. In the second chapter, The fractional endolymph equation is considered, the existence and uniqueness of solution for the fractional-order Endolymph equation is given. The analytical solution of the fractional-order endolymph equation is derived by the corresponding Green's function. And we can obtain its exact solution by the operational method. Obviously, it is difficult showing its analytical solution and its exact solution numerically. Then the Predictor-Corrector method is proposed for solving the fractional differentional equations of endolymph for its numer- ical solution. In the third chapter, the nonlinear fractional dynamical systems with fractional damping is considered, the problem is transferred into a system of fractional-order ordinary differential equations. A computationally effective fractional Predictor-Corrector method is used to simulate it. Numerical examples are presented in every chapter, which verify the efficiency of the above numerical method. The techniques can also be applied to deal with other generic fracitonal-order ordinary equations.