Fractional Derivatives And Its Applications

In recent years, the fractional order calculus is widely applied in various fields, and it has solved many problems which integer order calculus can’t explain very well.Because of the wide application of fractional order calculus, it has become one of the important contents recently. In this article, We mainly discuss the contents of two aspects.First of all, we give a numerical method to solve the fractional derivative under one dimensional scattered noise data which on the basis of the integer order calculus. The problems of solving derivatives are well-known to be ill-posed in the sense that small errors in the data might induce large errors in the computed derivative. In. this paper, we use the function approximation (Radial Basis Function-RBF) to approximate the exact function; at the same time we use, the Tikhonov regularization method to solve this ill-posed problem effectively. We adopt a priori rule or a posteriori rule of Morozov’s discrepancy principle in choosing the regularization parameter, and the stability analysis under two choices is given.Second, we present the application of fractional order derivatives in mechanics. We add a fractional element-spring-pot on the exist classical viscoelastic model to fit the viscoelastic of biological tissues, then we compare the model we proposed with the existing model. At the same time, we give the analysis of sensitivity of all viscoelastic parameters, comparing the different fitting results of the models.When discuss the method of solving fractional derivatives, we give the stability analysis under the two choices. Under the theoretical analysis and numerical examples, we found the results in prior is superior to Morozov’s discrepancy principle’s when choosing regularization parameter.When give to illustrate the application of fractional derivatives in mechanics, we found the result is we can also be able to get a good fitting results although the model we proposed is less a viscoelastic parameter on the basis of the exist model. From a computational cost consider, the model we proposed can be used for mathematical modeling and simulation in practical application.