Abstract
We consider a ill-posed problem-fractional numerical differentiation with a new method. We propose Fourier truncation method to compute fractional numerical derivatives. A Holder-type stability estimate is obtained. A numerical implementation is described. Numerical examples show that the proposed method is effective and stable.
Highlights
In this paper we shall consider the problem of stably calculating the fractional derivative of a function f given in Lp R, D fx d dx f t dt x t (1.1)for 0,1
We propose Fourier truncation method to compute fractional numerical derivatives
The process of numerical fractional differentiation is well known to be an ill-posed problem [1,2,3], and it has been discussed by many authors, and a large number of different solution methods have been proposed
Summary
In this paper we shall consider the problem of stably calculating the fractional derivative of a function f given in Lp R , D f. The process of numerical fractional differentiation is well known to be an ill-posed problem [1,2,3], and it has been discussed by many authors, and a large number of different solution methods have been proposed. The idea of Fourier truncation method is very simple and natural: since the ill-posedness of fractional numerical differentiation is caused by the high frequency components, we cut off them. Such a similar idea of solving numerical differentiation has occurred in [10,11]. Fourier truncation method is more direct, natural and simple
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