Numerical Fractional Differentiation: Stability Estimate and Regularization
Ailin Qian *
Department of Mathematics and Statistics, Hubei University of Science and Technology, Xianning, Hubei, 437100, People's Republic of China.
Guangfu Wang
School of Basic Science, East China Jiaotong University, Nanchang, Jiangxi, 330013, People's Republic of China.
*Author to whom correspondence should be addressed.
Abstract
It is well known that the problem of fractional differentiation n is an ill-posed problem. So far there exists many approximation methods for solving this problem. In this paper we prove a stability estimate for a problem of fractional differentiation. Based on the obtained stability estimate, we present a Tikhonov regularization method and obtain the error estimate. According to the optimality theory of regularization, the error estimates are order optimal. Numerical experiment shows that the regularization works well.
Keywords: Fractional differentiation, ill-posed problems, Tikhonov regularization, stability, estimate, error estimate