Abstract
A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.
Appendix A Appendices
A.1. Monotonicity of functions fi(t). Let us investigate the following functions for monotonicity:
At the left point of the interval
Therefore, the functions fk(t) are strictly monotone and reach the smallest values, which are the largest in absolute values, as
where cik are defined by formulas (4.12).
A.2. Estimate of sum
Since the common term of the series Σ can be represented as |(m+1)γ–mγ–(mγ–(m –1)γ)|, we consider the function
Examining the monotonicity of g(x) we have
Since 1 + 1/x >1 for x > 0, then for γ ϵ (0,1) the inequality
holds. Consequently, g'(x) < 0, which means g(x) decreases monotonically, and g(m + 1) – g(m) < 0. Then the expression for the series Σ can be rewritten without the modulus sign in the form
Now, we write the elements of this series
then we obtain a formula for partial sums:
Transforming the expression for Sm and using the equivalence of infinitesimal functions, we calculate the limit
this implies
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