Abstract
In this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.
Highlights
Fractional calculus has become a powerful mathematical tool to model a lot of physical processes
The branch of fractional calculus, that is devoted to differential equations containing simultaneously both the left and right fractional derivatives, is a research area where exists a large number of an unsolved/unexplored issues [15, 21, 29]
The errors generated by Method I are the biggest one, while for Method III are the smallest one
Summary
Fractional calculus has become a powerful mathematical tool to model a lot of physical processes. In this paper we focus on the Riesz-Caputo fractional operator of variable order with fixed memory and its application in a strong form of the space-fractional continuum (1D case), which is just a straightforward extension of the results presented in [45]. Such approach allows, when adopted to the mechanical applications, for clear physical interpretation and gives smooth passage to classical approach as a limit case [5, 48]. All presented numerical sachems were implemented in Python using library for real and complex floating-point arithmetic with arbitrary precision (http:// mpmath.org/)
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