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Acta Mechanica Sinica

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01-04-2015 | Research Paper

An inverse problem to estimate an unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid

Authors: Bo Yu, Xiaoyun Jiang, Haitao Qi

Published in: Acta Mechanica Sinica | Issue 2/2015

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Abstract

In this paper, we propose a numerical method to estimate the unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid. The implicit numerical method is employed to solve the direct problem. For the inverse problem, we first obtain the fractional sensitivity equation by means of the digamma function, and then we propose an efficient numerical method, that is, the Levenberg–Marquardt algorithm based on a fractional derivative, to estimate the unknown order of a Riemann–Liouville fractional derivative. In order to demonstrate the effectiveness of the proposed numerical method, two cases in which the measurement values contain random measurement error or not are considered. The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid.

Graphical Abstract

In this paper, we propose a numerical method to estimate the unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes first problem for a heated generalized second grade fluid. The implicit numerical method is employed to solve the direct problem. For the inverse problem, we obtain the fractional sensitivity equation by means of the digamma function. Numerical simulations demonstrate the effectiveness of the proposed method.

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Title: An inverse problem to estimate an unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid
Authors: Bo Yu
Xiaoyun Jiang
Haitao Qi
Publication date: 01-04-2015
Publisher: The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences
Published in: Acta Mechanica Sinica / Issue 2/2015
Print ISSN: 0567-7718
Electronic ISSN: 1614-3116
DOI: https://doi.org/10.1007/s10409-015-0408-7

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