Acta Mechanica Sinica

, Volume 31, Issue 2, pp 153–161

# 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

• Bo Yu
• Xiaoyun Jiang
• Haitao Qi
Research Paper

### 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.

## Keywords

Riemann–Liouville fractional derivative Generalized second grade fluid Inverse problem  Implicit numerical method Fractional sensitivity equation

## Notes

### Acknowledgments

The project was supported by the National Natural Science Foundation of China (Grants 11472161, 11102102, and 91130017), the Independent Innovation Foundation of Shandong University (Grant 2013ZRYQ002), and the Natural Science Foundation of Shandong Province (Grant ZR2014AQ015)

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