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

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.