Numerical method for the estimation of the fractional parameters in the fractional mobile/immobile advection–diffusion model

ABSTRACT

In this paper, we mainly investigate the numerical identification of the fractional parameters in the fractional mobile/immobile advection–diffusion model. For the direct problem, two efficient numerical schemes with second-order spatial accuracy and fourth-order spatial accuracy are derived, respectively. The unconditional stability and convergence of the numerical schemes are analysed rigorously by the Fourier analysis method. For the inverse problem, we first propose a novel numerical scheme to compute the fractional sensitivity matrix, and then employ the nonlinear Levenberg–Marquardt iterative method to estimate the fractional parameters. Finally, numerical examples are given to illustrate the effectiveness and accuracy of the proposed numerical schemes, and the numerical identification of the fractional parameters is also presented in tabular form, which demonstrate the effectiveness of the proposed numerical method for the identification of the fractional parameters in the fractional mobile/immobile advection–diffusion model.

KEYWORDS:

• Fractional mobile/immobile advection–diffusion model
• finite difference
• stability and convergence
• fractional sensitivity matrix
• parameter estimation

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M32
• 26A33

Acknowledgements

The authors thank the referees and the editors for their valuable detailed comments and helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The project was supported by China Postdoctoral Science Foundation [Grant No. 2016M602127], Shandong Provincial Natural Science Foundation, China [Grant No. ZR2017MA030 and ZR2015AM011] and the National Natural Science Foundation of China [Grant No. 11771254 and 11672163].