Derivation and numerical validation of the fundamental solutions for constant and variable-order structural derivative a
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Derivation and numerical validation of the fundamental solutions for constant and variable-order structural derivative advection–dispersion models Fajie Wang , Wei Cai, Bin Zheng and Chao Wang
Abstract. Various non-local structural derivative diffusion models have been proposed based on different kernel functions to describe the anomalous time dependence of the mean-squared displacements. In the present study, the fundamental solutions for constant and variable-order structural derivative advection–dispersion models are achieved via scaling transformation and the generalized non-Euclidean Hausdorff fractal distance. Comparative numerical investigations of the structural derivative models have been conducted to reveal the influences of various kernels via the meshless method of fundamental solutions. Numerical results verify the validity of the derived fundamental solutions and the rationality of the employed numerical method for structural derivative advection–dispersion models. Mathematics Subject Classification. 35R11, 74S40, 65L80, 60K50. Keywords. Anomalous diffusion, Structural derivative, Fundamental solutions, Kernel functions.
1. Introduction The diffusion processes have been widely observed in various fields, such as underground solute transport [1], magnetic resonance imaging [2] and seawater invasion [3]. Thus, it is of great importance to understand the diffusion process to make full use of the solute transport or to control the contaminant diffusion. The classical diffusion process is believed to follow the Fick’s law, and the mean-squared displacement (MSD) exhibits as a linear function to the time t, i.e., x2 (t) ∝ t. Recently, the anomalous diffusion phenomena, known as non-Markovian processes, have been widely found in the pollutant and gas transport in porous media. Experimental results reveal that the porous media with heterogeneity will lead to the non-Gaussian distribution and power law tailing. The MSD for such anomalous diffusion behaves in the form of power law function of t as [4–7] x2 (t) ∝ tα ,
(1)
where the sub-diffusion is characterized as 0 < α < 1, while the super-diffusion corresponds to α > 1. It has been witnessed that a couple of mathematical and physical models, such as the fractional derivative and Hausdorff derivative diffusion models, have been proposed to describe the power law dependence of MSD [15–17]. Ultraslow diffusion is widely investigated in recent years and known to diffuse more slowly than subdiffusion. Usually, the logarithmic function is employed to characterize such diffusion processes [8–10], namely, λ
x2 (t) ∝ (ln t) ,
(2)
where λ > 0. If λ = 4, it reduces back into the classical Sinai diffusion law [9], while λ = 0.5 correlates with the well-known Harris law [11]. Subsequently, the inverse Mittag–Leffler function has also been 0123456789().: V,-vol
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applied to characterize ultraslow diffusion [12] λ x2 (t) ∝ Eα−1 (t) ,
(3)
where Eα−1 (t) is the inverse function of the
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