Inverse Problem of an Anomalous Diffusion Model Employing Lightning Optimization Algorithm

The classical concept of diffusion characterized by Fick’s law is well suited for describing a wide class of practical problems of interest. Nevertheless, it has been observed that it is not enough to properly represent other relevant applications of practical interest. When in a system of particles their spreading is slower or faster than predicted by the classical diffusion model, such a phenomenon is referred to as anomalous diffusion. Time fractional, space fractional and even space-time fractional equations are widely used to model phenomena such as solute transport in porous media, financial modelling and cancer tumor behavior. Considering the effects of partial and temporary retention in dispersion processes a new analytical formulation was derived to simulate anomalous diffusion. The new approach leads to a fourth-order partial differential equation (PDE) and assumes the existence of two concomitant fluxes. This work investigates the behavior of the bi-flux approach in one dimensional (1D) medium evaluating the mean square displacement for different cases in order to classify the diffusion process in normal, sub-diffusive or super-diffusive.

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Direct and Inverse Problems for a Fourth Order Anomalous Diffusion Model

Junior

Bevilácqua

Neto

2020

DDF

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The second order equation (also known as Fick’s equation) is derived from a classical well-known theory, but it is not enough to model all applications of interest. Recently, fractional equations and higher order equations began to receive more attention, demanding increased research efforts. They are used to simulate the diffusion process in many important applications in sciences, such as chemistry, heat and mass transfer, biology and ecology. In this work, the sensitivity analysis is performed for a recently developed anomalous diffusion model in order to evaluate the possibility of estimating a set of parameters that are part of the fourth order equation model, including the parameters representing the variation of the fraction of particles that are allowed to diffuse using a sigmoid function. Finally, after the sensitivity analysis the Inverse Problem approach is used to estimate viable parameters that are necessary for simulation in the cases considered. The differential equation was approximated using the Finite Difference Method, and that solution was implemented in the RStudio platform. The Sensitivity Matrix was calculated and the Inverse Problem was solved using the same RStudio platform, and the Simulated Annealing Method.

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Inverse Problem of an Anomalous Diffusion Model Employing Lightning Optimization Algorithm

Cited by 3 publications

References 16 publications

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Direct and Inverse Problems for a Fourth Order Anomalous Diffusion Model

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