Numerical restorability of parameter values of space-time fractional soil consolidation model

Abstract

At present, a large number of fractional differential models of migration processes in soils are developed. Their practical application largely depends on the possibility to determine the values of their parameters. In this regard, we study the possibility of recovering the values of parameters for one such generalized model from noised data in order to assess the threshold of measurement accuracy, beyond which the complication of a model leads to an inability to distinguish its solutions from the solutions of simpler models. We consider the 1D fractional-order model of water head dissipation in water-saturated soil with linear deformation that includes the Caputo–Fabrizio derivative with respect to the time variable and the Riemann–Liouville derivative with respect to the space variable. Direct problems for this model are proposed to be solved by an optimized computational procedure based on a finite-difference scheme. Inverse problems of model’s parameter identification are solved using a multi-threaded Particle Swarm Optimization technique. The results of computational experiments showed that the values of model parameters can be restored with less than \(10\%\) relative error for the number of input water head values equal to 1000 and the level of noise less than \(5\%\). Our results also show that the order of the Riemann–Liouville derivative can be with an average relative error of less than \(3\%\) restored even at \(10\%\) level of noise and 40 input values, when the accuracy of other parameters’ restoration drops significantly.