Application of fractional calculus in ground heat flux estimation

Abstract

Ground (soil) heat flux is important physical factor primarily because of its role in surface energy balance, analysis of atmospheric boundary layer and land surface-atmosphere interaction. Direct measurement of this property is often associated with difficulties arising from need for adequate calibration of measuring devices, determination of proper depth for probes, upward water migration and accumulation below measuring plates to lack of understanding of the governing thermal processes occurring at the ground surface. In the following paper approach for inferring heat flux indirectly, from known ground surface temperature time-dependant functions, using previously developed fractional diffusion equation for ground heat conduction is elaborated. Fractional equation is solved for two, most frequently encountered harmonic surface temperature functions. Yielded results were compared with analytic solutions. Validation results indicate that solutions obtained with fractional approach closely correspond to analytic solutions with remark that former are more general, containing the term covering the transitional effect

Discrete fractional diffusion model with two memory terms

“…Particularly anomalous diffusion is superior to classical Darcy and Fourier laws. One of the most crucial models is the time fractional diffusion equation [9][10][11] and some efficient methods in both analytical and numerical methods were developed in the past ten years [12][13][14][15][16][17][18][19][20]. Some researchers pointed out that the fractal time or the discrete time can model anomalous diffusion or random walk.…”

Fractional Approach for Estimating Sap Velocity in Trees

“…Likewise, the heat transfer process was modelled in [14] considering a part of the heat flux dispersed in the air around the material under study, whereas the authors of [7] analyzed heat conduction through a sphere. Additional modeling of thermal processes can be found, e.g., in [1], [2], [6], [9], [13].…”