Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation

Abstract

Abstract:The authors present a model of heat conduction using the Caputo fractional derivative with respect to time. The presented model was used to reconstruct the thermal conductivity coefficient, heat transfer coefficient, initial condition and order of fractional derivative in the fractional heat conduction inverse problem. Additional information for the inverse problem was the temperature measurements obtained from porous aluminum. In this paper, the authors used a finite difference method to solve direct problems and the Real Ant Colony Optimization algorithm to find a minimum of certain functional (solve the inverse problem). Finally, the authors present the temperature values computed from the model and compare them with the measured data from real objects.

Comparison of the Probabilistic Ant Colony Optimization Algorithm and Some Iteration Method in Application for Solving the Inverse Problem on Model With the Caputo Type Fractional Derivative

“…Mathematical fundaments and results concerning the convergence of the different versions of the ACO algorithm are included in [ 24 , 25 , 26 , 27 ]. Based on the analysis of the papers of various authors, see for example [ 28 , 29 , 30 , 31 , 32 ], as well as on the computation carried out by us (for example [ 7 , 17 , 20 , 22 , 31 ]) we come to the conclusion that using the artificial intelligence algorithms (ACO, RACO, RACO, IRM, etc.) in most cases does not require the application of the explicit regulatory procedures.…”

“…Generally speaking, the solution of the inverse heat conduction problem consists in such a choice of the model parameters (model input) so the given output temperature distribution is obtained. More information regarding the heat processes modeling in the porous materials, inverse heat conduction problems, and use of the fractional calculus for modeling these kind of processes may be found in [ 5 , 7 , 19 , 20 , 21 ].…”

New Iterative Method of Solving Nonlinear Equations in Fluid Mechanics

“…Fractional partial differential equations (FPDEs) are useful tools to model many physical phenomena such as nonlinear seismic oscillations, hydrodynamic movement, decay laws, diffusion processes, electrostatics, electrodynamics, elasticity [1][2][3][4]. For example, FPDEs have been used to model filtration flow in a porous medium [5]. Solving FPDEs is quite a complicated process, which requires highly complex mathematical calculations and solution methods.…”

Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation

“…This number is L + M • I. More about the method and its application for the inverse problems (particularly in the field of heat conductivity) can be found in [5,[17][18][19].…”