Computational Methods for Parameter Identification in 2D Fractional System with Riemann–Liouville Derivative

Abstract

In recent times, many different types of systems have been based on fractional derivatives. Thanks to this type of derivatives, it is possible to model certain phenomena in a more precise and desirable way. This article presents a system consisting of a two-dimensional fractional differential equation with the Riemann–Liouville derivative with a numerical algorithm for its solution. The presented algorithm uses the alternating direction implicit method (ADIM). Further, the algorithm for solving the inverse problem consisting of the determination of unknown parameters of the model is also described. For this purpose, the objective function was minimized using the ant algorithm and the Hooke–Jeeves method. Inverse problems with fractional derivatives are important in many engineering applications, such as modeling the phenomenon of anomalous diffusion, designing electrical circuits with a supercapacitor, and application of fractional-order control theory. This paper presents a numerical example illustrating the effectiveness and accuracy of the described methods. The introduction of the example made possible a comparison of the methods of searching for the minimum of the objective function. The presented algorithms can be used as a tool for parameter training in artificial neural networks.

Analysis of the convective heat transfer through straight fin by using the Riemann-Liouville type fractional derivative: Probed by machine learning

“…They compared their results with VIM and HPM for validation. Brociek et al [47] used the Riemann–Liouville type fractional derivative to analyze the two dimensional system by considering the alternating direction implicit method (ADIM). They have also identified the unknown parameters by taking the inverse problem.…”

On Ψ-Hilfer Fractional Integro-Differential Equations with Non-Instantaneous Impulsive Conditions

“…The reason for this is FDEs accurately describe many real-world phenomena such as biology, physics, chemistry, signal processing, and many more (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]). Furthermore, it should be remarked that FDEs have interesting applications in solving inverse problems, and in the modeling of heat flow in porous material (see, e.g., [14][15][16][17][18][19][20][21][22][23][24][25]).…”

On Nonlinear Ψ-Caputo Fractional Integro Differential Equations Involving Non-Instantaneous Conditions

“…The reason for this is FDEs efficiently describe many real-world processes such as in chemistry, biology, signal processing, and many others (see, e.g., [4,[7][8][9]13,[17][18][19][20][21]). Additionally, FDEs have interesting applications in solving inverse problems, and in the modeling of heat flow in porous material (see, e.g., [22][23][24]).…”