The optimization problem for determining the functional dependence of the variable order of the fractional derivative of the Gerasimov-Caputo type

D. A. Tvyordyj   ; R. I. Parovik

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 47 (2024) no. 2, pp. 35-57

Résumé

When solving mathematical modeling problems, it is often necessary to turn to the theory of integral and differential calculus. This theory can be used to describe dynamic processes of various types. The use of fractional derivatives allows us to refine some models by taking into account the memory effect, which is expressed in the equations depending on the current state of the system from previous states. This effect is called non-locality and its intensity is determined by the value of the exponent in the fractional derivative. Classically, this value $\alpha$$\alpha$ a noninteger constant, but there are also generalizations for time-varying nonlocality and other functional dependencies. Fractional differential models are finding increasing application in the physical, mathematical, and technical sciences. However, given the nature of the modeled process, the selection of various parameters for such models must be carried out empirically. Model parameters are refined by iterating through values and comparing simulation results with experimental data representing the process. This process continues until the results begin to qualitatively approximate the data, which is a time-consuming process that inevitably leads to ideas about solving inverse problems. The purpose of this work is to demonstrate that it is possible to use methods of unconditional optimization to solve inverse problems and determine the type of functional dependence $\alpha$(t). The direct problem is formulated as a Cauchy problem for a fractional differential equation, where the derivative is interpreted in the sense of Gerasimov-Caputo with a variable exponent $\alpha$(t) for the fractional derivative. The direct problem is solved numerically using a nonlocal, implicit finite difference scheme. The inverse problem is defined as the problem of discrete minimization of the function $\alpha$(t) based on experimental data. To solve this problem, we have chosen the Levenberg-Marquardt iterative method. Through test examples, we have shown that this method can be used for unconstrained optimization to determine the shape of the function $\alpha$(t) and its optimal values in various models.

Détail

Keywords: Inverse problems, non-conditional optimization, Newton methods of function minimization, fractional derivatives, Gerasimov-Caputo, memory effect, non-locality, implicit finite-difference schemes.
Mots-clés : Levenberg-Marquardt algorithm

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