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Если Вы ощутили землетрясение, пожалуйста, сообщите о нёмRefinement of Variable Order Fractional Derivative of Gerasimov-Caputo Type by Multidimensional Levenberg-Marquardt Optimization Method
- Federal State Budget Research Institution Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences
Abstract When solving many fundamental and applied problems related to the
study of properties of dynamical systems, situations often arise when the parameters
of the model are not exactly defined. One can try to solve such a problem by
searching the parameters, taking into account the understanding of the nature of
the modelling process. On the other hand, it is possible to solve inverse problems—a
common type of problems in many scientific fields, where it is necessary to deter-
mine the values of the model parameters on the basis of experimental data, but it is
impossible to make direct measurements of the parameters. The paper considers an
inhomogeneous fractional equation with a Gerasimov-Caputo type operator of
variable α(t) order. The possibility of recovering the form of the function α(t) and
refining its values on the basis of test generated data is investigated. The direct
problem is defined as a Cauchy problem for a fractional equation solved numerically.
The solution of the inverse problem is reduced to the minimization of the inviscid
functional, and the minimization problem is solved using the iterative Levenberg-
Marquardt unconditional optimization method. Thus, the problem of determining the
optimal form α(t) of a fractional equation is reduced to 4 parameters that control the
course of the solution of the inverse problem and do not depend on the physical
meaning of the original problem in specific models. On test examples it was shown
that the Levenberg-Marquardt method can indeed be used for unconditional optimi-
zation to determine type of function α(t) and its optimal values.
Tverdyi, D. A. Refinement of Variable Order Fractional Derivative of Gerasimov-Caputo Type by Multidimensional Levenberg-Marquardt Optimization Method // Computing Technologies and Applied Mathematics, Komsomolsk-na-Amure, Russia, 07 октября 2024 года – 11 2025 года. – Springer Cham, 2025. – P. 159-173. – DOI 10.1007/978-3-031-90546-9_11
