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Generalized Memory: Fractional Calculus Approach

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991 Moscow, Russia
Fractal Fract 2018, 2(4), 23; https://doi.org/10.3390/fractalfract2040023
Received: 23 August 2018 / Revised: 20 September 2018 / Accepted: 21 September 2018 / Published: 24 September 2018
The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time. View Full-Text
Keywords: fractional dynamics; fractional calculus; long memory; economics fractional dynamics; fractional calculus; long memory; economics
MDPI and ACS Style

Tarasov, V.E. Generalized Memory: Fractional Calculus Approach. Fractal Fract 2018, 2, 23.

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