Fractional nonlinear dynamics of learning with memory. (English) Zbl 1459.34046
Nonlinear Dyn. 100, No. 2, 1231-1242 (2020); corrigendum ibid. 103, No. 2, 2163-2167 (2021).
Summary: In this paper, we consider generalization of the Lucas model of learning (learning-by-doing) that is described in the paper [R. E. Lucas jun., Econometrica 61, No. 2, 251–272 (1993; Zbl 0825.90148)], who was awarded the Nobel Prize in Economic Sciences in 1995. The model equation is nonlinear differential equation of the first order used in macroeconomics to explain effects of innovation and technical change. In the standard learning model, the memory effects and memory fading are not taken into account. We propose the learning models that take into account fading memory. Fractional differential equations of the suggested models contain fractional derivatives with the generalized Mittag-Leffler function (the Prabhakar function) in the kernel and their special case containing the Caputo fractional derivative. These nonlinear fractional differential equations, which describe the learning-by-doing with memory, and the expressions of its exact solutions are suggested. Based on the exact solution of the model equation, we show that the estimated productivity growth rate can be changed by memory effects.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
| 39A70 | Difference operators |
| 47B39 | Linear difference operators |
| 65Q10 | Numerical methods for difference equations |
Keywords:
nonlinear dynamics; fractional differential equation; growth model; learning-by-doing; fractional derivative; learning modelCitations:
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