Tomas Skovranek | Searcharxiv

arXiv2019

The Mittag-Leffler Fitting of the Phillips Curve

In this paper, a mathematical model based on the one-parameter Mittag-Leffler function is proposed to be used for the first time to describe the relation between unemployment rate and inflation rate, also known as the Phillips curve. The Phillips curve is in the literature often represented by an exponential-like shape. On the other hand, Phillips in his fundamental paper used a power function in the model definition. Considering that the ordinary as well as generalised Mittag-Leffler function behaves between a purely exponential function and a power function it is natural to implement it in the definition of the model used to describe the relation between the data representing the Phillips curve. For the modelling purposes the data of two different European economies, France and Switzerland, were used and an "out-of-sample" forecast was done to compare the performance of the Mittag-Leffler model to the performance of the power-type and exponential-type model. The results demonstrate that the ability of the Mittag-Leffler function to fit data that manifest signs of stretched exponentials, oscillations or even damped oscillations can be of use when describing economic relations and phenomenons, such as the Phillips curve.

econ.GN↗

arXiv2009

Matrix approach to discrete fractional calculus II: partial fractional differential equations

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach (Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of numerical solution of fractional diffusion equation with various combinations of time/space fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.

math.NA↗