We study properties of β-numeration systems, where β > 1 is the real root of the polynomial x −mx − x−1, m ∈ N, m ≥ 1. We consider arithmetic operations on the set of β-integers, i.e., on the set of numbers whose greedy expansion in base β has no fractional part. We show that the number of fractional digits arising under addition of β-integers is at most 5 for m ≥ 3 and 6 for m = 2, whereas und...

The stochastic solution to a diffusion equations with polynomial coefficients is called a Pearson diffusion. If the first time derivative is replaced by a Caputo fractional derivative of order less than one, the stochastic solution is called a fractional Pearson diffusion. This paper develops an explicit formula for the covariance function of a fractional Pearson diffusion in steady state, in t...

Fractional differential equations have extensively used in physics, chemistry as well as engineering fields. Therefore, approximating the solution of differential equations of fractional order is necessary. Consequently, it is essential to approximate the solution of differential equations of fractional order. The piecewise quadratic polynomial function based method has been presented in this p...

Fractional calculus became a vital tool in describing many phenomena appeared in physics, chemistry as well as engineering fields. Analytical solution of many applications, where the fractional differential equations appear, cannot be established. Therefore, cubic polynomial splinefunction-based method combined with shooting method is considered to find approximate solution for a class of fract...

This paper compares several optimization algorithms that can be used to generate exact D-optimal designs (i.e., designs for a specified number of runs) for any polynomial model. The merits and limitations of each algorithm are demonstrated on several low-order polynomial models, with numerical results verified against analytical results. The efficiencies – with respect to estimating model param...

One-dimensional Ginzburg–Landau equations with derivatives of noninteger order are considered. Using psi-series with fractional powers, the solution of the fractional Ginzburg–Landau (FGL) equation is derived. The leading-order behaviours of solutions about an arbitrary singularity, as well as their resonance structures, have been obtained. It was proved that fractional equations of order α wit...

In this work, we introduce a generalization of the differential polynomial neural network utilizing fractional calculus. Fractional calculus is taken in the sense of the Caputo differential operator. It approximates a multi-parametric function with particular polynomials characterizing its functional output as a generalization of input patterns. This method can be employed on data to describe m...