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The Lambert W function has utility for solving various exponential and logarithmic equations arranged in the form of $g(x)e^{g(x)}$. Using the Lambert W function and tetration, a variety of categorized inversion formulas are presented.…
Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and…
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
In this article, we present an iterative method to find simple roots of nonlinear equations, that is, to solving an equation of the form $f(x) = 0$. Different from Newton's method, the method we purpose do not require evaluation of…
Functional graphs (FGs) model the graph structures used to analyse the behaviour of functions from a discrete set to itself. In turn, such functions are used to study real complex phenomena evolving in time. As the systems involved can be…
This paper surveys recent analytical and numerical research on linear problems for the integral fractional Laplacian, fractional obstacle problems, and fractional minimal graphs. The emphasis is on the interplay between regularity,…
We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented…
Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…
We present a new tool to compute the number $\phi_\A (\b)$ of integer solutions to the linear system $$ \x \geq 0 \qquad \A \x = \b $$ where the coefficients of $\A$ and $\b$ are integral. $\phi_\A (\b)$ is often described as a \emph{vector…
This paper introduces a new numerical method for approximating the Lambert W function in the real domain. The method transforms the function into a simpler form that allows iterative refinement of an initial guess. Two iterative strategies…
A method is given for finding roots of a one-variable function using Taylor's expansion of that function and fractional derivative calculated at a suitable tangent point without using Newton's method, but is regarded as a variant of Halley…
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions.…
The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review 20 4], none of which…
In this work, the Lambert-Tsallis Wq function is used to provide analytical solutions of fractional polynomials of the type ax^r+bx^s+c = 0. This class of fractional polynomial appears in several areas of physics as well it is in the heart…
In this paper the singular Emden-Fowler equation of fractional order is introduced and a computational method is proposed for its numerical solution. For the approximation of the solutions we have used Boubaker polynomials and defined the…
Approximate solutions to functional evolution equations are constructed through a combination of series and conjugation methods, and relative errors are estimated. The methods are illustrated, both analytically and numerically, by…
In this short paper, the authors report a new computational approach in the context of Density Functional Theory (DFT). It is shown how it is possible to speed up the self-consistent cycle (iteration) characterizing one of the most…
We propose an approach to determine the continual progression of algorithmic efficiency, as an alternative to standard calculations of time complexity, likely, but not exclusively, when dealing with data structures with unknown maximum…
Some fractional Newton methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper we introduce a fractional Newton method with order $\alpha+1$ and compare with another fractional…
Prompted by an observation about the integral of exponential functions of the form $f(x)=\lambda e^{\alpha x}$, we investigate the possibility to exactly integrate families of functions generated from a given function by scaling or by…