Related papers: differint: A Python Package for Numerical Fraction…
Artificial intelligence has recently experienced remarkable advances, fueled by large models, vast datasets, accelerated hardware, and, last but not least, the transformative power of differentiable programming. This new programming…
Incremental computation has recently been studied using the concepts of change structures and derivatives of programs, where the derivative of a function allows updating the output of the function based on a change to its input. We…
We develop a calculus of variations for functionals which are defined on a set of non differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the…
This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional…
There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of…
Derivatives and integrals of non-integer order were introduced more than three centuries ago, but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most…
Integer-order differential operators were originally used to describe local and isotropic effects, in both space and time. However, in fields like biology, the modelling of complex phenomena with spatial heterogeneity necessitates more…
Using fractional calculus we define integrals of the form $% \int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued H\"{o}lder continuous functions of order $\displaystyle \beta \in (\frac13, \frac12)$ and $f$ is a continuously…
In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of…
We introduce a new approach for solving forward systems of differential equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the…
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness…
A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…
We compare traditional approach of computing logarithmic returns with the fractional differencing method and its tempered extension as methods of data preparation before their usage in advanced machine learning models. Differencing…
$\mathtt{qnm}$ is an open-source Python package for computing the Kerr quasinormal mode frequencies, angular separation constants, and spherical-spheroidal mixing coefficients. The $\mathtt{qnm}$ package includes a Leaver solver with the…
A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order $\alpha, 0 < \alpha \leq 1$, called $F^\alpha$-integral, is defined, which is suitable to integrate functions with fractal…
The aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The…
Algebraic differentiators have attracted much interest in recent years. Their simple implementation as classical finite impulse response digital filters and systematic tuning guidelines may help to solve challenging problems, including, but…
In this article, the existence and uniqueness about the solution for a class of stochastic fractional-order differential equation systems are investigated, where the fractional derivative is described in Caputo sense. The fractional…
We announce some Python classes for numerical solution of partial differential equations, or boundary value problems of ordinary differential equations. These classes are built on routines in \texttt{numpy} and \texttt{scipy.sparse.linalg}…
It is well known that certain fractional diffusion equations can be solved by the densities of stable L\'evy motions. In this paper we use the classical semigroup approach for L\'evy processes to define semi-fractional derivatives, which…