Related papers: Fuzzification of Fractal Calculus
This article deals with the complexity involved in fuzzy derivatives when both input and output are from nonempty, convex, and compact fuzzy space. Consider a fuzzy valued mapping, and for fuzzy differentiation of fuzzy valued function, we…
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…
A Compact Introduction to Fractional Calculus is presented including basic definitions, fractional differential equations and special functions.
Many mathematical models utilize limit processes. Continuous functions and the calculus, differential equations and topology, all are based on limits and continuity. However, when we perform measurements and computations, we can achieve…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
This article presents a theory of differential and integral calculus for mapping between Banach spaces formed by subsets of fuzzy numbers called A-linearly correlated fuzzy numbers, where both the domain and codomain are spaces composed of…
It is wellknown that the ordinary calculus is inadequate to handle fractal structures and processes and another suitable calculus needs to be developed for this purpose. Recently it was realized that fractional calculus with suitable…
In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$\mu$…
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 goal of this work is to introduce and study fuzzy limits of functions. Two approaches to fuzzy limits of a function are considered. One is based on the concept of a fuzzy limit of a sequence, while another generalizes the conventional…
In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for…
This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff's concepts of fractional dimension geometry. The…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we…
The study of fuzzy fractional variational problems in terms of a fractional Liouville-Caputo derivative is introduced. Necessary optimality conditions for problems of the fuzzy fractional calculus of variations with free end-points are…
The thesis deals with applications of fractional calculus to fractals. It introduces the notion of local fractional derivative (LFD). Fractal and multifractal functions have been studied in the thesis using LFD. New kind of equations are…
We consider fuzzy valued functions from two parametric representations of $\alpha$-level sets. New concepts are introduced and compared with available notions. Following the two proposed approaches, we study fuzzy differential equations.…
Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential…
In this paper, we revisit the diffusive representations of fractional integrals established in \cite{diethelm2023diffusive} to explore novel variants of such representations which provide highly efficient numerical algorithms for the…