Calculus on fractal subsets of real line - I: formulation
By A. Parvate, A. D. Gangal. 2003
195 citations recorded by the source corpus.
A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order α, 0 < α ≤ 1, called Fα-integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called Fα-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, "changing" only on a fractal set. The Fα-derivative is local unlike the classical fractional derivative. The Fα-calculus retains much of the simplicity of ordinary calculus. Several results including analogues of fundamental theorems of calculus are proved. The integral staircase function, which is a generalization of the functions like the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, the γ-dimension. Spaces of Fα-differentiable and Fα-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and Fα-differentiability is generalized using Sobolev-like construction. Fα-differential equations are equations involving Fα-derivatives. They can be used to model sublinear dynamical systems and fractal time processes, since sublinear behaviors are associated with staircase-like functions which occur naturally as their solutions. As examples, we discuss a fractal-time diffusion equation, and one-dimensional motion of a particle undergoing friction in a fractal medium.
Subfields: Modeling and Simulation, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/5612b8a5ce44c496d84fbac62502a42b1c1ab53c
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In Modeling and Simulation
In Statistical and Nonlinear Physics
Citations in the corpus
- Advanced Local Fractional Calculus and Its Applications (2012, unknown): “[30] A.”
- Generalizations of Hölder’s and Some Related Integral Inequalities on Fractal Space (2011, unknown): “Local fractional derivative had been introduced in [12–36]; that is the local fractional derivative was structured in [12–18, 26, 30–36], Jumarie gave the modified Riemann-Liouville derivative in [19, 20], the fractal derivative was considered in [21–25, 27–29], and the generalized fractal derivativ”
- Fractal Logistic Equation (2019, background): “Recently, in the seminal papers [57,58], generalized standard calculus was adopted to include functions with support on totally disconnected fractal sets and self-avoiding curves.”