On the Fundamental Theorem of Calculus for Fractal Sets

Bongiorno, Donatella; Corrao, Giuseppa

doi:10.1142/s0218348x15500085

Cited by 25 publications

(5 citation statements)

References 5 publications

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“…Numerous researchers have approached fractal analysis through various methods, including harmonic analysis [8,9], measure theory [10][11][12][13][14][15][16], probabilistic approaches [17], fractional space [18], fractional calculus [19], and unconventional techniques [20].…”

Section: Introductionmentioning

confidence: 99%

Fractal Schrödinger equation: implications for fractal sets

Khalili Golmankhaneh,

Pellis,

Zingales

2024

J. Phys. A: Math. Theor.

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This paper delves into the world of fractal calculus, investigating its implications for fractal sets. It introduces the Fractal Schrödinger Equation and provides insights into its consequences. The study presents a General Solution for the Time-Dependent Schrödinger Equation, unveiling its core aspects. Exploring quantum mechanics in the context of fractals, the paper analyzes the Probability Density of the Radial Hydrogen Atom, unveiling its behavior within fractal dimensions. The investigation extends to deciphering the intricate Energy Levels of the Hydrogen Atom, uncovering the interplay of quantum mechanics and fractal geometry. Innovatively, the research applies the Fractal Schrödinger Equation to Simple Harmonic Motion, leading to the introduction of the Fractal Probability Density Function for the Harmonic Oscillator. The paper employs a series of illustrative ﬁgures that enhance the comprehension of the ﬁndings. By intertwining quantum mechanics and fractal mathematics, this research paves the way for deeper insights into their relationship.

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Section: Introductionmentioning

confidence: 99%

Fractal Schrödinger equation: implications for fractal sets

Khalili Golmankhaneh,

Pellis,

Zingales

2024

J. Phys. A: Math. Theor.

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“…In this chapter, we present a method of standard calculus for fractal subsets of the real line, that was independently formulated by various authors (see Refs. [5][6][7][8][9]). Such formulation is aimed at those self-similar fractal subsets of the real line with finite and positive s-dimensional Hausdorff measure, briefly called s-sets.…”

Section: Introductionmentioning

confidence: 99%

Derivation and Integration on a Fractal Subset of the Real Line

Bongiorno¹

2023

Fractal Analysis - Applications and Updates

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The ordinary calculus is usually inapplicable to fractal sets, therefore we introduce the various approaches made so far to describe the theory of derivation and integration on a fractal set. In particular we study the Riemann type integrals (s-Riemann integral, s-HK integral, s-first return integral) defined on a closed fractal subset of the real line with finite positive s-dimensional Hausdorff measure (s-set) with particular attention to the Fundamental Theorem of Calculus. Moreover we pay attention to the relation between the s-HK integral, the s-first return integral and the Lebesgue integral respectively. Finally we give a descriptive characterization of the primitives of a s-HK integrable function.

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“…Researchers have tackled the problem of fractal analysis using approaches. These include analysis [14,15], measure theory [16,17,18,19,20,21,22], probabilistic methods [23], fractional space and nonstandard methods [24], fractional calculus [25,26,27] and non standard methods [28]. Essential topological characteristics such as connectivity, ramification, and loopiness were exhibited by fractals and can be quantified using six independent dimension values.…”

Section: Introductionmentioning

confidence: 99%

Non-standard analysis for fractal calculus

Golmankhaneh

Welch

Serpa

et al. 2023

J Anal

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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 solving α-order differential equations. Notably, we extend our analysis to solve Fractal Bernoulli differential equations. The applications of our findings are then showcased through the solutions of problems such as fractal compound interest, the escape velocity of the earth in fractal space and time, and the estimation of time of death incorporating fractal time. Visual representations of our results are also provided to enhance understanding.

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On the Fundamental Theorem of Calculus for Fractal Sets

Abstract: The aim of this paper is to formulate the best version of the Fundamental theorem of Calculus for real functions on a fractal subset of the real line. In order to do that an integral of HenstockKurzweil type is introduced.

Cited by 25 publications

References 5 publications

Fractal Schrödinger equation: implications for fractal sets

Fractal Schrödinger equation: implications for fractal sets

Derivation and Integration on a Fractal Subset of the Real Line

Non-standard analysis for fractal calculus

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