Fractal and Fractional

23 pages, 1649 KiB

Open AccessArticle

A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems

by Iman Malmir

Fractal Fract. 2019, 3(3), 46; https://doi.org/10.3390/fractalfract3030046 - 03 Sep 2019

Cited by 12 | Viewed by 3204

Fractional integration operational matrix of Chebyshev wavelets based on the Riemann–Liouville fractional integral operator is derived directly from Chebyshev wavelets for the first time. The formulation is accurate and can be applied for fractional orders or an integer order. Using the fractional integration [...] Read more.

Fractional integration operational matrix of Chebyshev wavelets based on the Riemann–Liouville fractional integral operator is derived directly from Chebyshev wavelets for the first time. The formulation is accurate and can be applied for fractional orders or an integer order. Using the fractional integration operational matrix, new Chebyshev wavelet methods for finding solutions of linear-quadratic optimal control problems and analysis of linear fractional time-delay systems are presented. Different numerical examples are solved to show the accuracy and applicability of the new Chebyshev wavelet methods. Full article

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11 pages, 335 KiB

Open AccessBrief Report

Characterization of the Local Growth of Two Cantor-Type Functions

by Dimiter Prodanov

Fractal Fract. 2019, 3(3), 45; https://doi.org/10.3390/fractalfract3030045 - 21 Aug 2019

Cited by 4 | Viewed by 2697

Abstract

The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor’s singular function by means of its fractional velocity. It is demonstrated that the [...] Read more.

The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor’s singular function by means of its fractional velocity. It is demonstrated that the Cantor function has finite one-sided velocities, which are non-zero of the set of change of the function. In addition, a related singular function based on the Smith–Volterra–Cantor set is constructed. Its growth is characterized by one-sided derivatives. It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2. Full article

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15 pages, 325 KiB

Open AccessArticle

Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions

by Bashir Ahmad, Madeaha Alghanmi, Ahmed Alsaedi and Sotiris K. Ntouyas

Fractal Fract. 2019, 3(3), 44; https://doi.org/10.3390/fractalfract3030044 - 18 Aug 2019

Cited by 3 | Viewed by 2522

Abstract

We discuss the existence of solutions for a Caputo type multi-term nonlinear fractional differential equation supplemented with generalized integral boundary conditions. The modern tools of functional analysis are applied to achieve the desired results. Examples are constructed for illustrating the obtained work. Some [...] Read more.

We discuss the existence of solutions for a Caputo type multi-term nonlinear fractional differential equation supplemented with generalized integral boundary conditions. The modern tools of functional analysis are applied to achieve the desired results. Examples are constructed for illustrating the obtained work. Some new results follow as spacial cases of the ones reported in this paper. Full article

12 pages, 4083 KiB

Open AccessArticle

Solving Helmholtz Equation with Local Fractional Derivative Operators

by Dumitru Baleanu, Hassan Kamil Jassim and Maysaa Al Qurashi

Fractal Fract. 2019, 3(3), 43; https://doi.org/10.3390/fractalfract3030043 - 01 Aug 2019

Cited by 39 | Viewed by 2965

Abstract

The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz [...] Read more.

The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz equations with local fractional derivative operators (LFDOs). The operators are taken in the local fractional sense. Two test problems are presented to demonstrate the efficiency and the accuracy of the proposed method. The approximate solutions obtained are compared with the results obtained by the local fractional Laplace decomposition method (LFLDM). The results reveal that the LFLVIM is very effective and convenient to solve linear and nonlinear PDEs. Full article

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21 pages, 19181 KiB

Open AccessArticle

Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

by L.K. Mork, Trenton Vogt, Keith Sullivan, Drew Rutherford and Darin J. Ulness

Fractal Fract. 2019, 3(3), 42; https://doi.org/10.3390/fractalfract3030042 - 12 Jul 2019

Cited by 8 | Viewed by 3656

Abstract

Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot [...] Read more.

Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points. Full article

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Graphical abstract

12 pages, 440 KiB

Open AccessEditor’s ChoiceArticle

Fractal Logistic Equation

by Alireza Khalili Golmankhaneh and Carlo Cattani

Fractal Fract. 2019, 3(3), 41; https://doi.org/10.3390/fractalfract3030041 - 11 Jul 2019

Cited by 37 | Viewed by 4497

Abstract

In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding [...] Read more.

In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics. Full article

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10 pages, 2441 KiB

Open AccessFeature PaperArticle

Cornu Spirals and the Triangular Lacunary Trigonometric System

by Trenton Vogt and Darin J. Ulness

Fractal Fract. 2019, 3(3), 40; https://doi.org/10.3390/fractalfract3030040 - 10 Jul 2019

Cited by 2 | Viewed by 3645

Abstract

This work is intended to directly supplement the previous work by Coutsias and Kazarinoff on the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals, specifically the Cornu spirals [Physica 26D (1987) 295]. These systems are intimately related to [...] Read more.

This work is intended to directly supplement the previous work by Coutsias and Kazarinoff on the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals, specifically the Cornu spirals [Physica 26D (1987) 295]. These systems are intimately related to incomplete Gaussian summations. The current work provides a focused look at the specific system built off of the triangular numbers. The special cyclic character of the triangular numbers modulo m carries through to triangular lacunary trigonometric systems. Specifically, this work characterizes the families of Cornu spirals arising from triangular lacunary trigonometric systems. Special features such as self-similarity, isometry, and symmetry are presented and discussed. Full article

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Graphical abstract

15 pages, 788 KiB

Open AccessArticle

Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives

by Ndolane Sene and José Francisco Gómez Aguilar

Fractal Fract. 2019, 3(3), 39; https://doi.org/10.3390/fractalfract3030039 - 07 Jul 2019

Cited by 15 | Viewed by 3388

Abstract

This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville–Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the [...] Read more.

This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville–Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the left generalized fractional derivative and the Liouville–Caputo left generalized fractional derivative were represented graphically and the effect of the orders of the fractional derivatives analyzed. We finish by analyzing the global asymptotic stability and the converging-input-converging-state of the unforced mass-damper system, the unforced spring-damper, the spring-damper system, and the mass-damper system. Full article

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9 pages, 237 KiB

Open AccessArticle

k-Fractional Estimates of Hermite–Hadamard Type Inequalities Involving k-Appell’s Hypergeometric Functions and Applications

by Muhammad Uzair Awan, Muhammad Aslam Noor, Marcela V. Mihai and Khalida Inayat Noor

Fractal Fract. 2019, 3(3), 38; https://doi.org/10.3390/fractalfract3030038 - 03 Jul 2019

Cited by 1 | Viewed by 2268

Abstract

The main objective of this paper is to obtain certain new k-fractional estimates of Hermite–Hadamard type inequalities via s-convex functions of Breckner type essentially involving k-Appell’s hypergeometric functions. We also present applications of the obtained results by considering particular examples. Full article

14 pages, 279 KiB

Open AccessArticle

Inequalities Pertaining Fractional Approach through Exponentially Convex Functions

by Saima Rashid, Muhammad Aslam Noor and Khalida Inayat Noor

Fractal Fract. 2019, 3(3), 37; https://doi.org/10.3390/fractalfract3030037 - 27 Jun 2019

Cited by 15 | Viewed by 2466

Abstract

In this article, certain Hermite-Hadamard-type inequalities are proven for an exponentially-convex function via Riemann-Liouville fractional integrals that generalize Hermite-Hadamard-type inequalities. These results have some relationships with the Hermite-Hadamard-type inequalities and related inequalities via Riemann-Liouville fractional integrals. Full article

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Fractal Fract., Volume 3, Issue 3 (September 2019) – 10 articles

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