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Fractal Fract, Volume 3, Issue 3 (September 2019)

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Open AccessArticle
A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems
Fractal Fract 2019, 3(3), 46; https://doi.org/10.3390/fractalfract3030046 - 03 Sep 2019
Viewed by 224
Abstract
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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Open AccessBrief Report
Characterization of the Local Growth of Two Cantor-Type Functions
Fractal Fract 2019, 3(3), 45; https://doi.org/10.3390/fractalfract3030045 - 21 Aug 2019
Viewed by 265
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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Open AccessArticle
Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions
Fractal Fract 2019, 3(3), 44; https://doi.org/10.3390/fractalfract3030044 - 18 Aug 2019
Viewed by 348
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
Open AccessArticle
Solving Helmholtz Equation with Local Fractional Derivative Operators
Fractal Fract 2019, 3(3), 43; https://doi.org/10.3390/fractalfract3030043 - 01 Aug 2019
Viewed by 422
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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Open AccessArticle
Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions
Fractal Fract 2019, 3(3), 42; https://doi.org/10.3390/fractalfract3030042 - 12 Jul 2019
Viewed by 464
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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Open AccessArticle
Fractal Logistic Equation
Fractal Fract 2019, 3(3), 41; https://doi.org/10.3390/fractalfract3030041 - 11 Jul 2019
Cited by 1 | Viewed by 595
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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Open AccessFeature PaperArticle
Cornu Spirals and the Triangular Lacunary Trigonometric System
Fractal Fract 2019, 3(3), 40; https://doi.org/10.3390/fractalfract3030040 - 10 Jul 2019
Viewed by 500
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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Open AccessArticle
Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives
Fractal Fract 2019, 3(3), 39; https://doi.org/10.3390/fractalfract3030039 - 07 Jul 2019
Viewed by 488
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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Open AccessArticle
k-Fractional Estimates of Hermite–Hadamard Type Inequalities Involving k-Appell’s Hypergeometric Functions and Applications
Fractal Fract 2019, 3(3), 38; https://doi.org/10.3390/fractalfract3030038 - 03 Jul 2019
Viewed by 439
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
Open AccessArticle
Inequalities Pertaining Fractional Approach through Exponentially Convex Functions
Fractal Fract 2019, 3(3), 37; https://doi.org/10.3390/fractalfract3030037 - 27 Jun 2019
Viewed by 470
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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