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Fractal Fract, Volume 3, Issue 2 (June 2019)

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Open AccessArticle
Green’s Function Estimates for Time-Fractional Evolution Equations
Fractal Fract 2019, 3(2), 36; https://doi.org/10.3390/fractalfract3020036 - 25 Jun 2019
Viewed by 515
Abstract
We look at estimates for the Green’s function of time-fractional evolution equations of the form D 0 + ν u = L u , where D 0 + ν is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with [...] Read more.
We look at estimates for the Green’s function of time-fractional evolution equations of the form D 0 + ν u = L u , where D 0 + ν is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y 1 β for β ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D 0 β u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D 0 β u = Ψ ( i ) u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α . Thirdly, we obtain local two-sided estimates for the Green’s function of D 0 β u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( ν , t ) u = L u , where D ( ν , t ) is a Caputo-type operator with variable coefficients. Full article
Open AccessArticle
A Criterion for Subfamilies of Multivalent Functions of Reciprocal Order with Respect to Symmetric Points
Fractal Fract 2019, 3(2), 35; https://doi.org/10.3390/fractalfract3020035 - 25 Jun 2019
Viewed by 479
Abstract
In the present research paper, our aim is to introduce a new subfamily of p-valent (multivalent) functions of reciprocal order. We investigate sufficiency criterion for such defined family. Full article
Open AccessArticle
Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions
Fractal Fract 2019, 3(2), 34; https://doi.org/10.3390/fractalfract3020034 - 21 Jun 2019
Viewed by 486
Abstract
In this paper, we investigate a new class of boundary value problems involving fractional differential equations with mixed nonlinearities, and nonlocal multi-point and Riemann–Stieltjes integral-multi-strip boundary conditions. Based on the standard tools of the fixed point theory, we obtain some existence and uniqueness [...] Read more.
In this paper, we investigate a new class of boundary value problems involving fractional differential equations with mixed nonlinearities, and nonlocal multi-point and Riemann–Stieltjes integral-multi-strip boundary conditions. Based on the standard tools of the fixed point theory, we obtain some existence and uniqueness results for the problem at hand, which are well illustrated with the aid of examples. Our results are not only in the given configuration but also yield several new results as special cases. Some variants of the given problem are also discussed. Full article
Open AccessArticle
A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems
Fractal Fract 2019, 3(2), 33; https://doi.org/10.3390/fractalfract3020033 - 18 Jun 2019
Viewed by 520
Abstract
In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments [...] Read more.
In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments are discussed and the approximate solutions are obtained to be more correct compared to the other obtained results in the literature. Full article
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Open AccessArticle
On Extended General Mittag–Leffler Functions and Certain Inequalities
Fractal Fract 2019, 3(2), 32; https://doi.org/10.3390/fractalfract3020032 - 18 Jun 2019
Viewed by 481
Abstract
In this paper, we introduce and investigate generalized fractional integral operators containing the new generalized Mittag–Leffler function of two variables. We establish several new refinements of Hermite–Hadamard-like inequalities via co-ordinated convex functions. Full article
Open AccessArticle
Random Variables and Stable Distributions on Fractal Cantor Sets
Fractal Fract 2019, 3(2), 31; https://doi.org/10.3390/fractalfract3020031 - 11 Jun 2019
Cited by 1 | Viewed by 783
Abstract
In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability [...] Read more.
In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability theory, defining concepts such as Shannon entropy on fractal thin Cantor-like sets. Stable distributions on fractal sets are suggested and related physical models are presented. Our work is illustrated with graphs for clarity of the results. Full article
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Open AccessArticle
A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets
Fractal Fract 2019, 3(2), 30; https://doi.org/10.3390/fractalfract3020030 - 03 Jun 2019
Cited by 1 | Viewed by 541
Abstract
In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is a [...] Read more.
In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is a combination of the local fractional Laplace transform (LFLT) and the homotopy perturbation method (HPM). The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique. Full article
Open AccessArticle
On Some Generalized Fractional Integral Inequalities for p-Convex Functions
Fractal Fract 2019, 3(2), 29; https://doi.org/10.3390/fractalfract3020029 - 20 May 2019
Viewed by 489
Abstract
In this paper, firstly we have established a new generalization of Hermite–Hadamard inequality via p-convex function and fractional integral operators which generalize the Riemann–Liouville fractional integral operators introduced by Raina, Lun and Agarwal. Secondly, we proved a new identity involving this generalized [...] Read more.
In this paper, firstly we have established a new generalization of Hermite–Hadamard inequality via p-convex function and fractional integral operators which generalize the Riemann–Liouville fractional integral operators introduced by Raina, Lun and Agarwal. Secondly, we proved a new identity involving this generalized fractional integral operators. Then, by using this identity, a new generalization of Hermite–Hadamard type inequalities for fractional integral are obtained. Full article
Open AccessArticle
Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses
Fractal Fract 2019, 3(2), 28; https://doi.org/10.3390/fractalfract3020028 - 18 May 2019
Viewed by 505
Abstract
The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time between two consecutive moments of impulses is the Erlang [...] Read more.
The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time between two consecutive moments of impulses is the Erlang distributed random variable. The study is based on Lyapunov functions. The fractional Dini derivatives are applied. Full article
Open AccessArticle
Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces
Fractal Fract 2019, 3(2), 27; https://doi.org/10.3390/fractalfract3020027 - 16 May 2019
Cited by 1 | Viewed by 619
Abstract
We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular [...] Read more.
We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results. Full article
Open AccessArticle
Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings
Fractal Fract 2019, 3(2), 26; https://doi.org/10.3390/fractalfract3020026 - 11 May 2019
Cited by 2 | Viewed by 584
Abstract
In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency [...] Read more.
In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency of the considered methods are illustrated by some examples. The results obtained by LFLVIM and LFLDM are compared with the results obtained by LFVIM. The results reveal that the suggested algorithms are very effective and simple, and can be applied for linear and nonlinear problems in sciences and engineering. Full article
(This article belongs to the Special Issue 2019 Selected Papers from Fractal Fract’s Editorial Board Members)
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Open AccessArticle
Analogues to Lie Method and Noether’s Theorem in Fractal Calculus
Fractal Fract 2019, 3(2), 25; https://doi.org/10.3390/fractalfract3020025 - 07 May 2019
Viewed by 646
Abstract
In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An [...] Read more.
In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An analogue for Noether’s Theorem on fractal sets is given, and a corresponding conservative quantity is suggested. Several examples are solved to illustrate the results. Full article
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Open AccessArticle
Some New Generalizations for Exponentially s-Convex Functions and Inequalities via Fractional Operators
Fractal Fract 2019, 3(2), 24; https://doi.org/10.3390/fractalfract3020024 - 28 Apr 2019
Cited by 2 | Viewed by 527
Abstract
The main objective of this paper is to obtain the Hermite–Hadamard-type inequalities for exponentially s-convex functions via the Katugampola fractional integral. The Katugampola fractional integral is a generalization of Riemann–Liouville fractional integral and Hadamard fractional integral. Some special cases and applications to [...] Read more.
The main objective of this paper is to obtain the Hermite–Hadamard-type inequalities for exponentially s-convex functions via the Katugampola fractional integral. The Katugampola fractional integral is a generalization of Riemann–Liouville fractional integral and Hadamard fractional integral. Some special cases and applications to special means are also discussed. Full article
Open AccessArticle
Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs
Fractal Fract 2019, 3(2), 23; https://doi.org/10.3390/fractalfract3020023 - 24 Apr 2019
Viewed by 513
Abstract
In the oil industry, many reservoirs produce from partially penetrated wells, either to postpone the arrival of undesirable fluids or to avoid problems during drilling operations. The majority of these reservoirs are heterogeneous and anisotropic, such as naturally fractured reservoirs. The analysis of [...] Read more.
In the oil industry, many reservoirs produce from partially penetrated wells, either to postpone the arrival of undesirable fluids or to avoid problems during drilling operations. The majority of these reservoirs are heterogeneous and anisotropic, such as naturally fractured reservoirs. The analysis of pressure-transient tests is a very useful method to dynamically characterize both the heterogeneity and anisotropy existing in the reservoir. In this paper, a new analytical solution for a partially penetrated well based on a fractal approach to capture the distribution and connectivity of the fracture network is presented. This solution represents the complexity of the flow lines better than the traditional Euclidean flow models for single-porosity fractured reservoirs, i.e., for a tight matrix. The proposed solution takes into consideration the variations in fracture density throughout the reservoir, which have a direct influence on the porosity, permeability, and the size distribution of the matrix blocks as a result of the fracturing process. This solution generalizes previous solutions to model the pressure-transient behavior of partially penetrated wells as proposed in the technical literature for the classical Euclidean formulation, which considers a uniform distribution of fractures that are fully connected. Several synthetic cases obtained with the proposed solution are shown to illustrate the influence of different variables, including fractal parameters. Full article
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Open AccessArticle
Hankel Determinants of Non-Zero Modulus Dixon Elliptic Functions via Quasi C Fractions
Fractal Fract 2019, 3(2), 22; https://doi.org/10.3390/fractalfract3020022 - 17 Apr 2019
Viewed by 457
Abstract
The Sumudu transform of the Dixon elliptic function with non-zero modulus α ≠ 0 for arbitrary powers N is given by the product of quasi C fractions. Next, by assuming the denominators of quasi C fractions as one and applying the Heliermanncorrespondence relating [...] Read more.
The Sumudu transform of the Dixon elliptic function with non-zero modulus α ≠ 0 for arbitrary powers N is given by the product of quasi C fractions. Next, by assuming the denominators of quasi C fractions as one and applying the Heliermanncorrespondence relating formal power series (Maclaurin series of the Dixon elliptic function) and the regular C fraction, the Hankel determinants are calculated for the non-zero Dixon elliptic functions and shown by taking α = 0 to give the Hankel determinants of the Dixon elliptic function with zero modulus. The derived results were back-tracked to the Laplace transform of Dixon elliptic functions. Full article
Open AccessArticle
Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions
Fractal Fract 2019, 3(2), 21; https://doi.org/10.3390/fractalfract3020021 - 17 Apr 2019
Viewed by 495
Abstract
In this paper, we discuss the existence and uniqueness of solutions for a new class of single and multi-valued boundary value problems involving both Riemann–Liouville and Caputo fractional derivatives, and nonlocal fractional integro-differential boundary conditions. Our results rely on modern tools of functional [...] Read more.
In this paper, we discuss the existence and uniqueness of solutions for a new class of single and multi-valued boundary value problems involving both Riemann–Liouville and Caputo fractional derivatives, and nonlocal fractional integro-differential boundary conditions. Our results rely on modern tools of functional analysis. We also demonstrate the application of the obtained results with the aid of examples. Full article
Open AccessArticle
Statistical Mechanics Involving Fractal Temperature
Fractal Fract 2019, 3(2), 20; https://doi.org/10.3390/fractalfract3020020 - 17 Apr 2019
Cited by 1 | Viewed by 599
Abstract
In this paper, the Schrödinger equation involving a fractal time derivative is solved and corresponding eigenvalues and eigenfunctions are given. A partition function for fractal eigenvalues is defined. For generalizing thermodynamics, fractal temperature is considered, and adapted equations are defined. As an application, [...] Read more.
In this paper, the Schrödinger equation involving a fractal time derivative is solved and corresponding eigenvalues and eigenfunctions are given. A partition function for fractal eigenvalues is defined. For generalizing thermodynamics, fractal temperature is considered, and adapted equations are defined. As an application, we present fractal Dulong-Petit, Debye, and Einstein solid models and corresponding fractal heat capacity. Furthermore, the density of states for fractal spaces with fractional dimension is obtained. Graphs and examples are given to show details. Full article
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Open AccessArticle
New Estimates for Exponentially Convex Functions via Conformable Fractional Operator
Fractal Fract 2019, 3(2), 19; https://doi.org/10.3390/fractalfract3020019 - 15 Apr 2019
Cited by 3 | Viewed by 450
Abstract
In this paper, we derive a new Hermite–Hadamard inequality for exponentially convex functions via α-fractional integral. We also prove a new integral identity. Using this identity, we establish several Hermite–Hadamard type inequalities for exponentially convexity, which can be obtained from our results. [...] Read more.
In this paper, we derive a new Hermite–Hadamard inequality for exponentially convex functions via α-fractional integral. We also prove a new integral identity. Using this identity, we establish several Hermite–Hadamard type inequalities for exponentially convexity, which can be obtained from our results. Some special cases are also discussed. Full article
Open AccessArticle
Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation
Fractal Fract 2019, 3(2), 18; https://doi.org/10.3390/fractalfract3020018 - 09 Apr 2019
Viewed by 481
Abstract
We study a class of conformable time-fractional stochastic equation T α , t a u ( x , t ) = σ ( u ( x , t ) ) W ˙ t , x R , t [ a , [...] Read more.
We study a class of conformable time-fractional stochastic equation T α , t a u ( x , t ) = σ ( u ( x , t ) ) W ˙ t , x R , t [ a , T ] , T < , 0 < α < 1 . The initial condition u ( x , 0 ) = u 0 ( x ) , x R is a non-random function assumed to be non-negative and bounded, T α , t a is a conformable time-fractional derivative, σ : R R is Lipschitz continuous and W ˙ t a generalized derivative of Wiener process. Some precise condition for the existence and uniqueness of a solution of the class of equation is given and we also give an upper bound estimate on the growth moment of the solution. Unlike the growth moment of stochastic fractional heat equation with Riemann–Liouville or Caputo–Dzhrbashyan fractional derivative which grows in time like t c 1 exp ( c 2 t ) , c 1 , c 2 > 0 ; our result also shows that the energy of the solution (the second moment) grows exponentially in time for t [ a , T ] , T < but with at most c 1 exp ( c 2 ( t a ) 2 α 1 ) for some constants c 1 , and c 2 . Full article
Open AccessArticle
Determination of the Fractal Dimension of the Fracture Network System Using Image Processing Technique
Fractal Fract 2019, 3(2), 17; https://doi.org/10.3390/fractalfract3020017 - 08 Apr 2019
Viewed by 549
Abstract
Fractal dimension (FD) is a critical parameter in the characterization of a rock fracture network system. This parameter represents the distribution pattern of fractures in rock media. Moreover, it can be used for the modeling of fracture networks when the spatial distribution of [...] Read more.
Fractal dimension (FD) is a critical parameter in the characterization of a rock fracture network system. This parameter represents the distribution pattern of fractures in rock media. Moreover, it can be used for the modeling of fracture networks when the spatial distribution of fractures is described by the distribution of power law. The main objective of this research is to propose an automatic method to determine the rock mass FD in MATLAB using digital image processing techniques. This method not only accelerates analysis and reduces human error, but also eliminates the access limitation to a rock face. In the proposed method, the intensity of image brightness is corrected using the histogram equalization process and applying smoothing filters to the image followed by revealing the edges using the Canny edge detection algorithm. In the next step, FD is calculated in the program using the box-counting method, which is applied randomly to the pixels detected as fractures. This algorithm was implemented in different geological images to calculate their FDs. The FD of the images was determined using a simple Canny edge detection algorithm, a manual calculation method, and an indirect approach based on spectral decay rate. The results showed that the proposed method is a reliable and fast approach for calculating FD in fractured geological media. Full article
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Open AccessArticle
An Application of the Sonine–Letnikov Fractional Derivative for the Radial Schrödinger Equation
Fractal Fract 2019, 3(2), 16; https://doi.org/10.3390/fractalfract3020016 - 04 Apr 2019
Viewed by 494
Abstract
The Sonine–Letnikov fractional derivative provides the generalized Leibniz rule and, some singular differential equations with integer order can be transformed into the fractional differential equations. The solutions of these equations obtained by some transformations have the fractional forms, and these forms can be [...] Read more.
The Sonine–Letnikov fractional derivative provides the generalized Leibniz rule and, some singular differential equations with integer order can be transformed into the fractional differential equations. The solutions of these equations obtained by some transformations have the fractional forms, and these forms can be obtained as the explicit solutions of these singular equations by using the fractional calculus definitions of Riemann–Liouville, Grünwald–Letnikov, Caputo, etc. Explicit solutions of the Schrödinger equation have an important position in quantum mechanics due to the fact that the wave function includes all essential information for the exact definition of a physical system. In this paper, our aim is to obtain fractional solutions of the radial Schrödinger equation which is a singular differential equation with second-order, via the Sonine–Letnikov fractional derivative. Full article
Open AccessArticle
Novel Fractional Models Compatible with Real World Problems
Fractal Fract 2019, 3(2), 15; https://doi.org/10.3390/fractalfract3020015 - 01 Apr 2019
Cited by 2 | Viewed by 526
Abstract
In this paper, some real world modeling problems: vertical motion of a falling body problem in a resistant medium, and the Malthusian growth equation, are considered by the newly defined Liouville–Caputo fractional conformable derivative and the modified form of this new definition. We [...] Read more.
In this paper, some real world modeling problems: vertical motion of a falling body problem in a resistant medium, and the Malthusian growth equation, are considered by the newly defined Liouville–Caputo fractional conformable derivative and the modified form of this new definition. We utilize the σ auxiliary parameter for preserving the dimension of physical quantities for newly defined fractional conformable vertical motion of a falling body problem in a resistant medium. The analytical solutions are obtained by iterating this new fractional integral and results are illustrated under different orders by comparison with the Liouville–Caputo fractional operator. Full article
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Open AccessArticle
Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation
Fractal Fract 2019, 3(2), 14; https://doi.org/10.3390/fractalfract3020014 - 27 Mar 2019
Cited by 4 | Viewed by 580
Abstract
In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the [...] Read more.
In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders α and ρ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided. Full article
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