Fractal Fract, Vol. 3, Pages 37: Inequalities Pertaining Fractional Approach through Exponentially Convex Functions Fractal and Fractional doi: 10.3390/fractalfract3030037 Authors: Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor 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...
Thu Jun 27, 2019 03:00
Fractal Fract, Vol. 3, Pages 36: Green’s Function Estimates for Time-Fractional Evolution Equations Fractal and Fractional doi: 10.3390/fractalfract3020036 Authors: Ifan Johnston Vassili Kolokoltsov 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 ...
Tue Jun 25, 2019 03:00
Fractal Fract, Vol. 3, Pages 35: A Criterion for Subfamilies of Multivalent Functions of Reciprocal Order with Respect to Symmetric Points Fractal and Fractional doi: 10.3390/fractalfract3020035 Authors: Shahid Mahmood Hari Mohan Srivastava Muhammad Arif Fazal Ghani Eman S. A. AbuJarad 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.
Tue Jun 25, 2019 03:00
Fractal Fract, Vol. 3, Pages 34: Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions Fractal and Fractional doi: 10.3390/fractalfract3020034 Authors: Bashir Ahmad Ahmed Alsaedi Sara Salem Sotiris K. Ntouyas 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...
Fri Jun 21, 2019 03:00
Fractal Fract, Vol. 3, Pages 32: On Extended General Mittag–Leffler Functions and Certain Inequalities Fractal and Fractional doi: 10.3390/fractalfract3020032 Authors: Marcela V. Mihai Muhammad Uzair Awan Muhammad Aslam Noor Tingsong Du Artion Kashuri Khalida Inayat Noor 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...
Tue Jun 18, 2019 03:00
Fractal Fract, Vol. 3, Pages 33: A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems Fractal and Fractional doi: 10.3390/fractalfract3020033 Authors: Ali Akgül Esra Karatas Akgül 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...
Tue Jun 18, 2019 03:00
Fractal Fract, Vol. 3, Pages 31: Random Variables and Stable Distributions on Fractal Cantor Sets Fractal and Fractional doi: 10.3390/fractalfract3020031 Authors: Alireza Khalili Golmankhaneh Arran Fernandez 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...
Tue Jun 11, 2019 03:00
Fractal Fract, Vol. 3, Pages 30: A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets Fractal and Fractional doi: 10.3390/fractalfract3020030 Authors: Dumitru Baleanu Hassan Kamil Jassim 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...
Mon Jun 03, 2019 03:00
Fractal Fract, Vol. 3, Pages 29: On Some Generalized Fractional Integral Inequalities for p-Convex Functions Fractal and Fractional doi: 10.3390/fractalfract3020029 Authors: Seren Salaş Yeter Erdaş Tekin Toplu Erhan Set 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,...
Mon May 20, 2019 03:00
Fractal Fract, Vol. 3, Pages 28: Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses Fractal and Fractional doi: 10.3390/fractalfract3020028 Authors: Snezhana Hristova Krasimira Ivanova 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...
Sat May 18, 2019 03:00
Fractal Fract, Vol. 3, Pages 27: Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces Fractal and Fractional doi: 10.3390/fractalfract3020027 Authors: Ayşegül Keten Mehmet Yavuz Dumitru Baleanu 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...
Thu May 16, 2019 03:00
Fractal Fract, Vol. 3, Pages 26: Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings Fractal and Fractional doi: 10.3390/fractalfract3020026 Authors: Dumitru Baleanu Hassan Kamil Jassim 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...
Sat May 11, 2019 03:00
Fractal Fract, Vol. 3, Pages 25: Analogues to Lie Method and Noether’s Theorem in Fractal Calculus Fractal and Fractional doi: 10.3390/fractalfract3020025 Authors: Alireza Khalili Golmankhaneh Cemil Tunç 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....
Tue May 07, 2019 03:00
Fractal Fract, Vol. 3, Pages 24: Some New Generalizations for Exponentially s-Convex Functions and Inequalities via Fractional Operators Fractal and Fractional doi: 10.3390/fractalfract3020024 Authors: Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor Ahmet Ocak Akdemir 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...
Sun Apr 28, 2019 03:00
Fractal Fract, Vol. 3, Pages 23: Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs Fractal and Fractional doi: 10.3390/fractalfract3020023 Authors: Ricardo Posadas-Mondragón Rodolfo G. Camacho-Velázquez 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,...
Wed Apr 24, 2019 03:00
Fractal Fract, Vol. 3, Pages 22: Hankel Determinants of Non-Zero Modulus Dixon Elliptic Functions via Quasi C Fractions Fractal and Fractional doi: 10.3390/fractalfract3020022 Authors: Rathinavel Silambarasan Adem Kılıçman 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...
Wed Apr 17, 2019 03:00
Fractal Fract, Vol. 3, Pages 21: Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions Fractal and Fractional doi: 10.3390/fractalfract3020021 Authors: Sotiris K. Ntouyas Ahmed Alsaedi Bashir Ahmad 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...
Wed Apr 17, 2019 03:00
Fractal Fract, Vol. 3, Pages 20: Statistical Mechanics Involving Fractal Temperature Fractal and Fractional doi: 10.3390/fractalfract3020020 Authors: Alireza Khalili Golmankhaneh 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....
Wed Apr 17, 2019 03:00
Fractal Fract, Vol. 3, Pages 19: New Estimates for Exponentially Convex Functions via Conformable Fractional Operator Fractal and Fractional doi: 10.3390/fractalfract3020019 Authors: Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor 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...
Mon Apr 15, 2019 03:00
Fractal Fract, Vol. 3, Pages 18: Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation Fractal and Fractional doi: 10.3390/fractalfract3020018 Authors: McSylvester Ejighikeme Omaba Eze R. Nwaeze 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 <...
Tue Apr 09, 2019 03:00
Fractal Fract, Vol. 3, Pages 17: Determination of the Fractal Dimension of the Fracture Network System Using Image Processing Technique Fractal and Fractional doi: 10.3390/fractalfract3020017 Authors: Rouhollah Basirat Kamran Goshtasbi Morteza Ahmadi 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...
Mon Apr 08, 2019 03:00
Fractal Fract, Vol. 3, Pages 16: An Application of the Sonine–Letnikov Fractional Derivative for the Radial Schrödinger Equation Fractal and Fractional doi: 10.3390/fractalfract3020016 Authors: Okkes Ozturk Resat Yilmazer 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...
Thu Apr 04, 2019 03:00
Fractal Fract, Vol. 3, Pages 15: Novel Fractional Models Compatible with Real World Problems Fractal and Fractional doi: 10.3390/fractalfract3020015 Authors: Ramazan Ozarslan Ahu Ercan Erdal Bas 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...
Mon Apr 01, 2019 03:00
Fractal Fract, Vol. 3, Pages 14: Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation Fractal and Fractional doi: 10.3390/fractalfract3020014 Authors: Ndolane Sene Aliou Niang Fall 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...
Wed Mar 27, 2019 02:00
Fractal Fract, Vol. 3, Pages 13: Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation Fractal and Fractional doi: 10.3390/fractalfract3010013 Authors: Melis Güneri Mustafa Saltan In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the...
Mon Mar 25, 2019 02:00
Fractal Fract, Vol. 3, Pages 12: Some New Fractional Trapezium-Type Inequalities for Preinvex Functions Fractal and Fractional doi: 10.3390/fractalfract3010012 Authors: Artion Kashuri Erhan Set Rozana Liko In this paper, authors the present the discovery of an interesting identity regarding trapezium-type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals are...
Sun Mar 24, 2019 02:00
Fractal Fract, Vol. 3, Pages 11: On the Fractal Langevin Equation Fractal and Fractional doi: 10.3390/fractalfract3010011 Authors: Alireza Khalili Golmankhaneh In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle- τ Cantor set. The fractal mean square displacement of different random walks on the middle- τ Cantor set are presented. Fractal under-damped and over-damped Langevin equations,...
Wed Mar 13, 2019 02:00
Fractal Fract, Vol. 3, Pages 10: On Analytic Functions Involving the q-Ruscheweyeh Derivative Fractal and Fractional doi: 10.3390/fractalfract3010010 Authors: Khalida Inayat Noor In this paper, we use concepts of q-calculus to introduce a certain type of q-difference operator, and using it define some subclasses of analytic functions. Inclusion relations, coefficient result, and some other interesting properties of these classes are studied.
Sun Mar 10, 2019 02:00
Fractal Fract, Vol. 3, Pages 9: Residual Power Series Method for Fractional Swift–Hohenberg Equation
Fractal Fract, Vol. 3, Pages 9: Residual Power Series Method for Fractional Swift–Hohenberg Equation Fractal and Fractional doi: 10.3390/fractalfract3010009 Authors: D. G. Prakasha P. Veeresha Haci Mehmet Baskonus In this paper, the approximated analytical solution for the fractional Swift–Hohenberg (S–H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique,...
Thu Mar 07, 2019 02:00
Fractal Fract, Vol. 3, Pages 8: The Fractal Calculus for Fractal Materials Fractal and Fractional doi: 10.3390/fractalfract3010008 Authors: Fakhri Khajvand Jafari Mohammad Sadegh Asgari Amir Pishkoo The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal...
Wed Mar 06, 2019 02:00
Fractal Fract, Vol. 3, Pages 7: Fractal Image Interpolation: A Tutorial and New Result Fractal and Fractional doi: 10.3390/fractalfract3010007 Authors: Chi Wah Kok Wing Shan Tam This paper reviews the implementation of fractal based image interpolation, the associated visual artifacts of the interpolated images, and various techniques, including novel contributions, that alleviate these awkward visual artifacts to achieve visually pleasant interpolated image. The fractal interpolation...
Sat Feb 23, 2019 02:00
Fractal Fract, Vol. 3, Pages 6: Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers Fractal and Fractional doi: 10.3390/fractalfract3010006 Authors: Vance Blankers Tristan Rendfrey Aaron Shukert Patrick D. Shipman Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers,...
Wed Feb 20, 2019 02:00
Fractal Fract, Vol. 3, Pages 5: On q-Uniformly Mocanu Functions Fractal and Fractional doi: 10.3390/fractalfract3010005 Authors: Rizwan S. Badar Khalida Inayat Noor Let f be analytic in open unit disc E = { z : | z | < 1 } with f ( 0 ) = 0 and f ′ ( 0 ) = 1 . The q-derivative of f is defined by: D q f ( z ) = f ( z ) − f ( q z ) ( 1 − q ) z , q ∈ ( 0 , 1 ) , z ∈ B − {...
Mon Feb 11, 2019 02:00
Fractal Fract, Vol. 3, Pages 4: Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions Fractal and Fractional doi: 10.3390/fractalfract3010004 Authors: Dimiter Prodanov The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions...
Fri Jan 25, 2019 02:00
Fractal Fract, Vol. 3, Pages 3: Study of Fractal Dimensions of Microcrystalline Cellulose Obtained by the Spray-Drying Method Fractal and Fractional doi: 10.3390/fractalfract3010003 Authors: Michael Ioelovich In this research, the fractal structure of beads of different sizes obtained by the spray-drying of aqueous dispersions of microcrystalline cellulose (MCC) was studied. These beads were formed as a result of the aggregation of rod-shaped cellulose nanocrystalline particles (CNP)....
Thu Jan 24, 2019 02:00
Fractal Fract, Vol. 3, Pages 2: Acknowledgement to Reviewers of Fractal Fract in 2018 Fractal and Fractional doi: 10.3390/fractalfract3010002 Authors: Fractal Fract Editorial Office Rigorous peer-review is the corner-stone of high-quality academic publishing [...]
Wed Jan 16, 2019 02:00
Fractal Fract, Vol. 3, Pages 1: Regularized Integral Representations of the Reciprocal Gamma Function Fractal and Fractional doi: 10.3390/fractalfract3010001 Authors: Dimiter Prodanov This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated....
Sat Jan 12, 2019 02:00
Fractal Fract, Vol. 2, Pages 30: Fractal Calculus of Functions on Cantor Tartan Spaces Fractal and Fractional doi: 10.3390/fractalfract2040030 Authors: Alireza Khalili Golmankhaneh Arran Fernandez In this manuscript, integrals and derivatives of functions on Cantor tartan spaces are defined. The generalisation of standard calculus, which is called F η -calculus, is utilised to obtain definitions of the integral and derivative of functions on Cantor tartan spaces of...
Tue Dec 18, 2018 02:00
Fractal Fract, Vol. 2, Pages 29: Approximate Controllability of Semilinear Stochastic Integrodifferential System with Nonlocal Conditions Fractal and Fractional doi: 10.3390/fractalfract2040029 Authors: Annamalai Anguraj K. Ramkumar The objective of this paper is to analyze the approximate controllability of a semilinear stochastic integrodifferential system with nonlocal conditions in Hilbert spaces. The nonlocal initial condition is a generalization of the classical initial condition...
Tue Nov 20, 2018 02:0
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