New classification of analytic functions with negative coefficients by Shigeyoshi Owa, Milutin Obradovic

“…The object of the present paper is to prove various distortion theorems for functions in A(p,n,Bk), and for fractional calculus of functions belonging to A(p,n,Bk). …”
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Notes on Local and Nonlocal Intuitionistic Fuzzy Fractional Boundary Value Problems with Caputo Fractional Derivatives by Ali El Mfadel, Said Melliani, M’hamed Elomari

“…In this paper, we investigate the existence and uniqueness results of intuitionistic fuzzy local and nonlocal fractional boundary value problems by employing intuitionistic fuzzy fractional calculus and some fixed-point theorems. As an application, we conclude this manuscript by giving an example to illustrate the obtained results.…”
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The Solvability and Optimal Controls for Some Fractional Impulsive Equation by Xianghu Liu, Zhenhai Liu, Jiangfeng Han

“…Firstly, we introduce the fractional calculus, Gronwall inequality, and Leray-Schauder’s fixed point theorem. …”
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Generalized classes of starlike and convex functions of order α by Shigeyoshi Owa, C. Y. Shen

“…We have introduced, in this paper, the generalized classes of starlike and convex functions of order α by using the fractional calculus. We then proved some subordination theorems, argument theorems, and various results of modified Hadamard product for functions belonging to these classes. …”
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-Dimensional Fractional Lagrange's Inversion Theorem by F. A. Abd El-Salam

“…The required basic definitions, lemmas, and theorems in the fractional calculus are presented. A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. …”
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Existence of Solutions for Fractional Differential Inclusions with Separated Boundary Conditions in Banach Space by Mabrouk Bragdi, Amar Debbouche, Dumitru Baleanu

“…In order to obtain necessary conditions for the existence result, we apply the fixed point technique, fractional calculus, and multivalued analysis.…”
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Adaptation of Differential Transform Method for the Numeric-Analytic Solution of Fractional-Order Rössler Chaotic and Hyperchaotic Systems by Asad Freihat, Shaher Momani

“…The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.…”
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New Expansion Formulas for a Family of the λ-Generalized Hurwitz-Lerch Zeta Functions by H. M. Srivastava, Sébastien Gaboury

“…These expansion formulas are obtained by making use of some important fractional calculus theorems such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. …”
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Numerical Solution of a Kind of Fractional Parabolic Equations via Two Difference Schemes by Abdon Atangana, Dumitru Baleanu

“…A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. …”
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On the Existence and Uniqueness Results for Fuzzy Linear and Semilinear Fractional Evolution Equations Involving Caputo Fractional Derivative by Ali El Mfadel, Said Melliani, M’hamed Elomari

“…The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach contraction principle. …”
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Signal Processing for Nondifferentiable Data Defined on Cantor Sets: A Local Fractional Fourier Series Approach by Zhi-Yong Chen, Carlo Cattani, Wei-Ping Zhong

“…Our results can be observed as significant extensions of the previously known results for the Fourier series in the framework of the local fractional calculus. Some examples are given to illustrate the efficiency and implementation of the present method.…”
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Oscillation of Nonlinear Fractional Dynamic Equations with Forcing Term by S. Manikandan, V. Muthulakshmi, S. Harikrishnan, Porpattama Hammachukiattikul

“…Our approach is determined from the implementation of generalized Riccati transformation, some properties of conformable time-scale fractional calculus, and certain mathematical inequalities. …”
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Fractional Order Models of Industrial Pneumatic Controllers by Abolhassan Razminia, Dumitru Baleanu

“…In the light of fractional calculus, a fractional order derivative-derivative (FrDD) controller and integral-derivative (FrID) are remodeled. …”
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Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String by Ming-Sheng Hu, Ravi P. Agarwal, Xiao-Jun Yang

“…We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. …”
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The Use of Cubic Splines in the Numerical Solution of Fractional Differential Equations by W. K. Zahra, S. M. Elkholy

“…Fractional calculus became a vital tool in describing many phenomena appeared in physics, chemistry as well as engineering fields. …”
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Option Pricing under the Subordinated Market Models by Longjin Lv, Changjuan Zheng, Luna Wang

“…Firstly, we prove that the subordinated Brownian motion controlled by the fractional diffusion equation has many financial properties, such as self-similarity, leptokurtic, and long memory, which indicate that the fractional calculus can describe the financial data well. …”
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Analytical Solutions of Fractional Differential Equations Using the Convenient Adomian Series by Xiang-Chao Shi, Lan-Lan Huang, Zhen-Guo Deng, Dan Liu

“…Due to the memory trait of the fractional calculus, numerical or analytical solution of higher order becomes very difficult even impossible to obtain in real engineering problems. …”
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Multipoint BVP for the Langevin Equation under φ-Hilfer Fractional Operator by Mohammed A. Almalahi, Satish K. Panchal, Fahd Jarad

“…At first, we derive the equivalent solution to the proposed problem at hand by relying on the results and properties of the generalized fractional calculus. Next, we investigate and develop sufficient conditions for the existence and uniqueness of solutions by means of semigroups of operator approach and the Krasnoselskii fixed point theorems as well as Banach contraction principle. …”
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ABC Fractional Derivative for Varicella-Zoster Virus Using Two-Scale Fractal Dimension Approach with Vaccination by Jirong Yang, Farkhanda Afzal, Perpetual Appiah

“…At the end, we provide the graphical results showing the effectiveness of two-scale dimension and fractional calculus in the current model.…”
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Fractional Dynamics of Genetic Algorithms Using Hexagonal Space Tessellation by J. A. Tenreiro Machado

“…Besides the analysis of the spacial layout, a modelling of the time evolution is performed by adopting a distance measure and the modelling in the Fourier domain in the perspective of fractional calculus. The results reveal a consistent, and easy to interpret, set of model parameters for distinct operating conditions.…”
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