scholarly journals Study of Fractional Analytic Functions and Local Fractional Calculus

2021 ◽  
pp. 39-46
Author(s):  
Chii-Huei Yu
Keyword(s):  
Analytic Function ◽  
Fractional Calculus ◽  
Analytic Functions ◽  
Fundamental Theorem ◽  
Product Rule ◽  
Integration By Parts ◽  
Local Fractional Calculus ◽  
Change Of Variable ◽  
Variable Integration

In this present paper, the role of fractional analytic function in local fractional calculus is studied. Some important properties and theorems in local fractional calculus are discussed, such as product rule, quotient rule, chain rule, fundamental theorem of local fractional calculus, change of variable, integration by parts and so on. In addition, we propose several examples and formulas of local fractional calculus.

scholarly journals APPLICATIONS OF TWO IMPORTANT THEOREMS OF FRACTIONAL CALCULUS

2021 ◽  
Author(s):  
CHII-HUEI YU
Keyword(s):  
Fractional Calculus ◽  
Fundamental Theorem ◽  
Fractional Derivatives ◽  
The Other ◽  
Integration By Parts ◽  
Other Hand

Abstract. In this article, we study the fundamental theorem of fractional calculus and integration by parts for fractional calculus, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives. On the other hand, some examples are proposed to illustrate the applications of these two important theorems.

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

scholarly journals Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

2021 ◽  
Vol 19 (1) ◽  
pp. 329-337
Author(s):  
Huo Tang ◽  
Kaliappan Vijaya ◽  
Gangadharan Murugusundaramoorthy ◽  
Srikandan Sivasubramanian
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Lower Bounds ◽  
Analytic Functions ◽  
Function Class ◽  
Partial Sums ◽  
Generalized Function ◽  
Inclusion Relations ◽  
Alpha Beta

Abstract Let f k ( z ) = z + ∑ n = 2 k a n z n {f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( z ) / f ′ ( z ) } {\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\} , where f ( z ) f\left(z) belongs to the subclass J p , q m ( μ , α , β ) {{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean ( p , q ) \left(p,q) -differential operator. In addition, the inclusion relations involving N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.

scholarly journals Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ming-Sheng Hu ◽  
Ravi P. Agarwal ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Series Solution ◽  
Differential Operators ◽  
Fractional Wave Equation ◽  
Local Fractional Calculus ◽  
Vibrating String ◽  
Fractional Differential ◽  
Fractional Differential Operators

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. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

scholarly journals Fractal Dynamical Model of Vehicular Traffic Flow within the Local Fractional Conservation Laws

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Long-Fei Wang ◽  
Xiao-Jun Yang ◽  
Dumitru Baleanu ◽  
Carlo Cattani ◽  
Yang Zhao
Keyword(s):  
Fractional Calculus ◽  
Conservation Laws ◽  
Traffic Flow ◽  
Mathematical Theory ◽  
Conservation Equation ◽  
Vehicular Traffic ◽  
New Model ◽  
Local Fractional Calculus ◽  
Vehicular Traffic Flow ◽  
Fractal Network

We suggest a new model of the scale conservation equation in the mathematical theory of vehicular traffic flow on the fractal network based on the local fractional calculus.

scholarly journals Second Hankel determinant for a class of analytic functions defined by q-derivative operator

2019 ◽  
Vol 27 (2) ◽  
pp. 167-177
Author(s):  
Dorina Răducanu
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
Derivative Operator ◽  
Second Hankel Determinant

AbstractIn this paper, we obtain the estimates for the second Hankel determinant for a class of analytic functions defined by q-derivative operator and subordinate to an analytic function.

scholarly journals Mathematical Models Arising in the Fractal Forest Gap via Local Fractional Calculus

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Chun-Ying Long ◽  
Yang Zhao ◽  
Hossein Jafari
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Mathematical Models ◽  
Cantor Sets ◽  
Forest Gap ◽  
Local Fractional Derivative ◽  
Local Fractional Calculus ◽  
Growth Equations ◽  
Gap Models ◽  
Individual Trees

The forest new gap models via local fractional calculus are investigated. The JABOWA and FORSKA models are extended to deal with the growth of individual trees defined on Cantor sets. The local fractional growth equations with local fractional derivative and difference are discussed. Our results are first attempted to show the key roles for the nondifferentiable growth of individual trees.

LIPSCHITZ-TYPE INEQUALITIES FOR ANALYTIC FUNCTIONS IN BANACH ALGEBRAS

2019 ◽  
Vol 100 (3) ◽  
pp. 489-497
Author(s):  
SILVESTRU SEVER DRAGOMIR
Keyword(s):  
Analytic Function ◽  
Banach Algebra ◽  
Analytic Functions ◽  
Banach Algebras ◽  
Logarithmic Functions

In this paper we provide some bounds for the quantity $\Vert f(y)-f(x)\Vert$, where $f:D\rightarrow \mathbb{C}$ is an analytic function on the domain $D\subset \mathbb{C}$ and $x$, $y\in {\mathcal{B}}$, a Banach algebra, with the spectra $\unicode[STIX]{x1D70E}(x)$, $\unicode[STIX]{x1D70E}(y)\subset D$. Applications for the exponential and logarithmic functions on the Banach algebra ${\mathcal{B}}$ are also given.

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