Integral expressions for Mathieu-type power series and for the Butzer-Flocke-Hauss Ω-function

2011 ◽  
Vol 14 (4) ◽  
Author(s):  
Živorad Tomovski ◽  
Tibor Pogány
Keyword(s):  
Power Series ◽  
Fractional Order ◽  
Integral Representations ◽  
Type Power

AbstractIn this paper several integral representations for the generalized fractional order Mathieu type power series $S_\mu (r;x) = \sum\limits_{n = 1}^\infty {\frac{{2nx^n }} {{(n^2 + r^2 )^{\mu + 1} }}(r \in \mathbb{R},\mu > 0,|x| \leqslant 1)} $ are presented. Also new integral expressions are derived for the Butzer-Flocke-Hauss (BFH) complete Omega function.

scholarly journals On a new (p,q)-Mathieu-type power series and its applications

2019 ◽  
Vol 13 (1) ◽  
pp. 309-324 ◽  
Author(s):  
Khaled Mehrez ◽  
Zivorad Tomovski
Keyword(s):  
Power Series ◽  
Special Function ◽  
Integral Representations ◽  
Type Series ◽  
Type Integral ◽  
Type Power

Our aim in this paper, is to establish certain new integral representations for the (p,q)-Mathieu-type power series. In particular, we investigate the Mellin-Barnes type integral representations for a particular case of these special function. Moreover, we introduce the notion of the (p, q)-Mittag- Leffler functions and we present a relationships between these two functions. Some other applications are proved, in particular two Tur?n type inequalities for the (p,q)-Mathieu-type series are derived.

scholarly journals Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel

2021 ◽  
Vol 5 (4) ◽  
pp. 273
Author(s):  
Iván Area ◽  
Juan J. Nieto
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Fractional Order ◽  
Numerical Approximations ◽  
Formal Power

In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series.

scholarly journals A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Mohamed R. Ali
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Power Series ◽  
Fractional Order ◽  
Nonlinear Ordinary Differential Equation ◽  
Lie Symmetry ◽  
Series Method ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Bo Equation

We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order, we complete the solutions by utilizing the power series method (PSM).

scholarly journals The Study of Fractional Order Controller with SLAM in the Humanoid Robot

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Shuhuan Wen ◽  
Xiao Chen ◽  
Yongsheng Zhao ◽  
Ahmad B. Rad ◽  
Kamal Mohammed Othman ◽  
...  
Keyword(s):  
Transfer Function ◽  
Power Series ◽  
Fractional Order ◽  
Humanoid Robot ◽  
Power Series Expansion ◽  
Pi Controller ◽  
Fractional Order Pi Controller ◽  
Fractional Order Controller ◽  
Accumulated Error ◽  
Fopi Controller

We present a fractional order PI controller (FOPI) with SLAM method, and the proposed method is used in the simulation of navigation of NAO humanoid robot from Aldebaran. We can discretize the transfer function by the Al-Alaoui generating function and then get the FOPI controller by Power Series Expansion (PSE). FOPI can be used as a correction part to reduce the accumulated error of SLAM. In the FOPI controller, the parameters (Kp,Ki,  and  α) need to be tuned to obtain the best performance. Finally, we compare the results of position without controller and with PI controller, FOPI controller. The simulations show that the FOPI controller can reduce the error between the real position and estimated position. The proposed method is efficient and reliable for NAO navigation.

Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order

2021 ◽  
Vol 8 (3) ◽  
pp. 537-548
Author(s):  
S. E. Fadugba ◽  
◽  
F. Ali ◽  
A. B. Abubakar ◽  
◽  
...  
Keyword(s):  
Power Series ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Epidemic Model ◽  
Differential Transform Method ◽  
Host Community ◽  
Convergent Power Series ◽  
Transform Method ◽  
Differential Transform ◽  
Reduced Differential Transform Method

This paper proposes the Caputo Fractional Reduced Differential Transform Method (CFRDTM) for Susceptible-Exposed-Infected-Recovered (SEIR) epidemic model with fractional order in a host community. CFRDTM is the combination of the Caputo Fractional Derivative (CFD) and the well-known Reduced Differential Transform Method (RDTM). CFRDTM demonstrates feasible progress and efficiency of operation. The properties of the model were analyzed and investigated. The fractional SEIR epidemic model has been solved via CFRDTM successfully. Hence, CFRDTM provides the solutions of the model in the form of a convergent power series with easily computable components without any restrictive assumptions.

scholarly journals Memories of the Future. Predictable and Unpredictable Information in Fractional Flipping a Biased Coin

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 807 ◽  
Author(s):  
Dimitri Volchenkov
Keyword(s):  
Power Series ◽  
Strong Coupling ◽  
Fractional Order ◽  
Transition Matrix ◽  
Order Parameter ◽  
Destructive Interference ◽  
Fair Coin ◽  
Coin Flipping ◽  
Information Components ◽  
Biased Coin

Some uncertainty about flipping a biased coin can be resolved from the sequence of coin sides shown already. We report the exact amounts of predictable and unpredictable information in flipping a biased coin. Fractional coin flipping does not reflect any physical process, being defined as a binomial power series of the transition matrix for “integer” flipping. Due to strong coupling between the tossing outcomes at different times, the side repeating probabilities assumed to be independent for “integer” flipping get entangled with one another for fractional flipping. The predictable and unpredictable information components vary smoothly with the fractional order parameter. The destructive interference between two incompatible hypotheses about the flipping outcome culminates in a fair coin, which stays fair also for fractional flipping.

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