• Journal of applied mathematics & informatics
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• v.39 no.1_2
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• pp.197-214
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• 2021

In this paper, we have studied dynamics of fractional order mathematical model of malaria transmission for two groups of human population say semi-immune and non-immune along with growing stages of mosquito vector. The present fractional order mathematical model is the extension of integer order mathematical model proposed by Ousmane Koutou et al. For this study, Atangana-Baleanu fractional order derivative in Caputo sense has been implemented. In the view of memory effect of fractional derivative, this model has been found more realistic than integer order model of malaria and helps to understand dynamical behaviour of malaria epidemic in depth. We have analysed the proposed model for two precisely defined set of parameters and initial value conditions. The uniqueness and existence of present model has been proved by Lipschitz conditions and fixed point theorem. Generalised Euler method is used to analyse numerical results. It is observed that this model is more dynamic as we have considered all classes of human population and mosquito vector to analyse the dynamics of malaria.

• East Asian mathematical journal
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• v.21 no.2
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• pp.219-231
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• 2005

This paper investigates some results (Theorems 2.1-2.3, below) concerning certain classes of analytic and univalent functions, involving the familiar fractional derivative operators. We state interesting consequences arising from the main results by mentioning the cases connected with the starlikeness, convexity, close-to-convexity and quasi-convexity of geometric function theory. Relevant connections with known results are also emphasized briefly.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.929-967
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• 2016

Let p(t, x) be the fundamental solution to the problem $${\partial}^{\alpha}_tu=-(-{\Delta})^{\beta}u,\;{\alpha}{\in}(0,2),\;{\beta}{\in}(0,{\infty})$$. If ${\alpha},{\beta}{\in}(0,1)$, then the kernel p(t, x) becomes the transition density of a Levy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivatives $$D^n_x(-{\Delta}_x)^{\gamma}D^{\sigma}_tI^{\delta}_tp(t,x),\;{\forall}n{\in}{\mathbb{Z}}_+,\;{\gamma}{\in}[0,{\beta}],\;{\sigma},{\delta}{\in}[0,{\infty})$$, where $D^n_x$ x is a partial derivative of order n with respect to x, $(-{\Delta}_x)^{\gamma}$ is a fractional Laplace operator and $D^{\sigma}_t$ and $I^{\delta}_t$ are Riemann-Liouville fractional derivative and integral respectively.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.167-178
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• 2007

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.1
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• pp.63-74
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• 2018

In this paper, we study the initial value problem for neutral functional differential equations involving Caputo fractional derivative of order ${\alpha}{\in}(0,1)$ with infinite delay. Some sufficient conditions for the uniqueness and continuous dependence of solutions are established by virtue of fractional calculus and Banach fixed point theorem. Some results obtained showed that the solution was closely related to the conditions of delays and minor changes in the problem. An example is provided to illustrate the main results.

• Journal of Institute of Control, Robotics and Systems
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• v.21 no.8
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• pp.718-722
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• 2015

This paper proposes a fractional-order PID (Proportional-Integral-Derivative) control used electromagnetic strip stabilization controller in a continuous galvanizing line. Compared to a conventional PID controller, a fractional-order PID controller has integration-fractional-order and derivation-fractional-order as additional control parameters. Thanks to increased control parameters, more precise controller adjustment is available. In addition, accurate transfer function of a real system generally has a fractional-order form. Therefore, it is more adequate to use a fractional-order PID controller than a conventional PID controller for a real world system. Finite element models of a $1200{\times}2000{\times}0.8mm$ strip, which were extracted using a commercial software ANSYS were used as simulation plants, and Gaussian mixture models were used to find optimized control parameters that can reduce the strip vibrations to the lowest amplitude. Simulation results show that a fractional-order PID controller significantly reduces strip vibration and transient response time than a conventional PID controller.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.1-14
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• 2008

In this paper, A Riesz fractional diffusion equation with a nonlinear source term (RFDE-NST) is considered. This equation is commonly used to model the growth and spreading of biological species. According to the equivalent of the Riemann-Liouville(R-L) and $Gr\ddot{u}nwald$-Letnikov(G-L) fractional derivative definitions, an implicit difference approximation (IFDA) for the RFDE-NST is derived. We prove the IFDA is unconditionally stable and convergent. In order to evaluate the efficiency of the IFDA, a comparison with a fractional method of lines (FMOL) is used. Finally, two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.

• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.1115-1125
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• 2023

This paper deals with the controllability of linear and nonlinear generalized fractional dynamical systems in finite dimensional spaces. The results are obtained by using fractional calculus, Mittag-Leffler function and Schauder's fixed point theorem. Observability of linear system is also discussed. Examples are given to illustrate the theory.

• Nuclear Engineering and Technology
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• v.52 no.9
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• pp.2017-2024
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• 2020

In this study, a fractional order proportional integral derivative (FOPID) controller is designed to create the reference power trajectory and to conquer the uncertainties and external disturbances. A fractional nonlinear model was utilized to describe the nuclear reactor dynamic behaviour considering thermal-hydraulic effects. The controller parameters were tuned using optimization method in Matlab/Simulink. The FOPID controller was simulated using Matlab/Simulink and the controller performance was evaluated for Hard variation of the reference power and compared with that of integer order a proportional integral derivative (IOPID) controller by two models of fractional neutron point kinetic (FNPK) and classical neutron point kinetic (CNPK). Also, the FOPID controller robustness was appraised against the external disturbance and uncertainties. Simulation results showed that the FOPID controller has the faster response of the control attempt signal and the smaller tracking error with respect to the IOPID in tracking the reference power trajectory. In addition, the results demonstrated the ability of FOPID controller in disturbance rejection and exhibited the good robustness of controller against uncertainty.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.15-27
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• 2008

We present and discuss an algorithm for the numerical solution of initial value problems of the form $D_*^\alpha$y(t) = f(t, y(t)), y(0) = y0, where $D_*^\alpha$y is the derivative of y of order $\alpha$ in the sense of Caputo and 0<${\alpha}{\leq}1$. The algorithm is based on the fractional Euler's method which can be seen as a generalization of the classical Euler's method. Numerical examples are given and the results show that the present algorithm is very effective and convenient.