• Smart Structures and Systems
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• 제18권4호
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• pp.707-731
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• 2016

A unified mathematical model of the equations of generalized magneto-thermoelasticty based on fractional derivative heat transfer for isotropic perfect conducting media is given. Some essential theorems on the linear coupled and generalized theories of thermoelasticity e.g., the Lord- Shulman (LS) theory, Green-Lindsay (GL) theory and the coupled theory (CTE) as well as dual-phase-lag (DPL) heat conduction law are established. Laplace transform techniques are used. The method of the matrix exponential which constitutes the basis of the state-space approach of modern theory is applied to the non-dimensional equations. The resulting formulation is applied to a variety of one-dimensional problems. The solutions to a thermal shock problem and to a problem of a layer media are obtained in the present of a transverse uniform magnetic field. According to the numerical results and its graphs, conclusion about the new model has been constructed. The effects of the fractional derivative parameter on thermoelastic fields for different theories are discussed.

• Nuclear Engineering and Technology
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• 제52권9호
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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.

• Structural Engineering and Mechanics
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• 제52권5호
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• pp.937-956
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• 2014

The MEMS structures usually are made from silicon; consideration of the viscoelastic effect in microbeams duo to the phenomena of silicon creep is necessary. Application of the fractional model of microbeams made from viscoelastic materials is studied in this paper. Quasi-static and dynamical responses of an electrically actuated viscoelastic microbeam are investigated. For this purpose, a nonlinear finite element formulation of viscoelastic beams in combination with the fractional derivative constitutive equations is elucidated. The four-parameter fractional derivative model is used to describe the constitutive equations. The electric force acting on the microbeam is introduced and numerical methods for solving the nonlinear algebraic equation of quasi-static response and nonlinear equation of motion of dynamical response are described. The deflected configurations of a microbeam for different purely DC voltages and the tip displacement of the microbeam under a combined DC and AC voltages are presented. The validity of the present analysis is confirmed by comparing the results with those of the corresponding cases available in the literature.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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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.

• Geomechanics and Engineering
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• 제17권4호
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• pp.385-392
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• 2019

The present fractional-order plasticity models for granular soil are mainly established under the triaxial compression condition, due to its difficult in analytically solving the fractional differentiation of the third stress invariant, e.g., Lode's angle. To solve this problem, a three dimensional fractional-order elastoplastic model based on the transformed stress method, which does not rely on the analytical solution of the Lode's angle, is proposed. A nonassociated plastic flow rule is derived by conducting the fractional derivative of the yielding function with respect to the stress tensor in the transformed stress space. All the model parameters can be easily determined by using laboratory test. The performance of this 3D model is then verified by simulating multi series of true triaxial test results of rockfill.

• 한국구조물진단유지관리공학회 논문집
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• 제6권1호
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• pp.171-177
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• 2002

The viscoelastic dampers are considered to be one of the most efficient means of upgrading existing structures against seismic loads. Generally in the dynamic analysis of a structure with added viscoelastic dampers the internal forces of the dampers are represented by constants that are linearly proportional to displacement and velocity. The purpose of this study is to verify the validity of the linear Kelvin model by comparing the results from the linear analysis with those obtained from the more rigorous nonlinear model such as fractional derivative model. According to the results the structural responses of 1-DOF structure obtained using the linear model are very close to those obtained from nonlinear model. However for multi-D0F structure the difference between the results from both models is enlarged as a results of the assumptions associated with the linear modeling of the viscoelastic dampers.

• Nonlinear Functional Analysis and Applications
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• 제29권2호
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• pp.597-620
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• 2024

In this study, the conformable fractional derivative(CFD) of order 𝝔 in conjunction with the LC operator of orderρ is used to develop the model of the transmission of the A(H1N1) influenza infection. A brand-new A(H1N1) influenza infection model is presented, with a population split into four different compartments. Fixed point theorems were used to prove the existence of the solutions and uniqueness of this model. The basic reproduction number R₀ was determined. The least and most sensitive variables that could alter R₀ were then determined using normalized forward sensitivity indices. Through numerical simulations carried out with the aid of the Adams-Moulton approach, the study also investigated the effects of numerous biological characteristics on the system. The findings demonstrated that if 𝝔 < 1 and ρ < 1 under the CFD, also the findings demonstrated that if 𝝔 = 1 and ρ = 1 under the CFD, the A(H1N1) influenza infection will not vanish.

• Advances in nano research
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• 제16권2호
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• pp.141-154
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• 2024

This study aims to develop explicit models to investigate thermo-mechanical interactions in moving nanobeams. These models aim to capture the small-scale effects that arise in continuous mechanical systems. Assumptions are made based on the Euler-Bernoulli beam concept and the fractional Zener beam-matter model. The viscoelastic material law can be formulated using the fractional Caputo derivative. The non-local Eringen model and the two-phase delayed heat transfer theory are also taken into account. By comparing the numerical results to those obtained using conventional heat transfer models, it becomes evident that non-localization, fractional derivatives and dual-phase delays influence the magnitude of thermally induced physical fields. The results validate the significant role of the damping coefficient in the system's stability, which is further dependent on the values of relaxation stiffness and fractional order.

• 대한수학회보
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• 제55권4호
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• pp.1241-1261
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• 2018

We derive discrete time model of the geometric fractional Brownian motion. It provides numerical pricing scheme of financial derivatives when the market is driven by geometric fractional Brownian motion. With the convergence analysis, we guarantee the convergence of Monte Carlo simulations. The strong convergence rate of our scheme has order H which is Hurst parameter. To obtain our model we need to convert Wick product term of stochastic differential equation into Wick free discrete equation through Malliavin calculus but ours does not include Malliavin derivative term. Finally, we include several numerical experiments for the option pricing.

• 호남수학학술지
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• 제44권4호
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• pp.572-593
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• 2022

Crime committed by civilians and criminals using legal and illegal firearms and conversion of legal firearms into illegal ones has become a common practice around the world. As a result, policies to control civilian gun ownership have been debated in several countries. The issue arose because the linkages between firearm-related mortality, weapon accessibility, and violent crime data can imply diverse options for addressing criminality. In this paper, we have projected a mathematical model in terms of the Caputo fractional derivative to address the issues viz. input of legal guns, crime committed by legal and illegal guns, and strict government policies to monitor the license of legal guns, strict action against violent crime. The boundedness, existence and uniqueness of solutions and the stability of points of equilibrium are examined. It is observed that violent crime increases with the increase of crime committed by illegal guns, crime committed by legal guns and, decreases with the increase of legal guns, the deterrent effect of civilian gun ownership, and action of law against crime. Further, legal guns increase with the increase of the limitation of trade of illegal guns and decrease with the increase of conversion of legal guns into illegal guns and increase of the growth rate of illegal guns. Again, as crime is committed by legal guns also, the policy of illegal gun control does not assure a crime-free society. Weak gun control can lead to a society with less crime. Theoretical aspects are numerically verified in the present work.