• 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.

• Communications of the Korean Mathematical Society
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• v.31 no.4
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• pp.869-878
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• 2016

This paper presents an ecient fractional shifted Legendre polynomial method to solve the fractional Volterra's model for population growth model. The fractional derivatives are described based on the Caputo sense by using Riemann-Liouville fractional integral operator. The theoretical analysis, such as convergence analysis and error bound for the proposed technique has been demonstrated. In applications, the reliability of the technique is demonstrated by the error function based on the accuracy of the approximate solution. The numerical applications have provided the eciency of the method with dierent coecients of the population growth model. Finally, the obtained results reveal that the proposed technique is very convenient and quite accurate to such considered problems.

• East Asian mathematical journal
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• v.36 no.1
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• pp.81-90
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• 2020

Many compartment models assume a kinetically homogeneous amount of materials that have well-stirred compartments. However, based on observations from such processes, they have been heuristically fitted by exponential or gamma distributions even though biological media are inhomogeneous in real environments. Fractional differential equations using a specific kernel in Pharmacokinetic/Pharmacodynamic (PKPD) model are recently introduced to account for abnormal drug disposition. We discuss a tumor growth inhibition (TGI) model using fractional-order derivative from it. This represents a tumor growth delay by cytotoxic agents and additionally show variations in the equilibrium points by the change of fractional order. The result indicates that the equilibrium depends on the tumor size as well as a change of the fractional order. We find that the smaller the fractional order, the smaller the equilibrium value. However, a difference of them is the number of concavities and this indicates that TGI over time profile for fitting or prediction should be determined properly either fractional order or tumor sizes according to the number of concavities shown in experimental data.

• Kyungpook Mathematical Journal
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• v.60 no.4
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• pp.753-765
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• 2020

Smoking is one of the main causes of health problems and continues to be one of the world's most significant health challenges. In this paper, we use the multi-step Adomian decomposition method (MSADM) to obtain approximate analytical solutions for a mathematical fractional model of the evolution of the smoking habit. The proposed MSADM scheme is only a simple modification of the Adomian decomposition method (ADM), in which ADM is treated algorithmically with a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding problems. A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically. The results reveal that the method is effective and convenient for solving linear and nonlinear differential equations of fractional order.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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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.

• Korean Journal of Mathematics
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• v.17 no.4
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• pp.399-409
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• 2009

We study the weak convergence of various models to Fractional Brownian motion. First, we consider arima process and ON/OFF source model which allows for long packet trains and long inter-train distances. Finally, we figure out power spectrum density as a Fourier transform of autocorrelation function of arima model and binary signal model.

• 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.

• Honam Mathematical Journal
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• v.37 no.4
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• pp.411-421
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• 2015

This paper investigates the issue of analytic travelling wave solutions for some important coupled models of fractional order. Analytic travelling wave solutions of the considered model are found by means of the Q-function method. The results give us that the Q-function method is very simple, reliable and effective for searching analytic exact solutions of complex nonlinear partial differential equations.

• Korean Journal of Mathematics
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• v.15 no.1
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• pp.71-78
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• 2007

We consider arrival process and ON/OFF source model which allows for long packet trains and long inter-train distances. We prove the weak convergence of theses processes to Fractional Brownian motion. Finally, we figure out the coefficients of $B_H(t)$ and time $t$ when ON/OFF periods have the Pareto distribution.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.695-708
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• 2024

In this paper we present a non-standard finite difference method for solving a fractional decay model. The proposed NSFDM is constructed by incorporating a non-standard denominator function, resulting in an explicit numerical scheme as easy as the conventional Euler method, but it provides very accurate solutions and has unconditional stability. Two examples from the literature are presented to demonstrate the performance of the proposed numerical scheme, which is compared to three methods from the literature. It is found that the method's estimated errors are extremely minimal, such as within the machine precision.