It is well known that the Black-Scholes model is used to establish the behavior of the option pricing in the financial market. In this paper, we propose the modified version of Black-Scholes model with two assets based on the Liouville-Caputo fractional derivative. The analytical solution of the proposed model is investigated by the Laplace transform homotopy perturbation method.
Abstract:The Black Scholes model is a well-known and useful mathematical model in financial markets. In this paper, the two-dimensional Black Scholes equation with European call option is studied. The explicit solution of this problem is carried out in the form of a Mellin-Ross function by using Laplace transform homotopy perturbation method. The solution example demonstrates that the proposed scheme is effective.
The recent outbreak of mosquito-borne Zika virus (ZIKV) in Brazil, with estimated cases surpassing 1.5 million, has gained attention due to its rapid spread and neurological complications associated with the infection such as microcephaly and Guillain-Barré syndrome. Beside vector transmission, primarily via Aedes mosquitoes, sexual transmission also contributes to the virus outbreak. The epidemiological patterns of these viruses suggest that Zika virus could cause other outbreaks, particularly in tropical regions with high vector concentration. To plan and prepare for a counter-control measure, it is important to study the previous cases. Hence, the control model is proposed from the deterministic model of Zika virus infection. Regarding control measures, the control functions for these strategies; vector elimination, vector-to-human contact reduction, and human-to-human contact reduction are introduced into the system. The necessary conditions for the optimal controls are determined using Pontryagin's maximum principle and the optimality system is solved using Rung-Kutta fourth order scheme. Consequently, numerical results of the system with control and the system without control are shown and discussed.
A static mixer is a fidelity engineered device for the continuous mixing of fluid/solid materials. To meet the requirements for mixing purpose, a proper design of the static mixer is important. In this study, we proposed various designs of static mixer based on the blade geometry. Six blade patterns including four twisted blades, four elliptical blades and other four combined geometries of two twisted blades and two elliptical blades are chosen to investigate its mixing performance. Mathematical model of the two-phase flow with fluid/solid interaction in the static mixer is presented. Numerical solution of the flow patterns of particulate solids, and the velocity and the pressure fields of the fluid in the static mixer with different blade geometries are carried out. To evaluate the quality of the mixing performance, the results obtained from six blade designs are compared via the relative standard deviation and the amount of pressure drop along the mixing path.
This paper studies the transient slip flow and heat transfer of a fluid driven by the oscillatory pressure gradient in a microchannel of elliptic cross section. The boundary value problem for the thermal-slip flow is formulated based on the assumption that the fluid flow is fully developed. The semi-analytical solutions of velocity and temperature fields are then determined by the Ritz method. These solutions include some existing known examples as special cases. The effects of the slip length and the ratio of minor to major axis of the elliptic cross section on the velocity and temperature distribution in the microchannel are investigated.
Abstract:The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We prove the existence and uniqueness of a solution for the FAFDEs. Furthermore, the stability analysis of the model is investigated and the numerical simulation is accordingly performed to support the proposed model.
This study concerns the COVID-19 pandemic in Thailand related to social isolation and vaccination policies. The behavior of disease spread is described by an epidemic model via a system of ordinary differential equations. The invariant region and equilibrium point of the model, as well as the basic reproduction number, are also examined. Moreover, the model is fitted to real data for the second wave and the third wave of the pandemic in Thailand by a sum square error method in order to forecast the future spread of infectious diseases at each time. Furthermore, the model predictive control technique with quadratic programming is used to investigate the schedule of preventive measures over a time horizon. As a result, firstly, the plan results are proposed to solve the limitation of ICU capacity and increase the survival rate of patients. Secondly, the plan to control the outbreak without vaccination shows a strict policy that is difficult to do practically. Finally, the vaccination plan significantly prevents disease transmission, since the populations who get the vaccination have immunity against the virus. Moreover, the outbreak is controlled in 28 weeks. The results of a measurement strategy for preventing the disease are examined and compared with a control and without a control. Thus, the schedule over a time horizon can be suitably used for controlling.
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