Search

Black-Scholes (BS) equation is a popular mathematical model for determining the value of option in financial derivatives. To predict the option value during the contract of the option is a big problem. Several studies have been shown that the option price value can be determined by applying different methods. In this paper, we have discussed three finite difference methods: Explicit, Implicit and Crank-Nicolson for solving Black-Scholes equation for European call option and compared the obtained results with the exact value. It is found that the Crank-Nicolson method is more accurate and cost effective in comparison with explicit and implicit methods.

Water movement in an unsaturated porous medium (soil) can be expressed by Richards equation with the mass conservation law and Darcy-Buckingham's law. This equation can be expressed in three different forms as pressure head-based, moisture content based and mixed from. In this study, we solve one dimensional Richards Equation in mixed form numerically using finite difference method with various time-stepping schemes: Forward Euler, Backward Euler, Crank-Nicolson and a Stabilized Runge-Kutta-Legendre Super Time-Stepping and we compare their performances using Dirichlet boundary condition on an isotropic homogeneous vertical soil column.

In this work both linear and nonlinear advection-diffusion equations are considered and discussed their analytical solutions with different initial and boundary conditions. The work of Ogata and Banks, Harleman and Rumer, Cleary and Adrian, Atul Kumar et al., Mojtabi and Deville are reviewed for linear advection-diffusion equations and for nonlinear, we have chosen the work of Sakai and Kimura. Some enthusiastic functions used in the articles, drawbacks and applications of the results are discussed. Reduction of the advection-diffusion equations into diffusion equations make the governing equation solvable by using integral transform method for analytical solution. For nonlinear advection-diffusion equations, the Cole-Hopf transformation is used to reduce into the diffusion equation. Different dispersion phenomena in atmosphere, surface and subsurface area are outlined.

Flow movement in unsaturated soil can be expressed by Richards equation. This equation can be obtained by applying the mass conversation law and the Darcy law. In this work, we solve one-dimensional Kirchhoff transformed Richards equation with loss of water due to the evaporation of unsaturated porous media (soils) and transpiration of plants numerically using Crank-Nicolson method. The result has compared with evapotranspiration function and without it in the governing equation. It has found that an additional work in time and flow movement is needs to reach the given boundary condition for the model without evapotranspiration.

The human thermal comfort is the state of mind, which is affected not only by the physical and body’s internal physiological phenomena but also by the clothing properties such as thermal resistance of clothing, clothing insulation, clothing area factor, air insulation, and relative humidity. In this work, we extend the one-dimensional Pennes’ bioheat transfer equation by adding the protective clothing layer. The transient temperature profile with the clothing layer at the different time steps has been carried out using a fully implicit Finite Difference (FD) Scheme with interface condition between body and clothes. Numerically computed results are bound to agree that the clothing insulation and air insulation provide better comfort and keep the body at the thermal equilibrium position. The graphical representation of the results also verifies the effectiveness and utility of the proposed model.

We solve the Black - Scholes equation for option pricing numerically using an Explicit finite difference method. To overcome the stability restriction of the explicit scheme for parabolic partial differential equations in the time step size Courant-Friedrichs-Lewy (CFL) condition, we employ a Super Time Stepping (STS) strategy based on modified Chebyshev polynomial. The numerical results show that the STS scheme boasts of large efficiency gains compared to the standard explicit Euler method.

Numerical methods form an important part of the pricing of financial derivatives where there is no closed form analytical formula. Black-Scholes equation is a well known partial differential equation in financial mathematics. In this paper, we have studied the numerical solutions of the Black-Scholes equation for European options (Call and Put) as well as American options with dividends. We have used different approximate to discretize the partial differential equation in space and explicit (Forward Euler’s), fully implicit with projected Successive Over-Relaxation (SOR) algorithm and Crank-Nicolson scheme for time stepping. We have implemented and tested the methods in MATLAB. Finally, some numerical results have been presented and the effects of dividend payments on option pricing have also been considered.

Clothing plays a major role in protecting the human body from cold and hot environment by working as an insulator. The physical factors such as conduction, convection, radiation, evaporation and the physiological factors such as blood flow and metabolism within the body together with clothing system help to maintain human thermal comfort and the human thermoregulatory system. On the other hand, body is continuously losing a small amount of heat from the air gap between skin surface and protective clothes. This paper studies the role and effect of the insulations of clothes and air for dry heat loss and the moisture transfer due to air velocity along with walking speed and observes how they work together for temperature distribution in human body. Pennes’ bioheat equation is taken as a model and Backward in Time and Central in Space (BTCS) scheme is used for obtaining the solution of the model. Clothing parameters, wind speed and walking speed have been added in the boundary condition in Pennes’s model among them the clothing insulation helps prevent the heat loss whereas air velocity escalate heat loss from the body. The numerical results are implemented in Python.

The human thermal comfort is affected by the body’s heat exchange mechanism conduction, convection, radiation, and evaporation. The mode of heat transfer between the body and environment depends upon the human internal physiological phenomena, together with the boundary conditions. The present paper provides the comprehensive overview of the thermoregulatory system of human body and studies the numerical solution of unsteady-state one dimensional Pennes bio-heat equation with appropriate boundary conditions. The solution is used to observe the temperature profiles at different thermal conductivities, and different heat transfer coefficients in the living tissue at the various time steps. Various physical and physiological factors across the cylindrical living tissue have been incorporated in the model.

To study the spreading nature of Delta variant (B.1.617.2) dominated COVID-19 in Nepal to help the policymakers assess and manage health care facilities and vaccination programs.Deterministic mathematical models in the form of systems of ordinary differential equations were developed to describe the COVID-19 transmission in the high- and the low-risk regions of Nepal. The models were validated using the multiple data sets containing daily new cases in the whole country, the high-risk region, the low-risk region, and cases needing medical care, ICU, and ventilator.We found the reproduction number of RThe data-driven mathematical models are useful to assess control programs in resource-limited countries. The appropriate combination of vaccination and social distancing are necessary to keep the pandemic under-control and manage the medical care facilities in Nepal.

We solve one-dimensional Kirchhof transformed Richards equation numerically using finite difference method with various time-stepping schemes, forward in time central in space (FTCS), backward in time central in space (BTCS), Crank–Nicolson (CN), and a stabilized Runge–Kutta–Legendre super time-stepping (RKL), and compare their performances.

In this work, we develop a mathematical model for transport and growth of microbes by natural (rain) water infiltration and flow through unsaturated porous soil along the vertical direction under gravity and capillarity by coupling a system of advection diffusion equations (for concentration of microbes and their growth-limiting substrate) with the Richards equation. The model takes into consideration several major physical, chemical, and biological mechanisms. The resulting coupled system of PDEs together with their boundary conditions is highly nonlinear and complicated to solve analytically. We present both a partial analytic approach towards solving the nonlinear system and finding the main type of dynamics of microbes, and a full-scale numerical simulation. Following the auxiliary equation method for nonlinear reaction-diffusion equations, we obtain a closed form traveling wave solution for the Richards equation. Using the propagating front solution for the pressure head, we reduce the transport equation to an ODE along the moving frame and obtain an analytic solution for the history of bacteria concentration for a specific test case. To solve the system numerically, we employ upwind finite volume method for the transport equations and stabilized explicit Runge–Kutta–Legendre super-time-stepping scheme for the Richards equation. Finally, some numerical simulation results of an infiltration experiment are presented, providing a validation and backup to the analytic partial solutions for the transport and growth of bacteria in the soil, stressing the occurrence of front moving solitons in the nonlinear dynamics.

This paper focuses on the modeling of indoor air pollution in a naturally ventilated kitchen based on the computational fluid dynamics (CFD) approach to assess its ventilation effectiveness. The 3D incompressible Navier-Stokes equations with conservation of total energy are solved numerically using ANSYS-Fluent software and the pollutant paths are investigated from the profiles of velocity, pressure, turbulent kinetic energy and temperature throughout different sections of the kitchen. Experimental verification is made through the measurement of indoor air contaminant in the same kitchen. The simulation results agree well with the on-site measured data.

This paper is the extension of the work originally presented in 2nd International Conference on Man and Machine Interfacing (MAMI 2017) and the paper of Yue et al. The present study aims at observing the effect of various parameters on temperature distribution profiles at various environmental temperature, tissue thermal conductivities, metabolic rates, blood perfusion rates, and heat transfer coefficients. The analytic solution of Pennes’ bioheat equation in the steady-state case is obtained by using the Modified Bessel’s equation incorporating the effect of sweating and non-sweating state of the body. From the study, it was observed that the variation of atmospheric temperature and heat transfer coefficients have a significant effect for the temperature distribution in the body towards the skin surface.

In this work, an incompressible viscous Couette flow is derived by simplifying the Navier-Stokes equations and the resulting one dimensional linear parabolic partial differential equation is solved numerically employing a second order finit difference Crank-Nicolson scheme. The numerical solution and the exact solution are presented graphically.Journal of the Institute of Engineering, 2016, 12(1): 105-113