Jacob Emmanuel

This study focuses on the numerical solution of the Black-Scholes model, a key framework in financial mathematics for pricing European-style options. The model describes the behavior of option prices in relation to asset price, volatility, interest rate, and time to maturity. While exact analytical solutions exist for simple cases, numerical methods offer greater adaptability for real-world applications. In this work, we implement an explicit finite difference scheme to approximate the solution of the Black-Scholes partial differential equation. A stability criterion is derived to ensure numerical reliability, and accuracy is measured using the L1-norm by comparing results with the analytical solution. MATLAB simulations are used to compute the price of a European call option with a strike price of $100, a 12% risk-free interest rate, 10% volatility, and a one-year maturity. The generated graph (Figure 1) illustrates how the option value increases as the stock price moves from $70 to $130, notably rising when it exceeds the strike price. A comparative study with a semi-implicit scheme from existing literature confirms the enhanced precision of our explicit approach. These findings demonstrate the accuracy, efficiency, and practical utility of the explicit finite difference method for solving the Black-Scholes model.

This study presents a comprehensive numerical investigation of radiative heat transfer and flow dynamics involving Maxwell and Casson nanofluids flowing past a vertically aligned plate, incorporating the effects of internal heat generation. These non-Newtonian fluids exhibiting memory and yield stress behaviors are examined to understand their influence on velocity, temperature, and concentration profiles under coupled thermal and solutal gradients. The governing partial differential equations representing momentum, energy, and mass conservation are transformed into coupled nonlinear ordinary differential equations using similarity variables. These are solved using the fourth-order Runge-Kutta method with a MATLAB-based shooting technique. Key dimensionless parameters such as the Prandtl number, Grashof number, Biot number, heat source parameter, Nusselt number, Sherwood number, and skin friction coefficient are analyzed. The results show that internal heat generation significantly increases fluid temperature and velocity while reducing nanoparticle concentration and thermal transfer efficiency. Furthermore, elevated Biot numbers and thermal radiation broaden the thermal boundary layer, adversely affecting heat transfer. These findings have practical implications for engineering applications such as electronic cooling systems, chemical reactors, biomedical devices, and thermal insulation design. The insights gained enable improved modeling and thermal regulation in systems requiring efficient heat and mass transfer under non-Newtonian fluid behavior and internal heat sources.

In this paper, A 6 (six) compartmental (S, I U , I S , I A , Q, R) model was presented to examine the dynamical behavior of disease transmission in the system with quarantine effect on the symptomatic infected, asymptomatic infected and Reproduction number R 0 within a given population. The parameters model was analyzed and estimated experimentally using the real data of COVID-19 confirmed cases for Ethiopia via MATLAB 2021a. Reproduction number R 0 which is a key indicator to whether a disease outbreak spread force will persist or die out within population. R 0 was found using the next generation matrix with Gaussian elimination method to obtain the inverse of the transitive matrix. The model also aims at reducing R 0 owning to the fact that when the basic reproduction number is less than 1 infected person, disease dies out and when the reproduction number is greater than 1 infected person, the disease persists. The facts about R 0 geared us to mathematically check for the Routh-Hurwitz stability criteria and Lyapunov Functions to concisely establish the necessary and sufficient conditions for the Local and Global stability of model. results show that, when R 0 < 1 and R 0 > 1 the diseases free equilibrium and endemic equilibrium points are locally and globally asymptotically stable respectively. In order to interpret results and recommend possible control measure of disease, The dynamics of the Quarantine compartment in model was tested via sensitivity analysis to experimentally investigate transition/ transmission pattern. The effect of quarantine analysis on the model shows that preventive measures such as increase in quarantine with treatments during disease outbreak will significantly decrease the Reproduction number. Hence, increase in Quarantine compartment will flatten the curve of (S, I U , I S , I A , Q, R) dynamic model correspondingly.

Coronavirus (COVID-19) is the virus which has killed so many people in the world. The spread of COVID-19 within a region in South East Asia has been modelled using a compartment model called SEIR (Susceptible, Exposed, Infected, Recovered). Actual number of sick people needing treatments, or the number active case data was used to obtain realistic values of the model parameter such as the reproduction number (R0), incubation, and recovery periods. It is shown that at the beginning of the pandemic where most people were still not aware, the R0 was very high as seen by the steep increase of people got infected and admitted to the hospitals. Few weeks after the lockdown of the region was in place and people were obeying the regulation and observing safe distancing, the R0 values dropped significantly and converged to a steady value of about 3. Using the obtained model parameters, fitted on a daily basis, the maximum number of active cases converged to a certain value of about 2500 cases. It is expected that in the early June 2020 that the number of active cases will drop to a significantly low level. We Implement SEIR model to enumerate the infected Population and the number of casualties of this pandemic. Numerical Simulations was carried out to set off the analytical results in investigating the effect of the implementation of Isolation and social distancing as a function of time.