Naveed Zaman2025-11-182025-11-182016https://escholar.umt.edu.pk/handle/123456789/10442The NLBC is calculated by using Simpson’s 1/3 formula, while the derivatives involved are solved by adopting higher-order formulas of finite difference (FD). By the method of lines (MOL) and semi-discretization approximation, we transform the PDEs into a linear system of first-order differential equations whose result fulfills a recurrence relation including the exponential function of a matrix. The techniques used are L0-stable, factual (reliable), and do not compel the application of complicated computation. The refined parallel algorithm is also applied to a problem and is seen to be highly factual, with less error than the methods already existing.enThesis