This work presents an efficient procedure based on Chebychev spectral collocation method for computing the 2D Laplace’s equation on a rectangular domain. The numerical results and comparison of finite difference and finite element methods are presented. We obtained a satisfactory result when compared with other numerical solutions.

This work focuses on valuation scheme of European and American options of single asset with meshless radial basis approximations. The prices are governed by Black – Scholes equations. The option price is approximated with three infinitely smooth positive definite radial basis functions (RBFs), namely, Gaussian (GA), Multiquadrics (MQ), Inverse Multiquadrics (IMQ). The RBFs were used for discretizing the space variables while Runge-Kutta method was used as a time-stepping marching method to integrate the resulting systems of differential equations. Numerical examples are shown to illustrate the strength of the method developed. The findings show that the RBFs has proven to be adaptable interpolation method because it does not depend on the locations of the approximation nodes which have overcome frequently evolving problems in computational finance such as slow convergent numerical solutions. Thus, the results allow concluding that the RBF-FD-GA and RBF-FD-MQ methods are well suited for modeling and analyzing Black and Scholes equation.

This study investigates the continuous linear multistep techniques utilized for solving first-order initial value problems in ordinary differential equations. Specifically, the study focuses on step k = 9, utilizing Hermite polynomials as basis functions. This study effectively constructs the Adams-Bashforth, Adams-Moulton, and optimal order methods by applying collocation and interpolation methodologies. The methods are thoroughly examined using various numerical instances to demonstrate their efficacy and validity. Notably, the optimal order method exhibits superior accuracy and efficiency compared to the traditional Adams-Bashforth and Adams-Moulton methods. The research results contribute novel and improved methodologies for solving initial value problems in differential equations, which have extensive applications across diverse mathematical and scientific domains.