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This paper presents an examination of a Second Order Convergence Numerical Method (SOCNM) for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs). The SOCNM has been derived via the interpolating function comprises of polynomial and exponential forms. The analysis and the properties of SOCNM were discussed. Three numerical examples have been solved successfully to examine the performance of SOCNM in terms of accuracy and stability. The comparative study of SOCNM, Improved Modified Euler Method (IMEM), Fadugba and Olaosebikan Scheme (FOS) and the Exact Solution (ES) is presented. By varying the step length, the absolute relative errors at the final nodal point of the associated integration interval are computed. Furthermore, the analysis of the properties of SOCNM shows that the method is consistent, stable, convergent and has second order accuracy. Moreover, the numerical results show that SOCNM is more accurate than IMEM and FOS and also compared favourably with the ES. By varying the step length, there are two-order decrease in the values of the final absolute relative errors generated via SOCNM. Hence, SOCNM is found to be accurate, stable and a good tool for the numerical solutions of IVPs of different characteristics in ODEs.

This paper proposes a fractional numerical study on Advection-Dispersion Equation (ADE) with Fractional Order (FO) via the Caputo Fractional Reduced Differential Transform Method (CFRDTM). CFRDTM is the combination of the Caputo Fractional Derivative (CFD) and the well known Transform Method (RDTM). A convergent series solution for ADE with FO is obtained via CFRDTM. The performance of CFRDTM is tested on two illustrative examples. Hence, CFRDTM is found to be accurate and efficient.

A multi–step numerical method for the solution of second order ordinary differential equation was developed by interpolating in a finite range with a basis function. The basis function consists of a combination of exponential and trigonometric functions to ensure that such problems possess unique and continuously differentiable solutions. The method has been tested and found to be reliable, efficient and less tedious than other multi-step methods which require reduction of higher order equations into several first order equations. The method was applied to some special second order equations arising from mechanics and engineering problems. The requisite numerical properties were obtained.

This paper presents a direct solution of Black-Scholes PDE models with non-integer order via the non-integer iterative method. The Black-Scholes PDE models of non-integer order became popular and accepted globally by option traders for the valuation of financial derivatives over the period of time and the study of its numerical approaches has a wide range of applications in the real world.. The performance of the method has been confirmed and measured.