On the Application of an Intertwined Difference Equations and Newton-Lieberstein Algorithmic to Solve Mildly Non-linear Boundary Value Problems (MNBVP)

Over the years, numerous researchers have developed numerical or classical methods or the other to obtain solutions for Boundary Value Problems; many of these methods have not been adopted to solve Mildly Non-Linear Boundary Value Problems (MNBVP) basically because of their peculiarities. The MNBVP are problems that are neither entirely linear nor exclusively nonlinear rather for multi-variable functions, not all the constants and the function derivative is greater or equal to zero. This paper discusses the process of intertwining the Approximation Difference Equation with the Newton-Lieberstein method in solving mildly non-linear boundary value problems. The technique requires dual independent processes that involve different Methods. The first technique is aimed at reducing the BVP to a tridiagonal system, while the other is employed to solve the system of equations by applying the Newton-Lieberstein algorithm. The resulting equations that is formed from the nonlinear boundary problems will be nonlinear systems but it will be structurally related to tridiagonal systems. In practical settings, mathematical modelling problems can be expressed in the form of BVPs that arise frequently and majorly in the fields of science and engineering, such as electric circuits, fluid dynamics, the motion of rockets or satellites, and other areas of engineering applications. The intertwined methods will be used to evaluate the approximate solutions without much ado on the number of equations the problem has. The technique was employed to solve some problems with numerical results obtained showed the flexibility, the robustness, and how efficient the coined technique is. The results compare favourably with exact or analytical results.