Numerical Solution of the Mildly Non-linear Boundary Value Problems (MNBVP) Using Newton-Lieberstein Algorithm

Abstract

Several numerical and classical approaches have been employed to determine the Boundary Value Problems solutions, but not many of these methods have been adopted to work out Mildly Non-Linear Boundary Value Problems (MNBVP); hence, there is a necessity to think of ways of achieving this feat. The MNBVP are problems that are neither linear nor explicitly nonlinear. This paper discusses the process of intertwining the Newton-Lieberstein method into the upwind differencing technique in solving mildly non-linear boundary value problems. The method requires two independent stages that involve different techniques. First, transform the BVP to a tri-diagonal system. Secondly, solve the system of equations by applying the Newton-Lieberstein Algorithm. The resulting equations from the nonlinear boundary problems considered in this paper will generate a nonlinear but will relate structurally to tridiagonal equations. Here, the authors developed a method described as a hybridized iterative algorithm, for which the number of steps cannot be determined a priori. It is a variant of the Newton’s technique. Unlike other classical methods, it applies to nonlinear systems and to systems with more than a thousand equations, after which round off error tends to accumulate. In real life, mathematical problems can now be expressed in the BVPs that sprang largely in the fields of physics, science, and engineering, such as fluid dynamics, electric circuits, the motion of rockets or satellites, and other areas of application. The algorithm proposed will estimate the approximate solutions irrespective of the number of equations involved. Numerical results showed the robustness, efficiency, and flexibility of the intertwined technique.