Iterative Solutions for Non-Linear Systems

Abstract

The research work presented in this thesis deals with the study of the “Iterative solutions for non-linear systems”. A non-linear system is a set of simultaneous ‘n’ non-linear equations in which each equation is a function in ‘n’ unknown variables. Such type of non-linear systems have perceived several significant contributions in mathematics and allied engineering areas, for example, electrical circuits, chemical reactions, physical law, biological phenomena, computational economics etc. Due to the wider variety of behavior, finding solutions of non-linear systems is much more tedious than a scalar case. Moreover it is extremely hard to solve non-linear systems analytically. In this context, numerical techniques provide a fruitful way to solve non-linear systems. On account of this reality, one needs to rely on numerical techniques for solving non-linear systems. Therefore, a reliably expanding extent of present-day numerical research is focused on the analysis of the approximate solutions of non-linear systems. Among all numerical techniques, Newton’s technique is the most basic and outstanding iterative method for solving non-linear systems. In literature, numerous adjustments have been incorporated in Newton’s technique (known as Newton’s variants), which have either equivalent or better efficiency over Newton’s technique. In the present thesis, an endeavor has been made to the construct more variants of Newton’s method to solve non-linear systems. The essential standard of numerical algorithms has been followed to attain computational efficiency, which is always proportional to the quality of an algorithm and inversely proportional to its computational cost. The quality of an algorithm concerns with the convergence speed of algorithm along with its structure. Computational cost concerns with the amount of calculation work required to evaluate functions, derivatives, matrix inversions during the entire process. The primary focus of present research work is to address the construction of iterative schemes to propose solutions for systems of non-linear equations arising in different disciplines of science and engineering. The present work also sheds light on the development of iterative schemes for systems of non-linear equations associated with ordinary and partial differential equations. The development of iterative schemes consist of two parts: the first one is the ‘construction part’ and the second part establishes the proof of local convergence in Banach settings. Finally, a variety of problems involving non-linear systems have been numerically tested in order to demonstrate the exactness and the computational efficiency of the proposed iterative algorithms.