Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter August 28, 2013

Solving Nonlinear Problems with Singular Initial Conditions Using A Perturbed Scalar Homotopy Method

  • Cheng-Yu Ku EMAIL logo and Yung-Hsien Tsai

Abstract

In this paper, a novel method, named the perturbed scalar homotopy method, is proposed to solve nonlinear systems with a singular Jacobian matrix. The concept of the proposed perturbed scalar homotopy method roots from the conventional homotopy method but it takes the advantages of converting a vector function to a scalar function by using the square norm of the vector function to conduct a scalar-based homotopy method. Then, a small parameter, which is similar to the perturbation theory, is introduced to the singular systems of nonlinear equations such that the modified singular systems of nonlinear equations become nonsingular and the asymptotic solutions may be found. As a result, the proposed novel method does not need to calculate the inverse of the Jacobian matrix and thus has great numerical stability. In addition, the formulation of the proposed method reveals that this new method is exponentially convergent with the use of the exponential time function. Results obtained show that the proposed novel method can be used to solve singular systems of nonlinear equations with high accuracy as well as the convergence and it may be a better alternative for solving a system of non-linear algebraic equations.

Received: 2013-03-15
Accepted: 2013-06-19
Published Online: 2013-08-28
Published in Print: 2013-10-25

©[2013] by Walter de Gruyter Berlin Boston

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/ijnsns-2013-0029/html
Scroll to top button