Solving Nonlinear Problems with Singular Initial Conditions Using A Perturbed Scalar Homotopy Method
-
Cheng-Yu Ku
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.
©[2013] by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Masthead
- Mathematical Models for Erosion and the Optimal Transportation of Sediment
- Comments on “He's Homotopy Perturbation Method for Calculating Adomian Polynomials”
- Biomechanical Analysis of Eyring Prandtl Fluid Model for Blood Flow in Stenosed Arteries
- Consensus Recovery from Intentional Attacks in Directed Nonlinear Multi-agent Systems
- Exp-function Method for Fractional Differential Equations
- Solving Nonlinear Problems with Singular Initial Conditions Using A Perturbed Scalar Homotopy Method
- Tri-integrable Couplings by Matrix Loop Algebras
- Extended Reductive Perturbation Method and Its Relation to the Re-normalization Method
- Neural Network Approaches to Data Fusion Calculation and Correction in Wireless Sensor Network – A Case Study Comparison between BPN, RBFN and GRNN
- A Catfish Effect Inspired Harmony Search Algorithm for Optimization
- An Exact, Fully Nonlinear Solution of the Poisson-Boltzmann Equation with Anti-symmetric Electric Potential Profiles
- Study on Nonlinear Oscillations of Gear Systems with Parametric Excitation Solved by Homotopy Analysis Method
