A Novel Dynamical Method for Solving Ill-Conditioned Nonlinear Ordinary Differential Equations
Abstract:For solving nonlinear engineering problems, numerical methods including the finite element method, the boundary element method, the distinct element method, and the meshless method used in CAD, CAE, and CAM usually need to solve a system of a non-linear algebraic equation system. In this paper, a novel dynamical method, named the Dynamical Jacobian-Inverse Free Method (DJIFM), based on the construction of a scalar homotopy function to transform a vector function of non-linear algebraic equations (NAEs) into a time-dependent scalar function is proposed. With the introduction of a transformation matrix, the proposed dynamical method can be transformed into the DJIFM and other dynamical Newton-like methods. In this study, the DJIFM is adopted for the solution of an ill-conditioned nonlinear ordinary differential equation. Results reveal that the proposed DJIFM can improve the convergence and increase the numerical stability for solving NAEs especially for the system of nonlinear problems involving ill-conditioned Jacobian or poor initial values which cause convergence problems.
Document Type: Research Article
Publication date: July 1, 2012
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