Search Results (4)

In this article, we examine the local convergence analysis of an extension of Newton’s method in a Banach space setting. Traub introduced the method (also known as the Arithmetic-Mean Newton’s Method and Weerakoon and Fernando method) with an order of convergence of three. All the previous works either used higher-order Taylor series expansion or could not derive the desired order of convergence. We studied the local convergence of Traub’s method and two of its modifications and obtained the convergence order for these methods without using Taylor series expansion. The radii of convergence, basins of attraction, comparison of iterations of similar iterative methods, approximate computational order of convergence (ACOC), and a representation of the number of iterations are provided. Full article

Rubber isolators are usually used to protect high-precision equipment of autonomous underwater vehicles (AUVs), avoiding damage from overlarge dynamic excitation. Considering the nonlinear properties of the rubber material, the nonlinear behavior of rubber isolators under shock exaltation is hard to be predict accurately without the available modal and accurate parameters. In view of this, the present study proposes a nonlinear model and parameter identification method of rubber isolators to present their transient responses under shock excitation. First, a nonlinear model of rubber isolators is introduced for simulating their amplitude and frequency-dependent deformation under shock excitation. A corresponding dynamic equation of the isolation system is proposed and analytically solved by the Newmark method and the Newton-arithmetic mean method. Secondly, a multilayer feed-forward neural network (MFFNN) is constructed with the current model to search the parameters, in which the differences between the estimated and tested responses are minimized. The sine-sweep and drop test are planned with MFFNN to build the parameter identification process of rubber isolators. Then, a T-shaped isolator composed of high-damping silicon rubber is selected as a sample, and its parameters were determined by the current identification process. The transient responses of the isolation system are reconstructed by the current mode with the identified parameter, which show good agreement with measured responses. The accuracy of the proposed model and parameter identification method is proved. Finally, the errors between the reconstructed responses and tested responses are analyzed, and the main mode of energy attenuation in the rubber isolator is discussed in order to provide an inside view of the current model. Full article

The mathematical theory behind the porous medium type equation is well developed and produces many applications to the real world. The research and development of the fractional nonlinear porous medium models also progressed significantly in recent years. An efficient numerical method to solve porous medium models needs to be investigated so that the symmetry of the designed method can be extended to future fractional porous medium models. This paper contributes a new numerical method called Newton-Modified Weighted Arithmetic Mean (Newton-MOWAM). The solution of the porous medium type equation is approximated by using a finite difference method. Then, the Newton method is applied as a linearization approach to solving the system of nonlinear equations. As the system to be solved is large, high computational complexity is regulated by the MOWAM iterative method. Newton-MOWAM is formulated technically based on the matrix structure of the system. Some initial-boundary value problems with a different type of nonlinear diffusion term are presented. As a result, the Newton-MOWAM showed a significant improvement in the computation efficiency compared to the developed standard Weighted Arithmetic Mean iterative method. The analysis of efficiency, measured by the reduced number of iterations and computation time, is reported along with the convergence analysis. Full article

To answer the question stated in the title, we present and compare two approaches: first, a standard approach for solving dual fuzzy nonlinear systems (DFN-systems) based on Newton’s method, which uses 2D FN representation and second, the new approach, based on multidimensional fuzzy arithmetic (MF-arithmetic). We use a numerical example to explain how the proposed MF-arithmetic solves the DFN-system. To analyze results from the standard and the new approaches, we introduce an imprecision measure. We discuss the reasons why imprecision varies between both methods. The imprecision of the standard approach results (roots) is significant, which means that many possible values are excluded. Full article