Solution of a Subclass of Lane-Emden Differential Equation by Variational Iteration Method

In this paper we apply He's variational iteration method to find out an appropriate solution to a class of singular differential equation under imposed conditions by introducing and inducting in a polynomial pro satisfying the given subject to conditions at the outset as selective function to the solution extracting process. As for as application part is concerned, Illustrative examples from the available literature when treated all over reveal and out show that the solution deduced by proposed method is exact and again polynomial. Overall, a successful produce of exact solutions by proposed process itself justify the effectiveness and efficiency of the method so very much. Keywords: He's variational iteration method, Lane-Emden differential equation, exact solution, polynomial, Lagrange multiplier

Higher-order approximation of cubic–quintic duffing model

We apply an Artificial Parameter Lindstedt-Poincaré Method (APL-PM) to find improved approximate solutions for strongly nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. This approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution which makes it a unique solution. It is demonstrated that this method works very well for the whole range of parameters in the case of the cubic-quintic oscillator, and excellent agreement of the approximate frequencies with the exact one has been observed and discussed. Moreover, it is not limited to the small parameter such as in the classical perturbation method. Interestingly, This study revealed that the relative error percentage in the second-order approximate analytical period is less than 0.042% for the whole parameter values. In addition, we compared this analytical solution with the Newton– Harmonic Balancing Approach. Results indicate that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems. Utter simplicity of the solution procedure confirms that this method can be easily extended to other kinds of nonlinear evolution equations

Exp-Function Method for Duffing Equation and New Solutions of (2+1) Dimensional Dispersive Long Wave Equations

In this paper, the general solutions of the Duffing equation with third degree nonlinear term is obtain using the Exp-function method. Using the Duffing equation and its general solution, the new and general exact solution with free parameter and arbitrary functions of the (2+1) dimensional dispersive long wave equation are obtained. Setting free parameters as special values, hyperbolic as well as trigonometric function solutions are also derived. With the aid of symbolic computation, the Exp-function method serves as an effective tool in solving the nonlinear equations under study. Key words: Exp-function method; Duffing equation; Exact solutions; Nonlinear evolution equation

Spartan Daily, March 24, 1941

Volume 29, Issue 106https://scholarworks.sjsu.edu/spartandaily/3268/thumbnail.jp

Volume 77 - Issue 9 - November, 1967

https://scholar.rose-hulman.edu/technic/1006/thumbnail.jp

The Effects of Computer-Intensive Algebra on Students\u27 Understanding of the Function Concept.

The purpose of this study is to examine the effects of the Computer-Intensive Algebra (CIA) and traditional algebra (TA) curricula on students\u27 understanding of the function concept. CIA was developed recently in an effort to address some of the deficiencies in TA, namely, its over-emphasis of procedural skills and lack of attention to concepts and problem solving (Fey, 1992; Kieran, 1990). The major features of this innovative curriculum are (a) a problem solving approach based on the modelling and exploration of realistic problem situations, (b) an emphasis on conceptual knowledge, and (c) the extensive use of technology. This study hypothesizes that CIA students develop a richer understanding of functions than their TA counterparts while achieving at least the same level of procedural skill. A theoretical framework is proposed which describes a conceptual knowledge of functions in terms of the following components: (a) modelling a real-world situation using functions, (b) interpreting a function in terms of a realistic situation, (c) translating between different representations of functions, and (d) reifying functions. The subjects were university students enrolled in a College Algebra course. One section of this course was taught following the CIA curriculum and compared to the traditional sections. Instruments included pre and post tests on functions, attitude measures, researcher questionnaires, and the departmental final examination. Qualitative data were also collected via two sets of interviews conducted with students from each of the two curricula. The results indicated that the CIA students achieved a better overall understanding of functions and were better in the individual components of modelling, interpreting, and translating. No significant differences were found for reifying, which emerged as the most difficult in the function model for both groups. The data from the final examination were less definitive making it difficult to draw any definite conclusions about skill development. Other findings of interest were that the CIA students showed significant improvements in their attitudes and levels of anxiety toward mathematics. Also, the CIA class was rated as more interesting by the students and achieved a much higher percentage of students successfully completing the course

A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS

In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, which rigorously defines the infinitesimals

Higher-order approximation of cubic–quintic duffing model

We apply an Artificial Parameter Lindstedt-Poincaré Method (APL-PM) to find improved approximate solutions for strongly nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. This approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution which makes it a unique solution. It is demonstrated that this method works very well for the whole range of parameters in the case of the cubic-quintic oscillator, and excellent agreement of the approximate frequencies with the exact one has been observed and discussed. Moreover, it is not limited to the small parameter such as in the classical perturbation method. Interestingly, This study revealed that the relative error percentage in the second-order approximate analytical period is less than 0.042% for the whole parameter values. In addition, we compared this analytical solution with the Newton– Harmonic Balancing Approach. Results indicate that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems. Utter simplicity of the solution procedure confirms that this method can be easily extended to other kinds of nonlinear evolution equations