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A Family of Methods for Solving Nonlinear Equations with Twelfth-Order Convergence  [PDF]
Xilan Liu, Xiaorui Wang
Applied Mathematics (AM) , 2013, DOI: 10.4236/am.2013.42049

This paper presents a new family of twelfth-order methods for solving simple roots of nonlinear equations which greatly improves the order of convergence and the computational efficiency of the Newton’s method and some other known methods.

Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations  [PDF]
Pavel N. Ryabov,Dmitry I. Sinelshchikov,Mark B. Kochanov
Physics , 2011,
Abstract: The application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated.
A shooting method for singular nonlinear second order Volterra integro-differential equations
R. E. Shaw,L. E. Garey
International Journal of Mathematics and Mathematical Sciences , 1997, DOI: 10.1155/s016117129700080x
Abstract: The method of parallel shooting will be employed to solve nonlinear second order singular Volterra integro-differential equations with two point boundary conditions.
A New Newton-Type Method with Third-Order for Solving Systems of Nonlinear Equations  [PDF]
Zhongli Liu, Quanyou Fang
Journal of Applied Mathematics and Physics (JAMP) , 2015, DOI: 10.4236/jamp.2015.310154
Abstract: In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.
Some variants of Newton's method with fifth-order and four-order convergence for solving nonlinear equations
Yao-tang Li,Ai-quan Jiao
International Journal of Applied Mathematics and Computation , 2009,
Abstract: Chun's two-step predictor-corrector type iterative method for solving nonlinear equations (Iterative Methods Improving Newton's Method by the Decomposition Method, J . Comput. Math. Appl.50, 1559-1568 ) is improved and a kind of new iterative method is presented. Some present methods derived from Adomian Decomposition method are unified in a form and a series of new methods with high convergence order and the value of EFF are obtained by introducing parameter also. Those methods can be considered as a significant improvement of the Newton's method and its variant forms.
New Ninth Order J-Halley Method for Solving Nonlinear Equations  [PDF]
Farooq Ahmad, Sajjad Hussain, Sifat Hussain, Arif Rafiq
Applied Mathematics (AM) , 2013, DOI: 10.4236/am.2013.412233

In the paper [1], authors have suggested and analyzed a predictor-corrector Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which had a quantic convergence. We have compared this modified Halley method with some other iterative methods of ninth order, which shows that this new proposed method is a robust one. Some examples are given to illustrate the efficiency and the performance of this new method.

A New Efficient Optimal Eighth-Order Iterative Method for Solving Nonlinear Equations  [PDF]
J. P. Jaiswal,Neha Choubey
Mathematics , 2013,
Abstract: We established a new eighth-order iterative method, consisting of three steps, for solving nonlinear equations. Per iteration the method requires four evaluations (three function evaluations and one evaluation of the first derivative). Convergence analysis shows that this method is eighth-order convergent which is also substantiated through the numerical works.Computational results ascertain that our method is efficient and demonstrate almost better performance as compared to the other well known eighth-order methods.
A New Approach for the Exact Solutions of Nonlinear Equations of Fractional Order via Modified Simple Equation Method  [PDF]
Muhammad Younis
Applied Mathematics (AM) , 2014, DOI: 10.4236/am.2014.513186

In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.

Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations  [PDF]
Xiaobing Feng,Michael Neilan
Mathematics , 2007,
Abstract: This paper concerns with numerical approximations of solutions of second order fully nonlinear partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for second order fully nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called vanishing moment method, hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods with "guaranteed" convergence. The main idea of the proposed vanishing moment method is to approximate a second order fully nonlinear PDE by a higher order, in particular, a fourth order quasilinear PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.
A New Fifth-Order Iterative Method for Finding MultipleRoots of Nonlinear Equations
American Journal of Computational and Applied Mathematics , 2012, DOI: 10.5923/j.ajcam.20120206.04
Abstract: In this paper, we present a fifth-order method for finding multiple zeros of nonlinear equations. Per iteration, the new method requires two evaluations of functions and two of its first derivative. It is proved that the method has a convergence of order five. Finally, some numerical examples are given to show the performance of the presented method, and compared with some known methods.
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