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.
In the paper , 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.
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.