Development of derivative-free iterative methods for solving nonlinear ordinary and partial differential equations and their dynamical analysis

Abstract: The construction of fixed point iterative methods for solving nonlinear equations or systems of nonlinear equations is an interesting and challenging task in numerical analysis and many applied scientific disciplines. The importance of this subject has led to the development of many numerical methods, most frequently of iterative nature. In the past few decades, iterative techniques have been applied in many diverse fields as economics, engineering, physics, dynamical models, and so on. One of the best known iterative procedures for solving nonlinear system of equations is the quadratically convergent Newton’s method $$x^{(k+1)}=x^{(k)} [F^`(x^{(k)})]^{-1} F(x^{(k)}), k=0,1,2,...., $$ where $[F^`(x)]^{-1}$ is the inverse of first Frechet Deriative $F^`(x)$ of the function $F(x)$. This method has a quadratic rate of convergence. In many practical applications it is too complicated to calculate the derivative $F^′(x)$ of the function $F(x)\mbox{ of m }$ variables. In such situations it is preferable to use only the computed values of $F(x)$ and to approximate $F^′(x)$ by employing the values of $F(x)$ at suitable points. We also prefer in practical situation the derivative-free methods because the evaluation of derivatives may be costly and time consuming. So, \textbf {the main aim of this study is to introduce efficient derivative-free family of iterative methods, which possess higher order convergence and low computational costs and their applications for solving nonlinear O.D.E and P.D.E.} We stress that the increase of the convergence rate is attained without any additional function evaluations.