Some engineering applications of newly constructed algorithms for one-dimensional non-linear equations and their fractal behavior

Abstract

The aim of this paper is to develop some novel numerical algorithms for finding roots of one-dimensional non-linear equations. We derive these algorithms by utilizing the main and basic idea of the variational iteration technique. The convergence rate of the suggested algorithms is discussed. It is corroborated that the proposed numerical algorithms possess sixth-order convergence. To demonstrate the validity, applicability, and the performance of the proposed algorithms, we solved different test problems. These problems also include some real-life applications associated with the chemical engineering such as van der Wall’s equation, conversion of nitrogen-hydrogen feed to ammonia and the fractional-transformation in the chemical reactor problem. The numerical results of these problems show that the proposed algorithms are more effective against the other well-known similar nature existing methods. Finally, the dynamics of the suggested algorithms in the form of the polynomiographs of different complex polynomials have been analyzed that reveals the fractal nature and the other dynamical aspects of the suggested algorithms.