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.

本文的目的是开发一些新颖的数值算法来寻找一维非线性方程的根。我们利用变分迭代技术的主要思想和基本思想来推导这些算法。讨论了所建议算法的收敛速度。可以证明所提出的数值算法具有六阶收敛性。为了证明所提出算法的有效性，适用性和性能，我们解决了各种测试问题。这些问题还包括与化学工程相关的一些实际应用，例如范德华尔方程，氮氢进料到氨的转化以及化学反应器中的分步转化问题。这些问题的数值结果表明，所提出的算法相对于其他已知的类似性质的现有方法更为有效。最后，以不同复多项式的多项式符号形式对建议算法进行了动力学分析，揭示了建议算法的分形性质和其他动力学方面。