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The article discusses derivative‐free algorithms with and without memory for solving numerically nonlinear systems. We proposed a family of fifth‐ and sixth‐order schemes and extended them to algorithms with memory. We further discuss the convergence and computational efficiency of these algorithms. Numerical examples of mixed Hammerstein integral equation, discrete nonlinear ordinary differential equation, and Fisher's partial differential equation with Neumann's boundary conditions are discussed to demonstrate the convergence and efficiency of these schemes. Finally, some numerical results are included to examine the performance of the developed methods.

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In this paper, a new one-parameter class of fixed point iterative method is proposed to approximate the fixed points of contractive type mappings. The presence of an arbitrary parameter in the proposed family increases its interval of convergence. Further, we also propose new two-step and three-step fixed point iterative schemes. We also discuss the stability, strong convergence and fastness of the proposed methods. Furthermore, numerical experiments are performed to check the applicability of the new methods, and these have been compared with well-known similar existing methods in the literature.