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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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