A Comparative Study of Linearization Techniques With Adaptive Block Hybrid Method for Solving First‐Order Initial Value Problems

This paper investigates linearization methods used in the development of an adaptive block hybrid method for solving first‐order initial value problems. The study focuses on Picard, linear partition, simple iteration, and quasi‐linearization methods, emphasizing their role in enhancing the performance of the adaptive block hybrid method. The efficiency and accuracy of these techniques are evaluated through solving nonlinear differential equations. The study provides a comparative analysis focusing on convergence properties, computational cost, and the ease of implementation. Nonlinear differential equations are solved using the adaptive block hybrid method, and for each linearization method, we determine the absolute error, maximum absolute error, and number of iterations per block for different initial step‐sizes and tolerance values. The findings indicate that the four techniques demonstrated absolute errors, ranging from O (10 −12 ) to O (10 −20 ). We noted that both the Picard and quasi‐linearization methods consistently achieve the highest accuracy in minimizing absolute errors and enhancing computational efficiency. Additionally, the quasi‐linearization method required the fewest number of iterations per block to achieve its accuracy. Furthermore, the simple iteration method required fewer number of iterations than the linear partition method. Comparing minimal CPU time, the Picard method required the least. These results address a critical gap in optimizing linearization techniques for the adaptive block hybrid method, offering valuable insights that enhance the precision and efficiency of methods for solving nonlinear differential equations.