APPLICATION OF ITERATIVE ALGORITHMS TO PARTIAL DIFFERENTIAL EQUATIONS FOR VARIOUS OPERATORS

Abstract

The theory of operators—particularly expansive, non-expansive, and contraction operators—plays a crucial role in fixed-point approximation and the numerical treatment of differential equations. This research article investigates the mathematical behavior of these operators and evaluates the convergence efficiency of iterative algorithms such as Picard, Mann, Ishikawa, and Noor methods for solving linear and nonlinear partial differential equations (PDEs). By reformulating PDEs into operator equations, the study establishes comparative convergence behaviour, stability criteria, and error reduction capabilities of each operator class. The findings show that contraction operators ensure the fastest convergence, while non-expansive operators require additional constraints for stability, and expansive operators demand modified iterative structures to maintain bounded approximations. The study provides a unified analytical foundation for improving numerical schemes used in applied mathematics, engineering, and computational sciences.