A COMPARATIVE STUDY BETWEEN SIX ORDER RUNGE KUTTA METHOD AND SIX ORDER PREDICTOR CORRECTOR METHOD FOR SOLVING INITIAL VALUE PROBLEM OF BRATU TYPE EQUATION

Advisor: Mamo Abebe (PhD) Abstract This thesis conducts on a comparative study of two advanced numerical techniques—the six Runge-Kutta techniques in order (RK6) and the six predictor-corrector method in order (PC6) - to solve the initial value problem of Bratu-type formulas that have many applications in various science and engineering. The RK6 method, characterized by its high order of accuracy and stability, is evaluated for its performance in solving Bratu-type problems with varying step sizes and initial conditions. The method's capacity to maintain precision and its computational requirements are analyzed, with a focus on its ability to handle stiffness in the equations. Conversely, the PC6 method, which integrates predictive and corrective iterations to enhance solution accuracy, is assessed for its efficiency regarding point-by-point absolute error. This study involves a detailed comparison of the numerical results obtained from both methods against known analytical solutions and high-precision numerical benchmarks. The findings indicate that while RK6 generally provides higher accuracy with little iteration, PC6 offers improved computational efficiency for certain problem instances. This comparative study enhances the understanding of numerical techniques for solving problems, contributing valuable recommendations for efficient and accurate computational practices in applied mathematics and engineering.