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Predictor - corrector methods and their applications (Master thesis)

Τοπούζης, Παναγιώτης/ Topouzis, Panagiotis/ Εφαρμοσμένα Μαθηματικά/ Applied Mathematics

Ordinary Differential Equations (ODEs) constitute a fundamental tool for the mathematical modeling of problems in fields such as physics, engineering, and the biological sciences. Their analytical solution is often complicated or even impossible, especially in the case of nonlinear equations. For this reason, numerical methods have emerged as an essential tool for approximating the solutions of ODEs. This thesis focuses on the application of the predictor–corrector method for the numerical solution of ordinary differential equations. This method, which combines the advantages of explicit and implicit methods, is typically based on the Adams–Bashforth and Adams–Moulton formulas and is known for its high accuracy, stability, and computational efficiency. The aim of this thesis is to present and analyze fundamental numerical methods for solving initial value problems. Specifically, one-step methods such as the Euler method and the Runge–Kutta methods are examined, as well as multistep methods including the Adams–Bashforth and Adams–Moulton families. Particular emphasis is placed on predictor–corrector methods, which combine the advantages of different numerical approaches. Finally, numerical examples and results are presented, highlighting the behavior, accuracy, and comparative performance of the methods under study.