Mathematical modeling

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This text systematically explains Mathematical Modeling for undergraduate and graduate students across various disciplines, including mathematics, engineering, economics, finance, biology, chemistry, and physics. It provides a comprehensive overview of modeling techniques, covering both deterministic and stochastic methods, as well as first-principle and empirical solutions. The book transitions from simple algebraic analysis to advanced simulation methods, aiding in understanding how to address research problems effectively. Real-world processes often involve uncertainty, making the inclusion of randomness crucial. While many students are familiar with deterministic methods, they typically lack exposure to stochastic techniques found in advanced probability theory texts. This book develops both deterministic and stochastic methods, illustrating how the former can be generalized by the latter. Additionally, it discusses various empirical approximations for modeling processes and addresses the validity of assumptions under specific conditions. The text bridges empirical modeling and first-principle methods, explaining how modeling principles can validate empirical assumptions. It also covers essential aspects of micro-scale and macro-scale modeling, highlighting significant current research challenges.