Mathematical Modeling Projects

Cylindrical Dragon Farm

This paper develops mathematical models to predict the relationship between a dragon’s size and the heat of its flames, assuming the dragons are cylindrical in shape. Beginning with the model 𝑦=𝐴𝑥^2 (temperature proportional to the square of the diameter under constant height), the study applies three estimation methods—direct averaging, least squares fitting, and linearized least squares regression. After observing limitations, the model is refined to 𝑦=𝐾𝑥^3 by assuming height is proportional to diameter, which better captures the exponential growth observed in the data. Finally, a generalized power-law model y=Ax^r is derived, and the parameters 𝐴 and 𝑟 are estimated using least squares with partial derivatives and Cramer’s rule, yielding 𝑟≈3.061, which validates the cubic model.

The mathematical techniques applied include algebraic manipulation, proportional reasoning, least squares approximation, logarithmic linearization, regression analysis, and multivariable optimization with partial derivatives. The study demonstrates that the cubic model provides the best predictive accuracy and offers a simple averaging approach as a practical tool for forecasting flame temperature from dragon size.

A Model on Fish Population

This paper develops and analyzes a mathematical model for the dynamics of a fish population in a lake, incorporating both birth and death rates. Beginning with a nonlinear differential equation for population growth, the study applies quadratic analysis, equilibrium and phase line analysis, inflection point determination, and partial fraction integration to derive explicit solutions for population over time. The model identifies two critical thresholds: an upper bound (carrying capacity, 𝑀) and a lower bound (extinction threshold, 𝑚), showing that populations starting below 𝑚 inevitably decline to extinction, while those above 𝑚 approach the carrying capacity 𝑀.

The analysis includes long-term behavior via limits, stability of equilibrium solutions, and analytical verification through integration and substitution of initial conditions. A safety threshold at (𝑚+𝑀)/2 is introduced to ensure resilience against overfishing and environmental shocks. Finally, applying least squares regression and Cramer’s rule to real population data provides concrete values for 𝑚, 𝑀, and the safety point, yielding thresholds of 5,029 (extinction), 19,888 (carrying capacity), and 12,459 (safety level).

The mathematical methods used include differential equations, quadratic factorization, phase plane analysis, inflection analysis, integration with partial fractions, stability analysis, least squares approximation, and linear algebra techniques (Cramer’s rule). Together, the study demonstrates how applied mathematics can guide sustainable fisheries management by balancing ecological dynamics with human harvesting.

Dimensional Analysis and Model Ship Experiment

This paper develops a mathematical model for predicting ship propeller thrust using dimensional analysis and the Buckingham π theorem. Starting from physical variables and applying linear algebra techniques to form dimensionless products, the model expresses thrust through an equation, simplifying further when viscosity is negligible. The method is tested with a 1/10-scale ship experiment, where scaling laws show that the full ship requires exactly 1,000 times the thrust of the model, consistent with cubic scaling.

Key mathematical tools include dimensional homogeneity, matrix methods, scaling laws, and proportional reasoning, demonstrating how scale experiments can accurately predict full-scale ship performance