  We use the multidimensional heat conduction and heat transfer equations to
model the temperature distribution of water in a bathtub by solving partial
differential equations. We address optimal water addition and bathtub design.
First, we establish a water surface cooling model using Newton's law of cooling
to simulate heat exchange between air and water. Without new heat sources, the
water temperature reaches a minimum in 40 minutes. We then simulate adding hot
water with a one-dimensional heat conduction model, including air cooling
effects. We determine that the optimal heat input is 80 Joules and the optimal
water velocity is 0.042 m/s to maintain temperature and save water. The ideal
bathtub dimensions are 1.5m length, 0.6m width, 0.42m depth, with rounded
corners. Using finite difference methods and MATLAB's Pdetool, we solve the
heat conduction equation and verify numerical stability, discussing the model's
pros and cons and suggesting improvements.
