5.2 Algebra of Euler's Method in 1 Dimension

Euler's method is an approximation technique for solving differential equations, illustrated through a geometric approach to state space.

State Space Definition: Represents populations in the context of species interaction, e.g., sharks (x-axis) and tuna (y-axis).
Change Vector: Describes the rate of change, expressed in units of animals per unit time (sharks per time, tuna per time).
Change vector is illustrated by a blue arrow, calculated as the product of the change vector and a small time step ( \Delta t = 0.01 \).

Initial Point: Start at an initial state (x_0, y_0).
Calculate Change Vector: Determine (x_0') using the differential equation for the system.
Apply Time Step: Multiply the change vector by ( \Delta t ) to find a new point:
- New position: ( x_1 = x_0 + x_0' imes \Delta t ).
Iterative Process: Repeat the calculations using the new state to find subsequent points (x_2, x_3, ...).
- For each new (x), compute (x') and update the position again.

Logistic Growth Equation: ( x' = b x \times (1 - \frac{x}{k}) ), where ( k ) is the carrying capacity.
Initial condition chosen: ( x_0 = 10 ), with parameters ( b = 0.2 ) and ( k = 100 ).
Calculation Steps:
- Calculate ( x' ): ( x' = 0.2 imes 10 imes (1 - \frac{10}{100}) = 0.18 ).
- Choose a time step: ( \Delta t = 0.1 ).
- Calculate change: ( x' imes \Delta t = 0.18 imes 0.1 = 0.018 ).
- Update position: ( x_1 = 10 + 0.018 = 10.018 ).
- Repeat for ( x_1): Update and recalculate for further iterations as described.

Reference to Katherine Johnson from "Hidden Figures" who utilized Euler's method for lunar trajectory calculations, showcasing the method's practical applications in solving complex equations.

Euler's method serves to approximate solutions to differential equations through iterative calculation, providing a computational approach to modeling dynamic systems.