5.3 Algebra of Euler's Method in 2 Dimension

Introduction to Euler's Method in 2D

  • Euler's method approximates the trajectory of curves defined by differential equations.

  • It is particularly useful when no formula exists for the curve.

Basic Concept

  • Initial Condition: Start with a specific point on the curve.

  • Change Vector: At each step, compute the change vector, denoted as S', at the current point and multiply it by a small time step, represented as ( \Delta T ).

  • Update Process: Move to a new point by adding this tiny step to the current position.

Geometric Interpretation

  • Visually, the method involves taking finite steps along the curve, approximating the true path.

  • Spreadsheet Method: Involves multiple columns to calculate each step iteratively, including:

    • Current time (T)

    • Current state (S)

    • Change vectors S' and T'

    • Updated state calculation for next iteration.

Example with Numerical Values

  • Starting Point: (2, 3) at time 0.

    • Current values: ( S_0 = 2 ), ( T_0 = 3 )

    • Change Vector Calculation:

      • ( S' = S \cdot T - S = 2 \cdot 3 - 2 = 4 )

      • ( T' = -S \cdot T + T = -6 + 3 = -3 )

  • Selecting ( \Delta T ): Chosen as 0.1.

  • Change Length:

    • ( S' \times \Delta T = 4 \times 0.1 = 0.4 )

    • ( T' \times \Delta T = -3 \times 0.1 = -0.3 )

  • New Values Calculation:

    • New ( S ): ( 2 + 0.4 = 2.4 )

    • New ( T ): ( 3 - 0.3 = 2.7 )

Continued Iteration

  • Next Step: Update values and repeat calculations for the new state, iterating through all required steps.

  • Spreadsheet Output: Continue using the spreadsheet model to achieve as many steps as necessary for accurate results.

Homework Assignment

  • Implement Euler's method for the shark tuna model:

    • Use ( \Delta T = 0.01 \) for better accuracy.

    • Start from point (2, 3).

    • Perform 5,000 steps.

    • Optionally, write a 5-line Sage program to automate the calculations.