Euler Integration with Decaying Polynomial

Calculus Level 3

Suppose we want to use explicit Euler integration to compute the function y = 1 t 2 \large{y = \frac{1}{t^2}} .

y k = y k − 1 + y ˙ k − 1 Δ t y = 1 t 2 \large{y_k = y_{k-1} + \dot{y}_{k-1} \, \Delta t \\ y = \frac{1}{t^2}}

The k k subscript denotes the present processing interval, and the k − 1 k-1 subscript denotes the previous processing interval. The processing takes place with a constant time step ( Δ t ) (\Delta t) .

Which of these quantities is also constant?

Note: Assume that t ≠ 0 t \neq 0

y k − 1 − y k 2 y k − 1 3 / 2 \large{\frac{y_{k-1} - y_k }{2 \, y_{k-1}^{3/2} }} y k − 1 + y k 2 y k − 1 3 / 2 \large{\frac{y_{k-1} + y_k }{2 \, y_{k-1}^{3/2} }} y k − 1 − y k 2 y k − 1 1 / 2 \large{\frac{y_{k-1} - y_k }{2 \, y_{k-1}^{1/2} }} y k − 1 + y k 2 y k − 1 1 / 2 \large{\frac{y_{k-1} + y_k }{2 \, y_{k-1}^{1/2} }}

Steven Chase by Steven Chase , 37, USA

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1 solution

Steven Chase
Jun 5, 2018

Write the derivative of y y in terms of y y .

y = 1 t 2 = t − 2 y ˙ = − 2 t − 3 = − 2 y 3 / 2 \large{y = \frac{1}{t^2} = t^{-2} \\ \dot{y} = -2 t^{-3} = -2 y^{3/2}}

Plug into Euler equation:

y k = y k − 1 + y ˙ k − 1 Δ t y k = y k − 1 − 2 y k − 1 3 / 2 Δ t \large{y_k = y_{k-1} + \dot{y}_{k-1} \, \Delta t \\ y_k = y_{k-1} -2 y_{k-1}^{3/2} \, \Delta t }

Since Δ t \Delta t is a constant, find another expression for it by re-arranging.

Δ t = y k − 1 − y k 2 y k − 1 3 / 2 \large{\Delta t = \frac{y_{k-1} - y_k}{2 y_{k-1}^{3/2}}}

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