Eulers Method Bernd Schr oder logo1 Bernd Schr oder Louisiana - - PowerPoint PPT Presentation

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Eulers Method Bernd Schr oder logo1 Bernd Schr oder Louisiana - - PowerPoint PPT Presentation

Oct 14, 2023 694 likes •1.02k views

Definition Visualization Implementation Eulers Method Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Eulers Method Definition Visualization Implementation Eulers Method

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SLIDE 1

logo1 Definition Visualization Implementation

Euler’s Method

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-2
SLIDE 2

logo1 Definition Visualization Implementation

Euler’s Method

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-3
SLIDE 3

logo1 Definition Visualization Implementation

Euler’s Method

  • 1. Euler’s Method formalizes the idea of tracing curves in a

direction field for the differential equation y′ = F(x,y).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 4

logo1 Definition Visualization Implementation

Euler’s Method

  • 1. Euler’s Method formalizes the idea of tracing curves in a

direction field for the differential equation y′ = F(x,y).

  • 2. For a fixed step length ∆x, we compute the slope F(x0,y0)
  • f the tangent to the solution curve through (x0,y0).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 5

logo1 Definition Visualization Implementation

Euler’s Method

  • 1. Euler’s Method formalizes the idea of tracing curves in a

direction field for the differential equation y′ = F(x,y).

  • 2. For a fixed step length ∆x, we compute the slope F(x0,y0)
  • f the tangent to the solution curve through (x0,y0). Then

we follow that tangent line for a distance of ∆x.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-6
SLIDE 6

logo1 Definition Visualization Implementation

Euler’s Method

  • 1. Euler’s Method formalizes the idea of tracing curves in a

direction field for the differential equation y′ = F(x,y).

  • 2. For a fixed step length ∆x, we compute the slope F(x0,y0)
  • f the tangent to the solution curve through (x0,y0). Then

we follow that tangent line for a distance of ∆x.

  • 3. At the end point (x1,y1) of this line segment we use

F(x1,y1) to compute the slope at this new point (x1,y1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 7

logo1 Definition Visualization Implementation

Euler’s Method

  • 1. Euler’s Method formalizes the idea of tracing curves in a

direction field for the differential equation y′ = F(x,y).

  • 2. For a fixed step length ∆x, we compute the slope F(x0,y0)
  • f the tangent to the solution curve through (x0,y0). Then

we follow that tangent line for a distance of ∆x.

  • 3. At the end point (x1,y1) of this line segment we use

F(x1,y1) to compute the slope at this new point (x1,y1) and then we follow this new line segment.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-8
SLIDE 8

logo1 Definition Visualization Implementation

Euler’s Method

  • 1. Euler’s Method formalizes the idea of tracing curves in a

direction field for the differential equation y′ = F(x,y).

  • 2. For a fixed step length ∆x, we compute the slope F(x0,y0)
  • f the tangent to the solution curve through (x0,y0). Then

we follow that tangent line for a distance of ∆x.

  • 3. At the end point (x1,y1) of this line segment we use

F(x1,y1) to compute the slope at this new point (x1,y1) and then we follow this new line segment.

  • 4. This process repeats for as far as we want to go.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 9

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 10

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 11

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 12

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-13
SLIDE 13

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-14
SLIDE 14

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-15
SLIDE 15

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-16
SLIDE 16

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-17
SLIDE 17

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-18
SLIDE 18

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-19
SLIDE 19

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-20
SLIDE 20

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-21
SLIDE 21

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-22
SLIDE 22

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-23
SLIDE 23

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-24
SLIDE 24

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-25
SLIDE 25

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-26
SLIDE 26

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

slide-27
SLIDE 27

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 28

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0, 400 pts.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 29

logo1 Definition Visualization Implementation

Euler’s Method from a Direction Field

y′ = y−x2, y(0) = 0, 400 pts., exact solution: y(x) = x2 +2x+2−2ex

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 30

logo1 Definition Visualization Implementation

Reminder for Spreadsheet Implementation

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method

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SLIDE 31

logo1 Definition Visualization Implementation

Reminder for Spreadsheet Implementation

Step length: ∆x. Initial values: (x0,y0). xn+1 := xn +∆x yn+1 = yn +F(xn,yn)∆x. The value yn will be an approximation for the value of the solution y at xn.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Euler’s Method