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

Definition Visualization Implementation Euler’s Method Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

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 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

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 ( x 0 , y 0 ) of the tangent to the solution curve through ( x 0 , y 0 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

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 ( x 0 , y 0 ) of the tangent to the solution curve through ( x 0 , y 0 ) . Then we follow that tangent line for a distance of ∆ x . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

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 ( x 0 , y 0 ) of the tangent to the solution curve through ( x 0 , y 0 ) . Then we follow that tangent line for a distance of ∆ x . 3. At the end point ( x 1 , y 1 ) of this line segment we use F ( x 1 , y 1 ) to compute the slope at this new point ( x 1 , y 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

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 ( x 0 , y 0 ) of the tangent to the solution curve through ( x 0 , y 0 ) . Then we follow that tangent line for a distance of ∆ x . 3. At the end point ( x 1 , y 1 ) of this line segment we use F ( x 1 , y 1 ) to compute the slope at this new point ( x 1 , y 1 ) and then we follow this new line segment. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

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 ( x 0 , y 0 ) of the tangent to the solution curve through ( x 0 , y 0 ) . Then we follow that tangent line for a distance of ∆ x . 3. At the end point ( x 1 , y 1 ) of this line segment we use F ( x 1 , y 1 ) to compute the slope at this new point ( x 1 , y 1 ) and then we follow this new line segment. 4. This process repeats for as far as we want to go. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 , 400 pts . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Euler’s Method from a Direction Field y ′ = y − x 2 , y ( 0 ) = 0 , 400 pts . , exact solution: y ( x ) = x 2 + 2 x + 2 − 2 e x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Reminder for Spreadsheet Implementation logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method

Definition Visualization Implementation Reminder for Spreadsheet Implementation Step length: ∆ x . Initial values: ( x 0 , y 0 ) . : = x n + ∆ x x n + 1 = y n + F ( x n , y n ) ∆ x . y n + 1 The value y n will be an approximation for the value of the solution y at x n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Euler’s Method