SE301:Numerical Methods Unit 7 Solution of Ordinary Differential Equations

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SE301Numerical MethodsUnit 7Solution of
Ordinary Differential Equations

• Lesson 6 Solution of Systems of ODEs

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Outlines of Unit 7
Solution of ODE

• Lessons 1-3
• Introduction to ODE, Euler Method,
• Taylor Series methods,
• Midpoint, Heuns Predictor corrector methods

• Lessons 4-5
• Runge-Kutta Methods (concept derivation)
• Applications of Runge-Kutta Methods To solve
  first order ODE

• Lessons 6
• Solving Systems of ODE

Lesson 7 Multi-step methods

• Lessons 8-9
• Boundary Value Problems
• Discretization method

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Outlines of Unit 7
Solution of ODE

• Lessons 6
• Solving Systems of ODE

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Outlines of Unit 7
Solution of ODE

• Lessons 6
• Solving Systems of ODE

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Learning Objectives of Lesson 6

• Convert a single (or a system of ) high order
  ODEs to a system of first order ODEs
• Use the methods discussed earlier in this unit
  to solve systems of first order ODEs.

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Outlines of Lesson 6

• Solution of a system of first order ODEs
• Conversion of a high order ODEs to a system of
  first order ODEs
• Conversion of a system of high order ODEs to a
  system of first order ODEs
• Use different methods to solve systems of first
  order ODEs.
• Use different methods to solve high order ODEs.
• Use different methods to solve systems of high
  order ODEs.

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Solving a system of first order ODEs

• Methods discussed earlier such as Euler,
  Runge-Kutta,are used to solve first order
  ordinary differential equations
• The same formulas will be used to solve a system
  of first order ODEs. In this case, the
  differential equation is a vector equation and
  the dependent variable is a vector variable.

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Euler method for solving a system of first order
ODEs

• Recall Euler method for solving first order ODE.

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Example Euler method

• Euler method to solve a system of n first order
  ODE.

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Solving a system of n first order ODEs

• Exactly the same formula is used but the scalar
  variables and functions are replaced by vector
  variables and vector values functions.
• Y is a vector of length n
• F(Y,x) is vector valued function

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Example Euler method for solving a system of
first order ODEs

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Example RK2 method for solving a system of
first order ODEs

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Example RK2 method for solving a system of
first order ODEs

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Method for solving a system of first order ODEs

• We have extended Euler and RK2 methods to solve
  systems of first order ODE
• Other methods used to solve first order ODE can
  be easily extended to solve systems of first
  order ODE

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High Order ODE

• How do solve second order ODE?
• How do solve high order ODE?

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The general approach to solve ODEs
convert
solve
high order ODE
System of first order ODE
convert
solve
Second order ODE
Two first order ODEs
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Conversion Procedure
convert
solve
high order ODE
System of first order ODE

• Select of dependent variables
• One way is to take the original dependent
  variable and its derivatives up to one degree
  less than the highest order derivative.
• Write the Differential Equations in terms of the
  new variables. The equations comes from the way
  the new variables are defined or from the
  original equation.
• Express the equations in matrix form

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Remarks on the Conversion Procedure
convert
solve
high order ODE
System of first order ODE

• Any nth order ODE is converted to a system of n
  first order ODE.
• There are infinite number of ways to select the
  new variables. As a result, for each high order
  ODE there are infinite number of set of
  equivalent first order systems of ODEs.
• Use a table to make conversion easier.

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Example of converting High order ODE to first
order ODEs
One degree less than the highest order derivative
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Example of converting High order ODE to first
order ODEs
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Example of converting High order ODE to first
order ODEs
One degree less than the highest order derivative
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Example of converting High order ODE to first
order ODEs
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Conversion Procedure for Systems of high order
ODEs
convert
solve
System of high order ODE
System of first order ODE

• Select of dependent variables
• take the original dependent variables and
  their derivatives up to one degree less than the
  highest order derivative for each variable.
• Write the Differential Equations in terms of the
  new variables. The equations comes from the way
  the new variables are defined or from the
  original equation.
• Express the equations in matrix form

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Example of converting High order ODE to first
order ODEs
One degree less than the highest order derivative
One degree less than the highest order derivative
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Example of converting High order ODE to first
order ODEs
One degree less than the highest order derivative
One degree less than the highest order derivative
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Example of converting High order ODE to first
order ODEs
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Solution of a second order ODE

• Solve the equation using Euler method. Use h0.1

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Solution of a second order ODE
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Summary

• Formulas used in solving a first order ODE are
  used to solve systems of first order ODEs.
  Instead of scalar variables and functions we have
  vector variables and vector functions.
• High order ODEs are converted to a set of first
  order ODEs

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Remaining Lessons in Unit 7
Solution of ODE
Lesson 7 Multi-step methods

• Lessons 8-9
• Boundary Value Problems
• Discretization method