1. SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36
KFUPM
(Term 101)
Section 04
Read , 26-2, 27-1
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2. Outline of Topic 8 Lesson 1: Introduction to ODEs
Lesson 2: Taylor series methods
Lesson 3: Midpoint and Heun’s method
Lessons 4-5: Runge-Kutta methods
Lesson 6: Solving systems of ODEs
Lesson 7: Multiple step Methods
Lesson 8-9: Boundary value Problems
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3. Lecture 33 Lesson 6: Solving Systems of ODEs
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4. Learning Objectives of Lesson 6
Convert a single (or a system of) high order ODE to a system of first order ODEs.
Use the methods discussed earlier in this topic to solve systems of first order ODEs.
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5. Outlines of Lesson 6 Solution of a system of first order ODEs.
Conversion of a high order ODE 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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6. 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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7. Euler Method for Solving a System of First Order ODEs
Recall Euler method for solving a first order ODE:
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8. Example - Euler Method
Euler method to solve a system of n first order ODEs.
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9. 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 a vector valued function.
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10. Example : Euler method for solving a system of first order ODEs.
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11. Example : RK2 method for solving a system of first order ODEs
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12. Example : RK2 method for solving a system of first order ODEs
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13. Methods for Solving a System of First Order ODEs
We have extended Euler and RK2 methods to solve systems of first order ODEs.
Other methods used to solve first order ODE can be easily extended to solve systems of first order ODEs.
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14. High Order ODEs How do solve a second order ODE?
How do solve high order ODEs?
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15. The General Approach to Solve ODEs
Convert
Solve
High order ODE
System of first order ODEs
Convert
Solve
Second order ODE
Two first order ODEs
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16. System of first order ODEs
Conversion Procedure
Convert
Solve
High order ODE
System of first order ODEs
Select the 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 come from the way the new variables are defined or from the original equation.
Express the equations in a matrix form.
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17. 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 ODEs.
There are an infinite number of ways to select the new variables. As a result, for each high order ODE there are an infinite number of set of equivalent first order systems of ODEs.
Use a table to make the conversion easier.
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18. Example of Converting a High Order ODE to First Order ODEs
One degree less than the highest order derivative
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19. Example of Converting a High Order ODE to First Order ODEs
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20. Example of Converting a High Order ODE to First Order ODEs
One degree less than the highest order derivative
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21. Example of Converting a High Order ODE to First Order ODEs
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22. Conversion Procedure for Systems of High Order ODEs
Convert
Solve
System of high order ODEs
System of first order ODE
Select the 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 come from the way the new variables are defined or from the original equation.
Express the equations in a matrix form.
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23. Example of Converting a 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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24. Example of Converting a High Order ODE to First Order ODEs
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25. Solution of a Second Order ODE
Solve the equation using Euler method. Use h=0.1
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26. Solution of a Second Order ODE
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27. 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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28. Remaining Lessons in Topic 8
Solution of ODEs
Lesson 7:
Multi-step methods
Lessons 8-9:
Boundary Value Problems
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