1. Dr. Mujahed AlDhaifallah ( Term 342)
EE 3561 : Computational Methods Unit 8 Part I Solution of Ordinary Differential Equations
Dr. Mujahed AlDhaifallah
( Term 342)
EE3561_Unit 8
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2. Lesson 1: Introduction to ODE
EE3561:Computational Methods Topic 8 Solution of Ordinary Differential Equations
Lesson 1: Introduction to ODE
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3. Learning Objectives of Topic 8
Solve Ordinary differential equation (ODE) problems.
Appreciate the importance of numerical method in solving ODE.
Assess the reliability of the different techniques.
Select the appropriate method for any particular problem.
Develop programs to solve ODE.
Use software packages to find the solution of ODE
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4. Computer Objectives of Topic 8
Develop programs to solve ODE.
Use software packages to find the solution of ODE
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5. Lessons in Topic 8 Lesson 1: Introduction to ODE
Lesson 2: Taylor series methods
Lesson 3: Midpoint and Heun’s method
Lesson 4: Runge-Kutta methods
Lesson 5: Applications of RK method
Lesson 6: Solving systems of ODE
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6. Learning Objectives of Lesson 1
Recall basic definitions of ODE,
order,
linearity
initial conditions,
solution,
Classify ODE based on( order, linearity, conditions)
Classify the solution methods
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7. Derivatives Derivatives Partial Derivatives Ordinary Derivatives
v is a function of one
independent variable
Partial Derivatives
u is a function of
more than one
independent variable
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8. Differential Equations
Ordinary Differential Equations
involve one or more
Ordinary derivatives of
unknown functions
Partial Differential Equations
involve one or more
partial derivatives of
unknown functions
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9. Ordinary Differential Equations
Ordinary Differential Equations (ODE) involve one or more ordinary derivatives of unknown functions with respect to one independent variable
x(t): unknown function
t: independent variable
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10. Example of ODE: Model of falling parachutist
The velocity of a falling parachutist is given by
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11. Ordinary differential equation
Definitions
Ordinary differential equation .
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12. (Dependent variable) unknown function to be determined
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13. (independent variable)
the variable with respect to which other variables are differentiated
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14. Order of a differential equation
The order of an ordinary differential equations is the order of the highest order derivative
First order ODE
Second order ODE
Second order ODE
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15. Solution of a differential equation
A solution to a differential equation is a function that satisfies the equation.
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16. Linear ODE Linear ODE Non-linear ODE An ODE is linear if
The unknown function and its derivatives appear to power one
No product of the unknown function and/or its derivatives
Linear ODE
Non-linear ODE
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17. Nonlinear ODE
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18. Solutions of Ordinary Differential Equations
Is it unique?
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19. Uniqueness of a solution
In order to uniquely specify a solution to an n th order differential equation we need n conditions
Second order ODE
Two conditions are needed to uniquely specify the solution
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20. Auxiliary conditions auxiliary conditions Boundary Conditions
The conditions are not at one point of the independent variable
Initial Conditions
all conditions are at one point of the independent variable
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21. Boundary-Value and Initial value Problems
Boundary-Value Problems
The auxiliary conditions are not at one point of the independent variable
More difficult to solve than initial value problem
Initial-Value Problems
The auxiliary conditions are at one point of the independent variable
same
different
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22. Classification of ODE ODE can be classified in different ways Order
First order ODE
Second order ODE
Nth order ODE
Linearity
Linear ODE
Nonlinear ODE
Auxiliary conditions
Initial value problems
Boundary value problems
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23. Analytical Solutions
Analytical Solutions to ODE are available for linear ODE and special classes of nonlinear differential equations.
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24. Numerical Solutions
Numerical method are used to obtain a graph or a table of the unknown function
Most of the Numerical methods used to solve ODE are based directly (or indirectly) on truncated Taylor series expansion
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25. Classification of the Methods
Numerical Methods
for solving ODE
Single-Step Methods
Estimates of the solution at a particular step are entirely based on information on the previous step
Multiple-Step Methods
Estimates of the solution at a particular step are based on information on more than one step
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26. Summary of Lesson 1 Recall basic definitions of ODE, order, linearity
initial conditions,
solution,
Classify ODE
First order ODE, Second Order ODE,…
Linear ODE, nonlinear ODE;
Initial value problems, boundary value problems
Classify the solution methods
Single step methods, multiple step methods
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27. More Lessons in this unit
Lesson 2: Taylor series methods
Lesson 3: Midpoint and Heun’s method
Lessons 4-5: Runge-Kutta methods
Lesson 6: Solving systems of ODE
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28. Lesson 2: Taylor Series Methods
SE301:Numerical Methods Topic 8 Solution of Ordinary Differential Equations
Lesson 2: Taylor Series Methods
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29. Lessons in Topic 8 Lesson 1: Introduction to ODE
Lesson 2: Taylor series methods
Lesson 3: Midpoint and Heun’s method
Lessons 4-5: Runge-Kutta methods
Lesson 6: Solving systems of ODE
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30. Learning Objectives of Lesson 2
Derive Euler formula using Taylor series expansion
Solve first order ODE using Euler method.
Assess the error level when using Euler method
Appreciate different types of error in numerical solution of ODE
Improve Euler method using higher-order Taylor Series.
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31. Taylor Series Method The problem to be solved is a first order ODE
Estimates of the solution at different base points
are computed using truncated Taylor series
expansions
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32. Taylor Series Expansion
nth order Taylor series method uses nth order Truncated Taylor series expansion
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33. Euler Method First order Taylor series method is known as Euler Method
Only the constant term and linear term are used in Euler method.
The error due to the use of the truncated Taylor series is of order O(h2).
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34. First Order Taylor Series Method (Euler Method)
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35. Euler Method
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36. Interpretation of Euler Methody2 y1 y0
x x x x
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37. Interpretation of Euler Method
Slope=f(x0,y0) y1
y1=y0+hf(x0,y0)
hf(x0,y0) y0
x x x x h
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38. Interpretation of Euler Methody2
y2=y1+hf(x1,y1)
Slope=f(x1,y1)
hf(x1,y1)
Slope=f(x0,y0) y1
y1=y0+hf(x0,y0)
hf(x0,y0) y0
x x x x h h
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39. Example 1 Use Euler method to solve the ODE
to determine y(1.01), y(1.02) and y(1.03)
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40. Example 1
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41. Example 1 Summary of the result i xi yi 1.00 -4.00 1 1.01 -3.98 2 1.02
1.00
-4.00 1
1.01
-3.98 2
1.02 3
1.03
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42. Example 1 Comparison with true value i xi yi 1.00 -4.00 1 1.01 -3.98
True value of yi
1.00
-4.00 1
1.01
-3.98 2
1.02 3
1.03
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43. Example 1 A graph of the solution of the ODE for 1<x<2
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44. Types of Errors Local truncation error:
error due to the use of truncated Taylor series to compute x(t+h) in one step.
Global Truncation error
accumulated truncation over many steps
Round off error:
error due to finite number of bits used in representation of numbers. This error could be accumulated and magnified in succeeding steps.
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45. Second Order Taylor Series methods
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46. Third Order Taylor Series methods
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47. High Order Taylor Series methods
Animate :
Convert to Speech:
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48. Higher Order Taylor Series methods
High order Taylor series methods are more accurate than Euler method
The 2nd, 3rd and higher order derivatives need to be derived analytically which may not be easy.
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49. Example 2 Second order Taylor Series Method
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50. Example 2
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51. Example 2
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52. Example 1 Summary of the results i ti xi 0.00 1 0.01 0.9901 2 0.02
0.00 1
0.01
0.9901 2
0.02
0.9807 3
0.03
0.9716
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53. Programming Euler Method
Write a MATLAB program to implement Euler method to solve
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54. Programming Euler Method
f=inline('1-2*v^2-t','t','v')
h=0.01
t=0
v=1
T(1)=t;
V(1)=v;
for i=1:100
v=v+h*f(t,v)
t=t+h;
T(i+1)=t;
V(i+1)=v;
end
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55. Programming Euler Method
f=inline('1-2*v^2-t','t','v')
h=0.01
t=0
v=1
T(1)=t;
V(1)=v;
for i=1:100
v=v+h*f(t,v)
t=t+h;
T(i+1)=t;
V(i+1)=v;
end
Definition of the ODE
Initial condition
Main loop
Euler method
Storing information
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56. Programming Euler Method
Plot of the
solution
plot(T,V)
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57. Summary of Lesson 2 Derive Euler formula
Solve first order ODE using Euler method.
Assess the error level when using Euler method
Error in numerical solution of ODE
Higher-order Taylor Series.
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58. More in this unit Lesson 3: Midpoint and Heun’s method
Provide the accuracy of the second order
Taylor series method without the need to
calculate second order derivative.
Lessons 4-5: Runge-Kutta methods
Provide the accuracy of high order
calculate high order derivative
Lesson 6: Solving systems of ODE
Formulas used in solving first order ODE are used to solve systems ODEs. Instead of scalar variables and functions we have vector variables and vector functions.
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