1. 525602:Advanced Numerical Methods for ME
School of Mechanical Engineering

2. Prescribed text :
Numerical Method for Engineering, Seventh Edition, Steven C.Chapra, Raymond P. Canale.,McGraw Hill 2014

3. Recommended reading :
Numerical Methods for Engineers and Scientists, Amos Gilat and Vish Subramaniam, Wiley,2008
Numerical Methods Using MATLAB, John.H.Methews., Kurtis D.Fink.,Prentice Hall , Fourth Edition,2004
ระเบียบวิธีเชิงตัวเลขในงานวิศวกรรม, ปราโมทย์ เดชะอำไพ., สำนักพิมพ์แห่งจุฬาลงกรณ์มหาวิทยาลัย พ.ศ.2556

4. Course Description Truncation errors and the Taylor series
Numerical Solutions for ODEs
Numerical Solutions for PDEs
Finite difference method
Optimization

5. Errors in Numerical Solutions
Truncation Errors
Round-Off Errors
Total Error
Local Error
Global Error

6. Truncation Errors :The Taylor series
Zero-order approximation
First-order approximation

7. The Taylor series
approximation

8. The Remainder for the Taylor Series Expansion

9. Truncation Errors: Example
Taylor’s series expansion:
The exact value:
The Zero-order approximation;
The truncation error:

10. Truncation Errors: Example
Taylor’s series expansion:
The exact value:
The first-order approximation;
The truncation error:

11. Round-Off Errors

12. Total Error
The true error:
The true relative error:

13. Total Error
The Global Error is the total discrepancy due to past as well as present steps
The Local Error refers to the error incurred over a single step. It is calculated with a Taylor series expansion.

14. The Global Error
The relative global error:

15. The Total Error: Example

16. The Total Error: Example
The percent relative global error (Step 1);
The percent relative global error (Step 2);

17. The Total Error: Example
Exact estimates of the errors in Euler’s method
Problem statement:

18. The Total Error: Example
The percent relative local error (Step 1);

19. The Total Error: Example
The percent relative local error (Step 2);

20. Numerical Solutions for ODEs
Introduction to Ordinary
Differential Equations (ODE)

21. Study Objectives 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.

22. Computer Objectives Develop programs to solve ODE.
Use software packages to find the solution of ODE

23. 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

24. 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

25. 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

26. 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

27. 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

28. Solution of a differential equation
A solution to a differential equation is a function that satisfies the equation.

29. 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

30. Nonlinear 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

31. 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

32. 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

33. 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

34. 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

35. Analytical Solutions
Analytical Solutions to ODE are available for linear ODE and special classes of nonlinear differential equations.

36. 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

37. 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

38. More Lessons in this unit
Taylor series methods
Midpoint and Heun’s method
Runge-Kutta methods
Multiple step Methods
Solving systems of ODE
Boundary value Problems

40. The Taylor series
Zero-order approximation
First-order approximation

41. The Taylor series
approximation

42. The Remainder for the Taylor Series Expansion

43. The sequence of events in the application of ODEs for engineering problem solving

44. Differential equations
Analytical methods
Exact solution
Numerical methods
Approximate solution