525602:Advanced Numerical Methods for ME

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 VishSubramaniam, Wiley,2008 • Numerical Methods Using MATLAB, John.H.Methews., Kurtis D.Fink.,Prentice Hall , Fourth Edition,2004 • ระเบียบวิธีเชิงตัวเลขในงานวิศวกรรม, ปราโมทย์เดชะอำไพ., 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 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 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 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 Boundary Conditions • The conditions are not at one point of the independent variable auxiliary conditions Initial Conditions • all conditions are at one point of the independent variable

33. same different 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

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