551614:Advanced Mathematics for Mechatronics. Numerical solution for ODEs School of Mechanical Engineering

Transcription

1 551614:Advanced Mathematics for Mechatronics Numerical solution for ODEs School of Mechanical Engineering 1

2 Prescribed text : Numerical Method for Engineering, Seventh Edition, Steven C.Chapra, Raymond P. Canale.,McGraw Hill 014

3 Recommended reading : Numerical Methods for Engineers and Scientists, Amos Gilat and Vish Subramaniam, Wiley,008 Numerical Methods Using MATLAB, John.H.Methews., Kurtis D.Fink.,Prentice Hall, Fourth Edition,004 ระเบ ยบว ธ เช งต วเลขในงานว ศวกรรม, ปราโมทย เดชะ อ าไพ., ส าน กพ มพ แห งจ ฬาลงกรณ มหาว ทยาล ย พ.ศ.556 3

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

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

6 Truncation Errors :The Taylor series f ( x ) = f ( x ) i+ 1 i + x 1! df dx x i + ( ) x d f ( x)! dx x i n n d f n n! dx x i Zero-order approximation f ( ) ( ) x f x i i +1 First-order approximation f ( x ) = f ( x ) i+ 1 i + x 1! df dx x i 6

7 The Taylor series approximation f ( ) 4 3 x = 0.1x 0.15x 0.5x 0.5x

8 The Remainder for the Taylor Series Expansion 8

9 Truncation Errors: Example Taylor s series expansion: sin( x) x x x x = x ! 5! 7! 9! The exact value: π sin = The Zero-order approximation; π π sin = = TR The truncation error: E = =

10 Truncation Errors: Example Taylor s series expansion: sin( x) x x x x = x ! 5! 7! 9! The exact value: π sin = The first-order approximation; π π π 6 sin = = ! TR The truncation error: E = =

11 Round-Off Errors 11

12 Total Error The true error: TrueError = TrueSolution NumericalSolution The true relative error: TrueRelativeError = TrueSolution NumericalSolution TrueSolution 1

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. 13

14 The Global Error The relative global error: TrueSolution NumericalSolution %TheGlobalError = 100 TrueSolution 14

15 The Total Error: Example 15

16 The Total Error: Example The percent relative global error (Step 1); %TheGlobalError = 100 = The percent relative global error (Step ); %TheGlobalError = 100 =

17 The Total Error: Example Exact estimates of the errors in Euler s method E local ( x) ( x) ( x) d f d f d f = + +! dx 3! dx 4! dx 3 4 x x x i i i Problem statement: 3 y = x + 1x 0x+ 8.5 y = 6x + 4x 0 y = 1x+ 4 4 d f dx 4 = 1 17

18 The Total Error: Example The percent relative local error (Step 1); E E E l l3 l 4 6(0) 4(0) 0 (0.5).5 + = = 1(0) + 4 (0.5) 3 = = (0.5) 4 = = local total l l l E = E + E 3 + E 4 =.0315 %TheLocalError= local E total TrueSolution = 18

19 The Total Error: Example The percent relative local error (Step ); E E E l l3 l 4 6(0.5) 4(0.5) 0 (0.5) = = 1(0.5) + 4 (0.5) 3 = = (0.5) 4 = = local total l l l E = E + E 3 + E 4 = %TheLocalError= 100 =

20 Numerical Solutions for ODEs Introduction to Ordinary Differential Equations (ODE) 0

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. 1

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 3

24 Derivatives Derivatives Ordinary Derivatives dv dt v is a function of one independent variable Partial Derivatives u y u is a function of more than one independent variable 4

25 Differential Equations Differential Equations Ordinary Differential Equations d dt v + 6tv = 1 involve one or more Ordinary derivatives of unknown functions Partial Differential Equations y u 0 involve one or more partial derivatives of unknown functions x u = 5

26 Ordinary Differential Equations Ordinary Differential Equations (ODE) involve one or more ordinary derivatives of unknown functions with respect to one independent variable Examples : dv( dt v( = e t x(: unknown function d x( dt 5 dx( dt + x( = cos( t: independent variable 6

27 Order of a differential equation The order of an ordinary differential equations is the order of the highest order derivative Examples : dx( t x( = e dt d x( dx( 5 dt dt d x( dt 3 + x( dx( dt + x = cos( 4 ( = 1 First order ODE Second order ODE Second order ODE 7

28 Solution of a differential equation A solution to a differential equation is a function that satisfies the equation. Example dx( dt : + x( = 0 Solution Proof dx( dt dx( dt : = e + x( x( t = = e e t t + e t = 0 8

29 An ODE is linear if Examples dx() t xt () = e dt : t Linear ODE The unknown function and its derivatives appear to power one No product of the unknown function and/or its derivatives Linear ODE 3 dxt () dxt () xt () 1 + = dt dt Non-linear ODE 9

30 An ODE is linear if Nonlinear ODE The unknown function and its derivatives appear to power one No product of the unknown function and/or its derivatives Examples of nonlinear ODE : dx( cos( x( ) dt d x( dx( 5 dt dt d x( dx( dt dt = 1 x( = + x( = 1 30

31 Uniqueness of a solution In order to uniquely specify a solution to an n th order differential equation we need n conditions d y( x) + dt y(0) = a y (0) = b 4y( x) = 0 Second order ODE Two conditions are needed to uniquely specify the solution 31

32 Auxiliary conditions auxiliary conditions Initial Conditions all conditions are at one point of the independent variable Boundary Conditions The conditions are not at one point of the independent variable 3

33 Boundary-Value and Initial value Problems Initial-Value Problems Boundary-Value Problems The auxiliary conditions are at one point of the independent variable The auxiliary conditions are not at one point of the independent variable More difficult to solve than initial value problem x + x + x = e t x(0) = 1, x (0) = same.5 x + x + x = e t x(0) = 1, x() = 1.5 different 33

34 Classification of ODE ODE can be classified in different ways Order First order ODE Second order ODE N th order ODE Linearity Linear ODE Nonlinear ODE Auxiliary conditions Initial value problems Boundary value problems 34

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

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 36

37 Classification of the Methods Numerical Methods for solving ODE One-Step Methods Multiple-Step Methods Estimates of the solution at a particular step are entirely based on information on the Estimates of the solution at a particular step are based on information on more than one step previous step 37

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 38

39 39

40 The sequence of events in the application of ODEs for engineering problem solving 40

41 Differential equations Analytical methods Exact solution Numerical methods Approximate solution 41

42 y y = 0.5x = x x + 1x 3 10x 0x + 8.5x

43 y y = 0.5x = x x + 1x 3 10x 0x + 8.5x