Dr. Jie Zou PHY 3320 1 Chapter 9 Ordinary Differential Equations: Initial-Value Problems Lecture (I) 1 1 Besides the main textbook, also see Ref.: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2nd ed., Ch. 20 , McGraw Hill, 2008.
Chapter 9. Ordinary Differential Equations: Initial-Value Problems Lecture (I) 1. 1 Besides the main textbook, also see Ref.: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2nd ed., Ch. 20 , McGraw Hill, 2008. Outline. Introduction: Some definitions - PowerPoint PPT Presentation
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Dr. Jie Zou PHY 3320 1
Chapter 9
Ordinary Differential Equations: Initial-Value
ProblemsLecture (I)11 Besides the main textbook, also see Ref.: “Applied Numerical
Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2nd ed., Ch. 20, McGraw Hill, 2008.
Dr. Jie Zou PHY 3320 2
Outline Introduction: Some definitions Engineering and Scientific
Differential equation: An equation involving the derivatives or differentials of the dependent variable.
Ordinary differential equation: A differential equation involving only one independent variable.
Example: For the bungee jumper,
Partial differential equation: A differential equation involving two or more independent variables (with partial derivatives).
Order of a differential equation: The order of the highest derivative in the equation.
Example: For an unforced mass-spring system with damping-a second-order equation:
2vcmgdtdvm d
02
2
kxdtdxc
dtxdm
Dr. Jie Zou PHY 3320 4
Introduction: Some definitions (cont.)
For an nth-order differential equation, n conditions are required to obtain a unique solution.
Initial-value problem: All conditions are specified at the same value of the independent variable (e.g., at x or t = 0).
Example: For the bungee jumper,
Boundary-value problem: Conditions are specified at different values of the independent variable.
Example: Particle in an infinite square well
00 ,2 vvcmgdtdvm d
Initial ConditionFig. PT6.3 (Ref. by
Chapra): Solutions for dy/dx = -2x3 + 12x2 – 20x + 8.5 with different constants of integration, C.
0 ,00 ,222
2
LψψmEdxd
Boundary Conditions
Dr. Jie Zou PHY 3320 5
Engineering and scientific applications
Fig. PT6.1 (Ref. by Chapra): The sequence of events in the development and solution of ODEs for engineering and science.
Dr. Jie Zou PHY 3320 6
Euler’s method Let’s look at the Bungee-Jumper’s
example: Solve an ODE-initial-value problem (1)
Step 1: Finite-difference approximation for dv/dt (2)
Step 2: Substitute Eq. (2) in Eq. (1) (3) Step 3: Notice that dv/dt at ti = g-
cdv(ti)2/m, (3) becomes
00 ,2 vvcmgdtdvm d
ii
ii
tttvtv
dtdv
1
1
iiid
ii tttvmcgtvtv
1
21
tdtdvvv
itii 1Euler’s method
(a one-step method)
Fig. 1.4 (Ref. by Chapra): Numerical solution by Euler’s method.
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Another look at Euler’s method
Solving ODE: dy/dt = f(t,y) All one-step methods (Runge-Kutta
methods) have the general form: : an increment function for
extrapolating from an old value yi to a new value yi+1.
One-step methods: use information from one pervious point i to extrapolate to a new value.
h: Step size = ti+1 – ti. Euler’s method:
= f(ti,yi), the 1st derivate of y at ti yi+1 = yi + f(ti,yi)h
yi1 yi h
Fig. 20.1 (Ref. by Chapra): Euler’s method
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Example: Euler’s method Example 20.1 (Ref.): Use Euler’s
method to integrate y’ = 4e0.8t – 0.5y from t = 0 to t = 4 with a step size of 1. The initial condition at t = 0 is y = 2. Note that the exact solution can be determined analytically as y = (4/1.3)(e0.8t – e-0.5t) + 2e-0.5t