Dr. Jie Zou PHY 33201 Chapter 9 Ordinary Differential Equations: Initial-Value Problems Lecture (I)...
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Transcript of Dr. Jie Zou PHY 33201 Chapter 9 Ordinary Differential Equations: Initial-Value Problems Lecture (I)...
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
Applications One-step Runge-Kutta (RK) Methods
(1) Euler’s Method The method (algorithm) Error analysis (next lecture) Stability (next lecture)
Dr. Jie Zou PHY 3320 3
Introduction: Some definitions
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:
2vcmgdt
dvm d
02
2
kxdt
dxc
dt
xdm
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 vvcmgdt
dvm 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 ,2
22
2
LψψmE
dx
d
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 vvcmgdt
dvm d
ii
ii
tt
tvtv
dt
dv
1
1
iiid
ii tttvm
cgtvtv
1
21
tdt
dvvv
itii 1
Euler’s method (a one-step method)
Fig. 1.4 (Ref. by Chapra): Numerical solution by Euler’s method.
Dr. Jie Zou PHY 3320 7
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
Dr. Jie Zou PHY 3320 8
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
Dr. Jie Zou PHY 3320 9
Resultst ytrue yEuler |t| (%)
0 2.00000 2.00000 ----
1 6.19463 5.00000 19.28
2 14.84392 11.40216 23.19
3 33.67717 25.51321 24.24
4 75.33896 56.84931 24.54
Fig. 20.2 (Ref. by Chapra)
Table 20.1 (Ref. by Chapra)