Engr/Math/Physics 25
Chp9: ODE’s
Numerical Solns
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Learning Goals
 List Characteristics of Linear,
MultiOrder, NonHomgeneous Ordinary
Differential Equations (ODEs)
 Solve ANALYTICALLLY, Linear, 2nd
Order, NonHomogeneous, Constant
Coefficient ODEs
 Use MATLAB to determine Numerical
Solutions to Ordinary Differential
Equations (ODEs)
Engineering/Math/Physics 25: Computational Methods
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Differential Equations
 Ordinary Diff Eqn
 Partial Diff Eqn
2 y
y
y
d2y
dy






 x, t 
x
,
t

b
x
,
t

b
y
x
,
t

t   a1 t   a2 yt   f (t ) 2
1
2
2
dt
dt
t
t
x
 PDE’s Not Covered in ENGR25
• Discussed in More Detail in ENGR45
 Examining the ODE, Note that it:
• is LINEAR → y, dy/dt, d2y/dt2 all raised to Power of 1
• 2nd ORDER → Highest Derivative is 2
• is NONhomogenous → RHS  0;
– i.e., y(t) has a FORCING Fcn f(t)
• has CONSTANT Coefficients
Engineering/Math/Physics 25: Computational Methods
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Numerical ODE Solution
 Today we’ll do some
MTH25
 We’ll “look under the
hood” of
NUMERICAL
Solutions to ODE’s
 The BASIC GamePlan for even
Sophisticated
Solvers:
• Given a STARTING
POINT, y(0)
• Use ODE to find
dy/dt at t=0
• ESTIMATE y1 as
dy
y1  y0  t 
dt t 0
Engineering/Math/Physics 25: Computational Methods
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Numerical Solution - 1
 Notation
n  Step Number
t  Time Step Length
tn  n  t
 Exact Numerical
Method (impossible
to achieve) by
Forward Steps
yn+1
y n  y (t n )
f n  f (t n , y n )
yn
 Now Consider
dy
 f (t , y )
dt
Engineering/Math/Physics 25: Computational Methods
5
tn
t
tn+1
t
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Numerical Solution - 2
yn+1
 The diagram at Left shows
that the relationship between
yn, yn+1 and the CHORD slope
Tangent
Slope
yn
Chord
Slope
tn
t
The Analyst
Chooses Δt
tn+1
y n1  y n
 chord slope
t
 The problem with this formula
is we canNOT calculate the
t
chord slope exactly
• We Know Only Δt & yn, but
NOT the NEXT Step yn+1
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Numerical Solution -3
 However, we can
calculate the
TANGENT slope at
any point FROM the
differential equation
itself
dy
mn 
 f (t n , yn )
dt t  t n
 Recognize dy/dt as
the Tangent Slope
tangent slope  f (t , y )
Engineering/Math/Physics 25: Computational Methods
7
 The Basic Concept
for all numerical
methods for solving
ODE’s is to use the
TANGENT slope,
available from the
R.H.S. of the ODE,
to approximate the
chord slope
yn 1  yn dy

 f t n , yn 
t
dt tn
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Euler Method – 1st Order
 Solve 1st Order ODE
with I.C.
dy
 f (t , y )
dt
y ( 0)  b
 Use: [Chord Slope]
 [Tangent Slope at
start of time step]
yn 1  yn dy

 f t n , yn 
t
dt tn
Engineering/Math/Physics 25: Computational Methods
8
 ReArranging
dy
yn 1  t
 yn
dt t n
or
yn1  yn  t  f n
 Then Start the
“Forward March”
with Initial
Conditions
t0  0
y0  b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Euler Example
 Consider 1st Order
ODE with I.C.
dy
  y 1
dt
y (0)  0
 Use The Euler
Forward-Step Reln
yn 1  yn  t f n
dy
 yn  t
dt tn
Engineering/Math/Physics 25: Computational Methods
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 But from ODE
dy
  yn  1
dt tn
 So In This Example:
yn1  yn  t (  yn  1)
 See Next Slide for
the 1st Nine Steps
For Δt = 0.1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Euler Exmpl Calc
dy
  y  1 t  0.1
dt
tn
yn
fn= – yn+1
yn+1= yn+t fn
0
0
0.000
1.000
0.100
1
0.1
0.100
0.900
0.190
2
0.2
0.190
0.810
0.271
3
0.3
0.271
0.729
0.344
4
0.4
0.344
0.656
0.410
5
0.5
0.410
0.590
0.469
6
0.6
0.469
0.531
0.522
7
0.7
0.522
0.478
0.570
8
0.8
0.570
0.430
0.613
9
0.9
0.613
0.387
0.651
n
Slope
Plot
Engineering/Math/Physics 25: Computational Methods
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Euler vs Analytical
0.8
 The Analytical
Solution
0.6
y  1 e
y
0.4
Numerical
0.2
Exact
1.25
1
0.75
0.5
0.25
0
0
t
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
t
Analytical Soln
dy
  y  1 y t  0   0
dt
 Let u = −y+1
 Integrate Both Sides
 Then
u  1 y
du dy  0  1
dy  du
 Sub for y & dy in
ODE  du
dt
u
du
 Separate
  dt
Variables u
Engineering/Math/Physics 25: Computational Methods
12
du
 u   1dt
 Recognize LHS as
Natural Log
ln u  t  C
 Raise “e” to the
power of both sides
e
ln u
e
t C
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Analytical Soln
dy
  y  1 y t  0   0
dt
 And
 Now use IC
e
e
ln u
1  0  Ke
u
t  C
C t
 e e  Ke
 Thus Soln u(t)
u  Ke
t
t
0
 K 1
 The Analytical Soln
1  y  1 e
t
 Sub u = 1−y
1  y  Ke
t
Engineering/Math/Physics 25: Computational Methods
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y  1 e
t
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Predictor-Corrector - 1
 Again Solve 1st
 Mathematically
Order ODE with I.C. y  y
n 1
n
 0.5 f (tn , yn )  f (tn 1 , yn 1 )
dy
dt
 f (t , y )
y ( 0)  b
 This Time Let:
Chord slope 
average of tangent
slopes at start and
END of time step
Engineering/Math/Physics 25: Computational Methods
14
t
Avg of the Tangent Slopes
at (tn,yn) & (tn+1,yn+1)
 BUT, we do NOT
know yn+1 and it
appears on the
RHS...
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Predictor-Corrector - 2
 Use Two Steps to
estimate yn+1
 First → PREDICT*
y

n 1
 yn  y
yn1  yn  t dy dt n
y

n 1
 yn  t f t n , yn 
yn1  yn  t f n
 Use y* in the
Avg Calc

 0.5t  f
y n1 yn
*

f
n
n 1

 Then Start the
“Forward March”
with the Initial
Conditions
 Then Correct
Engineering/Math/Physics 25: Computational Methods
15

y n1 yn  0.5t f tn , yn   f tn1 , yn1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt

Predictor-Corrector Example
dy
  y 1
dt
 Solve ODE with IC
y (0)  0

yn 1  yn  0.5t f n  f
 The Corrector step
 The next Step Eqn for dy/dt = f(t,y)= –y+1


yn 1  yn  0.5t  yn  1   y
*
n 1

1
 Numerical Results on Next Slide
Engineering/Math/Physics 25: Computational Methods
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt

n 1

Predictor-Corrector Example
f  dy dt   y  1
n
tn
yn

yn 1  yn  0.5t f n  f n1
fn
Slope
f n*1
yn1
Slope
0
0
0.000
1.000
0.100
0.900
0.095
1
0.1
0.095
0.905
0.186
0.815
0.181
2
0.2
0.181
0.819
0.263
0.737
0.259
3
0.3
0.259
0.741
0.333
0.667
0.329
4
0.4
0.329
0.671
0.396
0.604
0.393
Engineering/Math/Physics 25: Computational Methods
17
y n*1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt

Predictor-Corrector
0.8
 Greatly
Improved
Accuracy
0.6
y
0.4
Exact
0.2
Mod. Euler
t
Engineering/Math/Physics 25: Computational Methods
18
1.25
1
0.75
0.5
0.25
0
0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
dy
Example: dt  3.9 cos4.2 y   3.9 ln 5.1t  6
ODE
 Euler Solution with
∆t = 0.25
 The Solution Table
Euler Solution to dy/dt = 3.9cos(4.2y)-ln(5.1t+6)
38
36
34
y(t) by Euler
32
30
28
26
24
22
0
1
2
3
4
5
t
6
7
8
9
Engineering/Math/Physics 25: Computational Methods
19
10
n
t
y
dy/dt
dely
yn+1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
4.25
4.5
4.75
5
5.25
5.5
5.75
6
6.25
6.5
6.75
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
37.0000
36.5636
36.9143
36.5769
36.8872
36.5806
36.8418
36.6641
36.9608
36.3357
35.6768
35.2701
35.2882
35.2273
35.3380
35.0526
35.0491
35.0223
34.8909
34.2399
33.9524
33.1997
32.4496
31.6958
30.9492
30.1897
29.4564
28.6710
27.9981
27.1110
26.6745
25.9565
25.3424
25.2245
24.6604
24.6512
24.6268
24.5593
24.2973
23.3007
23.0678
-1.7457
1.4027
-1.3492
1.2410
-1.2264
1.0448
-0.7108
1.1868
-2.5004
-2.6357
-1.6265
0.0722
-0.2436
0.4430
-1.1420
-0.0139
-0.1072
-0.5255
-2.6041
-1.1497
-3.0108
-3.0006
-3.0151
-2.9862
-3.0384
-2.9328
-3.1419
-2.6916
-3.5484
-1.7458
-2.8722
-2.4562
-0.4717
-2.2562
-0.0369
-0.0977
-0.2699
-1.0481
-3.9863
-0.9318
-1.0551
-0.4364
0.3507
-0.3373
0.3103
-0.3066
0.2612
-0.1777
0.2967
-0.6251
-0.6589
-0.4066
0.0181
-0.0609
0.1107
-0.2855
-0.0035
-0.0268
-0.1314
-0.6510
-0.2874
-0.7527
-0.7502
-0.7538
-0.7466
-0.7596
-0.7332
-0.7855
-0.6729
-0.8871
-0.4365
-0.7180
-0.6141
-0.1179
-0.5641
-0.0092
-0.0244
-0.0675
-0.2620
-0.9966
-0.2329
-0.2638
36.5636
36.9143
36.5769
36.8872
36.5806
36.8418
36.6641
36.9608
36.3357
35.6768
35.2701
35.2882
35.2273
35.3380
35.0526
35.0491
35.0223
34.8909
34.2399
33.9524
33.1997
32.4496
31.6958
30.9492
30.1897
29.4564
28.6710
27.9981
27.1110
26.6745
25.9565
25.3424
25.2245
24.6604
24.6512
24.6268
24.5593
24.2973
23.3007
23.0678
22.8040
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Compare Euler vs. ODE45
Euler Solution
ODE45 Solution
Euler Solution to dy/dt = 3.9cos(4.2y)-ln(5.1t+6)
37.5
38
36
37
34
36.5
Y by ODE45
y(t) by Euler
32
30
36
28
35.5
26
35
24
22
0
1
2
3
4
5
t
6
7
8
9
10
34.5
0
1
2
3
4
6
5
T by ODE45
Euler is Much LESS accurate
Engineering/Math/Physics 25: Computational Methods
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
7
8
9
10
Compare Again with ∆t = 0.025
Euler Solution
ODE45 Solution
Euler Solution to dy/dt = 3.9cos(4.2y)-ln(5.1t+6)
37.5
37.2
37
37
36.8
Y by ODE45
y(t) by Euler
36.5
36.6
36.4
36
35.5
36.2
35
36
35.8
34.5
0
1
2
3
4
5
t
6
7
8
9
10
0
1
2
3
4
6
5
T by ODE45
Smaller ∆T greatly improves Result
Engineering/Math/Physics 25: Computational Methods
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
7
8
9
10
MatLAB Code for Euler
% Bruce Mayer, PE
% ENGR25 * 04Jan11
% file = Euler_ODE_Numerical_Example_1201.m
%
y0= 37;
delt = 0.25;
t= [0:delt:10];
n = length(t);
yp(1) = y0; % vector/array indices MUST start at 1
tp(1) = 0;
for k = 1:(n-1) % fence-post adjustment to start at 0
dydt = 3.9*cos(4.2*yp(k))^2-log(5.1*tp(k)+6);
dydtp(k) = dydt % keep track of tangent slope
tp(k+1) = tp(k) + delt;
dely = delt*dydt
delyp(k) = dely
yp(k+1) = yp(k) + dely;
end
plot(tp,yp, 'LineWidth', 3), grid, xlabel('t'),ylabel('y(t) by Euler'),...
title('Euler Solution to dy/dt = 3.9cos(4.2y)-ln(5.1t+6)')
Engineering/Math/Physics 25: Computational Methods
22
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
MatLAB Command Window for
ODE45
>>
>>
>>
by
dydtfcn = @(tf,yf) 3.9*(cos(4.2*yf))^2-log(5.1*tf+6);
[T,Y] = ode45(dydtfcn,[0 10],[37]);
plot(T,Y, 'LineWidth', 3), grid, xlabel('T by ODE45'), ylabel('Y
ODE45')
Engineering/Math/Physics 25: Computational Methods
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
All Done for Today
Carl
Runge
Carl David Tolmé Runge
Born: 1856 in
Bremen, Germany
Engineering/Math/Physics 25: Computational Methods
24
Died: 1927 in
Göttingen, Germany
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Engr/Math/Physics 25
Appendix
f x   2 x  7 x  9 x  6
3
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
2nd Order ODE SUMMARY-1
 If NonHomogeneous
Then find ANY
Particular Solution
 The Soln to the
Homog. Eqn
Produces the
Complementary
d2y
dy
5 2 (t )  7 (t )  3 y (t )  18
Solution, yc
dt
dt

y p  18 / 3  6 (a CONST)  Assume yc take this
 Next HOMOGENIZE
the ODE
2
d y
dy
5 2 (t )  7 (t )  3 y (t )  0
dt
dt
Engineering/Math/Physics 25: Computational Methods
26
form
yc t   Ae st


y c t   sAe
st

y c t   s 2 Ae st
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
2nd Order ODE SUMMARY-2
 Subbing yc = Aest
into the Homog. Eqn
yields the
Characteristic Eqn
5s  7 s  3  0
2
 Find the TWO roots
that satisfy the Char
Eqn by Quadratic
Formula
s1, 2
 If s1 & s2 → REAL &
UNequal
yc t   G1e  G2 e
s1t
• Decaying Contant(s)
 7  72  4  5  3

25
Engineering/Math/Physics 25: Computational Methods
27
 Check FORM of
Roots
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
s 2t
2nd Order ODE SUMMARY-3
 If s1 & s2 → REAL &
Equal, then s1 = s2
=s
st
yc t   e mt  b 
 m, b are constants
• Decaying Line
 If s1 & s2 →
Complex
Conjugates then
yc t   e
 at
 Add Particlular &
Complementary
Solutions to yield
the Complete
Solution
y t   yc  y p
B1 cos t  B2 sin t 
Engineering/Math/Physics 25: Computational Methods
28
• Decaying Sinusoid
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
2nd Order ODE SUMMARY-4
 To Find Constant
 Find Number-Values
Sets: (G1, G2), (m,
for the constants to
b), (B1, B2) Take for
complete the
COMPLETE solution
solution process
y t  0  IC 0  y0

dy
 IC1  y 0
dt t 0
• Yields 2 eqns in 2 for
the 2 Unknown
Constants
Engineering/Math/Physics 25: Computational Methods
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Finite Difference Methods - 1
 Another way of
thinking about
numerical methods
is in terms of finite
differences.
 Use the
Approximation
y n 1  yn  dy 
 
t
 dt  n
Engineering/Math/Physics 25: Computational Methods
30
 And From the
Differential Eqn
 dy 
 dt   f (tn , y n )
n
 From these two
equations obtain:
y n 1  y n
 f (t n , y n )
t
 Recognize as the
Euler Method
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt
Finite Difference Methods - 2
 Could make More Accurate by
Approximating dy/dt at the Half-Step as the
average of the end pts
y n 1  y n  dy 
1   dy 
 dy  
 
       
t
 dt  n  1 2   dt  n  dt  n 1 
2
 Then Again Use the
ODE to Obtain
y n 1  y n 1
  f n  f n 1 
t
2
Engineering/Math/Physics 25: Computational Methods
31
 Recognize as the
Predictor-Corrector
Method
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Lec-21_Integ_Diff.ppt