Chem 302 - Math 252

Chapter 6Differential Equations

Differential Equations

• Many problems in physical chemistry (eg. kinetics, dynamics, theoretical chemistry) require solution to a differential equation

• Many can not be solved analytically

• Deal only with first order ODE– Higher order equations can be reduced to a system of 1st order DE

2y k y

2z k y

y z

Differential Equations

• Simplest form y f x

dy f x dx

y f x dx

00

x

xy y f x dx

• Can integrate analytically or numerically (using techniques of Chapter 4)

Differential Equations

• General case ,y f x y

• Many simpler problems can be solved analytically

• Many involve ex

• However, in chemistry (physics & engineering) many problems have to be solved numerically (or approximately)

Picard Method ,y f x y

• Can not integrate exactly because integrand involves y

• Approximate iteratively by using approximations for y

,dy f x y dx ,y f x y dx

0

1 0 0,x

xy y f x y dx

0

2 0 1,x

xy y f x y dx

0

1 0 ,x

i ixy y f x y dx

• Continue to iterate until a desire level of accuracy is obtained in y

• Often gives a power series solution

Picard Method – Example 1 0,1y xy

0

1 0 0

0

212

1

1 1

1

x

x

x

y y xy dx

x dx

x x

• Continue to iterate until a desire level of accuracy is obtained in y

0

21

2 0 1

0

2 3 41 13 8

2

12

1

1 1

1

1

x

x

xx x

y y xy dx

x dx

x x x x

0

2 3 41 1 12 3 8

3 0 2

0

2 3 4 5 61 1 1 1 12 3 8 15 48

1

1 1 1

1

x

x

x

y y xy

x x x x

dx

x dx

x x x x x x

0

4 0 3

0

2 3 4 5 6 7 81 1 1 1 1 1 12 3 8 15

2 3 4 5 61 1 1 1 12 3 8 15 48

48 105 384

1

1 1 1

1

x

x

xx x x x x x

y y xy dx

x dx

x x x x x x x x

Picard 1.mws

Picard Method – Example 2 0,1

dyy

dt

0

1 0 0

0

,

1 1

1

t

t

t

y y f t y dt

dt

t

ty e

0

2 0 1

0

212

,

1 1

1

t

t

t

y y f t y dt

t dt

t t

0

3 0 2

2120

2 31 12 6

,

1 1

1

t

t

t

y y f t y dt

t t dt

t t t

0

4 0 3

2 31 12 60

2 3 41 1 12 6 24

,

1 1

1

t

t

t

y y f t y dt

t t t dt

t t t t

1!

0

nk

n kk

y t

Picard 2.mws

Euler Method 0 0, ,y f x y x y

• Assume linear between 2 consecutive points

• Between initial point and 1st (calculated) point

0 0

0 0

0 0

1 0 0 0

,

,

,

,

dyf x y

dxy

f x yxy f x y x

y y f x y x

1 ,i i i iy y f x y x

• User selects x

• Need to be careful - too big or too small can cause problems

Euler Method – Example

00,dC

kC Cdt

1 1

1 1 1 1,

1i i

i i i i it C

dCC C t C t kC C k t

dt

0kt

trueC C e

Euler 1.mwsAdobe Acrobat

Document euler_debug.txt euler_release.txt

Taylor Method• Based on Taylor expansion

0 0 0

0

2 32 3

0 0 0 02 3

00

1 1

2 3!

1

!

x x x

nn

nn x

df d f d ff x f x x x x x x x

dx dx dx

d fx x

n dx

2 31 11 , , ,2 3!i i i i i ii i x y x y x yy y y x y x y x

,

,, , ,, ,

,

,

i i

i i

i i i i i ii i i i

x y i i

x y i ix y x y x yx y x y

y f x y

f f dy f fy f x y

x y dx x y

Use chain rule

Euler method is Taylor method of order 1

Taylor Method – Example 00,

dCkC C

dt

2

2 3

iv 3 4

C kC

C C dCC k kC k C

t C dt

C C dCC k kC k C

t C dt

C C dCC k kC k C

t C dt

0kt

trueC C e

2 2 3 3 4 41 1 11 2 3! 4!i i i i i iC C kC x k C x k C x k C x

taylor.mws

Improved Euler (Heun’s) Method 0 0, ,y f x y x y

• Euler Method– Use constant derivative between points i & i+1

– calculated at xi

• Better to use average derivative across the interval 12 1

1 , ,i i iiy f x y f x y

1

112

1

1

,

, ,

i i i i

i

ii i i

i

y y f x y x

y f x y f x

y y y

y

x

• yi+1 is not knownPredict – Correct

(can repeat)

Improved Euler Method – Example

00,dC

kC Cdt

1 1

1 1 1 1,

112

1

1

*i i

i i i i it C

i i

i i

dCC C t C t kC C k t

dt

y kC k C

C C ty

0kt

trueC C e

Euler 1.mws Improved Euler.mws

Modified Euler Method 0 0, ,y f x y x y

• Modified Euler Method– Use derivative halfway between points i & i+1

12

1 12 2

1

,2

,

i i ii

i i

i i

xy y f x y

y f x y

y y y x

Modified Euler Method – Example

00,dC

kC Cdt

12

1 1

12

,

1

12 2 2

i i

i i i iit C

i

i i

t dC t tC C C kC C k

dt

y kC

C C ty

0kt

trueC C e

Euler 1.mws Improved Euler.mws Modified Euler.mws

Runge-Kutta Methods 0 0, ,y f x y x y

• Improved and Modified Euler Methods are special cases– 2nd order Runge-Kutta

• 4th order Runge-Kutta– Runge– Kutta– Runge-Kutta-Gill

Runge Methods 0 0, ,y f x y x y

1 1 2 3 42 26i i

xy y s s s s

1

2 12 2

3 22 2

3 3

,

,

,

,

i i

x xi i

x xi i

i i

s f x y

s f x y s

s f x y s

s f x x y xs

Runge.mws

Kutta Methods 0 0, ,y f x y x y

1 1 2 3 43 38i i

xy y s s s s

1

2 13 3

2 13 2 13 3

3 1 2 3

,

,

,

,

i i

x xi i

xi i

i i

s f x y

s f x y s

s f x y x s s

s f x x y x s s s

Kutta.mws

Runge-Kutta-Gill Methods 0 0, ,y f x y x y

1 11 1 2 3 42 2

2 1 2 16i i

xy y s s s s

1

2 12 2

1 1 13 1 22 2 2 2

1 13 2 32 2

,

,

, 1

, 1

i i

x xi i

xi i

i i

s f x y

s f x y s

s f x y x s s

s f x x y x s s

Runge Kutta Gill.mws

Systems of Equations

1 2, , , , 1, 2, ,jj n

dyf x y y y j n

dx

• All the previous methods can be applied to systems of differential equations

• Only illustrate the Runge method

, 1 , 1, 2, 3, 4,2 26j i j i j j j j

xy y s s s s

1, 1, 2, ,

2, 1, 1,1 2, 1,2 , 1,2 2 2 2

3, 1, 2,1 2, 2,2 , 2,2 2 2 2

4, 1, 3,1 2, 3,2 , 3,

, , , ,

, , , ,

, , , ,

, , , ,

j j i i i n i

x x x xj j i i i n i n

x x x xj j i i i n i n

j j i i i n i n

s f x y y y

s f x y s y s y s

s f x y s y s y s

s f x x y xs y xs y xs

Systems of Equations – Example 1

A 1

dAf A k A

dt

1 1A 2A 3A 4A2 26i i

tA A s s s s

1,A A 1

2,A A 1,A 1 1,A2 2

3,A A 2,A 1 2,A2 2

4,A A 3,A 1 3,A

i i

t ti i

t ti i

i i

s f A k A

s f A s k A s

s f A s k A s

s f A ts k A ts

1 2A B Pk k

B 1 2,dB

f A B k A k Bdt

P 2

dPf B k B

dt

1 1B 2B 3B 4B2 26i i

tB B s s s s

1 1P 2P 3P 4P2 2

6i i

tP P s s s s

1,B B 1 2

2,B B 1,A 1,B 1 1,A 2 1,B2 2 2 2

3,B B 2,A 2,B 1 2,A 2 2,B2 2 2 2

4,B B 3,A 3,B 1 3,A 2 3,B

,

,

,

,

i i i i

t t t ti i i i

t t t ti i i i

i i i i

s f A B k A k B

s f A s B s k A s k B s

s f A s B s k A s k B s

s f A ts B ts k A ts k B ts

1,P P 2

2,P P 1,B 2 1,B2 2

3,P P 2,B 2 2,B2 2

4,P P 3,B 2 3,B

i i

t ti i

t ti i

i i

s f B k B

s f B s k B s

s f B s k B s

s f B ts k B ts

Consecutive reactions Runge.mws

Systems of Equations – Example 2

A 1

dAf A k A

dt

31 2 4A B C E Fkk k k

B 1 2,dB

f A B k A k Bdt

F 4

dFf E k E

dt

C 2 3,dC

f B C k B k Cdt

E 3 4,dE

f C E k C k Edt

Consecutive reactions 2 Runge.mws

Systems of Equations – Example 3

A 1 2, ,dA

f A B C k AB k Cdt

1

3

2

A B C Ek

k

k

B 1 2, ,dB

f A B C k AB k Cdt

C 1 2 3, ,dC

f A B C k AB k C k Cdt

E 3

dEf C k C

dt Consecutive reactions 3 Runge.mws

Systems of Equations – Example 4

0dX

dt

X+A 2A

A+B 2B

B P

A 1 2,dA

f A B k A k ABdt

B 2 3,dB

f A B k AB k Bdt

P 3

dPf B k B

dt Lotka Volterra Runge.mws

Systems of Equations – Example 5

A 1 2

dAf B k k B

dt

B 3 4 5,dB

f A B k AB k C k Bdt

C 6 7,dC

f B C k BC k Cdt

oscillating reaction 2 Runge.mws