Some Techniques in
Deterministic Modeling for
Mathematical Biology
By: Ryan Borek, Dallas Hamann,
Heather Parsons, Carrie Ruda,
Carissa Staples, Erik Wolf
Content
I.
II.
III.
IV.
V.
Introduction
Pendulum Model
Nondimensionalization
Regular Perturbation Calculations
Solution and Conclusions
Introduction
This project was initiated from a study on
mathematical biology, which used the methods of
nondimensionalization and regular perturbation
theory. Last semester we studied biochemical
modeling, which uses the aforementioned
techniques in order to solve difficult bio-chemical
approximations.
Our presentation, however, uses a physics model to
describe how these techniques can be applied.
p
Pendulum Motion Model
θ
L
s
F1
F2
Earth’s Surface
Objective:
To form a differential
equation to model the
motion of a simple
pendulum.
p
Pendulum Motion Model
θ
What we know
L
L = Length of Weightless Rod
p = Fixed Point of Suspension
s
A “bob” of mass m
F1
F1 = tension in the Rod
F2 = mg where g is gravity
F2
s = distance from the “bob” to
the bottom of the arc
= ( t ) = Angle of Rod relative to
plane perpendicular to earth
(0) = a
Earth’s Surface
′(0) = 0
p
Pendulum Motion Model
θ
Find the Tangential Acceleration
L
s
Any Force on an arc can be broken down into
the sum of a tangential and normal force.
F1
m
F2T
F2N
F2=mg
θ
Earth’s Surface
F
2T
sin( ) =
F2
F
2T
sin( ) =
mg
F2T = mg sin( )
Pendulum Motion Model
p
Find Acceleration of change in the ArcLength
θ
Use the Arc Angle Formula
L
s
F1
m
F2T
F2N
F2=mg
θ
S=L
dS
d
=L
dt
dt
d 2S
d2
=L 2
2
dt
dt
So in this case, the acceleration of change
in traversed arc length is
2
Earth’s Surface
d
= −L 2
dt
Pendulum Motion Model
If you magnify an arc many times it begins to
look flat. For this reason we can set the tangential
force equal to the force derived from using the
acceleration of decrease in arc length.
Therefore:
2
d
F2T = mg sin( ) = − mL 2
dt
Pendulum Motion Model
A differential equation can now be formed.
d2
mg sin( ) = − mL 2
dt
2
d
g sin( ) = − L 2
dt
L
2
d
+ g sin( ) = 0
2
dt
d2
g
+ sin( ) = 0
2
dt
L
Let
2
=
g
and * denote a
L
variable with dimension.
d2 *
+
2
dt *
2
sin( *) = 0
(a non-linear differential equation)
Nondimensionalization
Nondimensionalization is a technique of dimensional
analysis that creates dimensionless variables, which
allows differential equations to be stated in a
dimensionless form. This is useful to scientists and
engineers because it simplifies complex equations
and puts the problem in a form amenable to the
techniques of perturbation theory.
The differential equation of our Pendulum Model is:
d 2θ *
2
+
ω
sin(θ *) = 0
2
dt *
First list variables and parameters with their dimensions
Variable
Dimension
*(dependent)
L (length)
t*(independent)
T (time)
ω
T-1
a
L
Conditions:
*(0) = a
and
dθ *
(0) = 0
dt *
Next for each variable v*, let p* (known as the intrinsic
reference quantity) be some combination of the parameters
that have the same dimension as v*. The the new
dimensionless variable, v is defined as
v*
v :=
p*
Our new dimensionless variables are now:
θ*
θ :=
a
L
L
With new conditions:
t :=t * ω TT −1
(
)
(0) = 1
and
dθ
(0) = 0
dt
Now rewrite the dimensional variable
* in terms of the parameter and the
dimensionless variable, leaving
the dimensionless variable t.
θ * =aθ
t =ωt *
Next we will change our dimensional differential equation to
a dimensionless equation by substitutions.
Next find the
derivative of *.
dθ
dθ *
=a
dt *
dt *
By the chain rule:
dθ dt
dθ
=a
a
dt dt *
dt *
Now with substitution:
dθ
dθ dt
= aω
a
dt
dt dt *
dθ
dθ *
= aω
dt
dt *
Next find the 2nd
derivative of *.
Rewrite as:
By the chain rule:
d 2θ *
d
dθ
d dθ
aω
=
= aω
2
dt *
dt *
dt
dt * dt
aω
(
d dθ
aω
)
dt
dt *
(
d dθ
)
d (dθ )
dt
dt = aω
dt *
dt
dt
dt *
Now with substitution:
aω
(
d dθ
dt
)
dt
2
dt
θ
d
2
= aω
dt *
dt 2
2
d
d 2θ *
θ
2
= aω
2
dt *
dt 2
Finally we can rewrite our original model:
d 2θ *
2
+
ω
sin(θ *) = 0
2
dt *
Substituting in:
2
d
θ
d 2θ *
2
= aω
2
2
dt
dt *
We get the new dimensionless model:
d 2θ
−1
+
a
sin( aθ ) = 0
2
dt
With initial conditions:
(0) = 1
and
dθ
(0) = 0
dt
The Series Method of Regular
Perturbation Calculations
• An effective method of approximating
solutions for scaled differential equations.
• 5 step process
Assumptions
The dependent variable can be expressed in a series
of the small parameter.
∞
θ = θ (t , a) =
i =0
Thus,
dθ
=
dt
d 2θ
=
2
dt
∞
i =0
∞
i =0
θ i (t )a i
θ ' (t )a i
i
θ '' (t )a i
i
Step A:
Substitute the power series into the differential equation.
Recall : The pendulum model differential equation.
d 2θ
−1
+
a
sin(aθ ) = 0
2
dt
Substituting:
∞
i =1
θ i'' (t )a i + a −1 sin[a
∞
i =1
θ i (t )a i ] = 0
Step B:
Expand all quantities so that each term is written as a
power series in a.
x3 x5 x7
+ − ............
Recall: sin( x ) = x −
3! 5! 7!
∞
i =1
θ (t )a + a {[a
''
i
i
−1
∞
i =0
θ i (t )a i ] −
[a
∞
i =0
θ i (t )a ]
i 3
3!
[a
+
∞
i =0
θ i (t )a i ]5
5!
.......} = 0
Step B (Continued):
(Note : We are using only powers of a no greater than 2)
The equation from the previous page simplifies into
the following:
1 3 2
θ (t) +θ (t)a +θ (t)a +[θo (t) +θ1(t)a +θ2 (t)a ] − θo a + O(a3 ) = 0
6
''
o
''
1
''
2
2
2
Step C:
Collect all terms in the equation and equate to zero the
successive coefficients in the series.
1 3
[θ (t ) + θ o (t )] + a[θ (t ) + θ1 (t )] + a [θ (t ) + θ 2 (t ) − θ o ] + O(a 3 ) = 0
6
''
o
''
1
3 Differential Equations :
2
''
2
θ +θo = 0
''
o
θ1'' + θ1 = 0
1 3
θ +θ2 − θo = 0
6
''
2
Step C (Continued):
Remark : The original non-linear differential equation
has been replaced by a sequence of linear differential
equations. This is a characteristic feature of
perturbation methods.
Step D:
Substitute the power series into the original initial (or
boundary) conditions, expand and equate the coefficients
to zero and obtain a set of new initial (or boundary)
conditions to supplement the sequence of differential
equations obtained in Step C.
θ (t , a) =
∞
i =0
dθ
=
dt
∞
i =0
θ i (t )a
θ (t )a
'
i
i
i
Step D (Continued):
We will now plug in our initial conditions and as a
result we get (a) and (b)
Initial conditions: θ (0) = 1 and θ ' (0) = 0
(a)
θ (0, a) =
∞
i =0
(b)
θ (0, a) =
'
∞
i =0
θ i (0)a = 1
i
θ (0)a = 0
'
i
i
Step D (Continued):
We will now expand and equate the coefficients to zero
and obtain a set of new initial or boundary conditions.
(a)
θ 0 (0) − 1 +
∞
i =1
θ i ( 0) a = 0
i
so, θ 0 (0) − 1 = 0, (θ 0 (0) = 1)
∴ θ i (0) = 0 ∀ i ≥ 1
Step D (Continued):
(b)
θ (0, a ) =
'
∞
i =0
θ (0)a = 0
'
i
so, θ (0) = 0 ∀ i
'
i
i
Step D (Continued):
We will now supplement the sequence of differential
equations obtained in Step C with our new boundary
conditions found in (a) and (b).
θ o'' + θ o = 0,
θ 0 (0) = 1, θ 0' (0) = 0
θ1'' + θ1 = 0,
θ1 (0) = 0, θ1' (0) = 0
1 3
θ + θ 2 − θ o = 0,
6
θ 2 (0) = 0, θ 2' (0) = 0, θ 0 (0) = 1
''
2
Step E:
Solve the linear differential equations in sequence.
1)
θ +θo = 0
2)
θ + θ1 = 0
3)
1 3
θ +θ2 − θo = 0
6
''
o
''
1
''
2
Step E (Continued):
Solving 1)
θ 0'' + θ 0 = 0 where θ 0 (0) = 1, θ 0' (0) = 0
θ 0 (t ) = A cos(t ) + B sin(t )
A = 1, B = 0
∴θ 0 (t ) = cos(t )
Step E (Continued):
Solving 2)
θ + θ1 = 0 where θ1 (0) = 0, θ (0) = 1
''
1
∴ θ1 (t ) = 0
'
1
Step E (Continued):
Solving 3)
1 3
θ + θ 2 − θ 0 = 0 where θ 2 = 0, θ 2' (0) = 0
6
''
2
1
= θ + θ 2 − cos3 (t ) = 0
6
''
2
1
= θ + θ 2 = cos 3 (t )
6
''
2
Step E (Continued):
This non-homogeneous differential equation
can be solved using the method of
“undetermined coefficients,” which gives
the following result:
1
1
1
θ 2 (t ) =
cos(t ) −
cos(3t ) + t sin(t )
192
192
16
Finally, an approximation:
By expanding the power series of θ = θ (t , a ) =
∞
i =0
θ i (t )a i
to powers of a no greater than 2, we achieve the following:
1
1
1
θ = θ (t , a) ≈ cos(t ) + a [
cos(t ) −
cos(3t ) + t sin(t )]
192
192
16
2
Conclusion
Nondimensionalization and Perturbation
methods are effective ways of approximating
solutions to differential equations. In the study of
enzyme kinetics (physical chemistry), chemical
reactions are modeled by non-linear differential
equations. Their solutions are approximated by the
Quasi-Steady-State Approximation and
Equilibrium Approximation, which utilize the
methods of nondimensionalization and
perturbation that were demonstrated earlier.
Bibliography
• Mathematics Applied to Deterministic
Problems in the Natural Sciences by Lin and
Segal, SIAM 1988
Special Thanks to:
Dr. Steve Deckelman
All for coming!