ordinary differential equations (odes)

1Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Ordinary Differential Equations (ODEs)

d ( )( , )

d

y tf t y

t

Daniel Baur

ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften

ETH Hönggerberg / HCI F128 – Zürich

E-Mail: [email protected]

http://www.morbidelli-group.ethz.ch/education/index


Problem Definition

We are going to look at initial value problems of explicit ODE systems, which means that they can be cast in the following form

0 0

d ( )( , )

d( )

y tf t y

ty t y

Higher Order ODEs Higher order explicit ODEs can be cast into the first order

form by using the following «trick»

Therefore, a system of ODEs of order n can be reduced to first order by adding n-1 variables and equations

Example:


21

12

d( , )

d

yf t y

t

12

21 1

2

d:

d

d dd

d d d

yy

t

y y

t t t

12

22 1

12

d

d

d d( , )

d d

yy

t

y yf t y

t t

21

12

d

d

ycy

t

12

21 1

2

d:

d

d dd

d d d

yy

t

y y

t t t

12

21

d

dd

d

yy

ty

cyt


ODEs as vector field

Since we know the derivatives as function of time and solution, we can calculate them at every point in time and space

2d

d

yy y

t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

Independent variable

Dep

ende

nt v

aria

ble

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

0.75

0.8

0.85

0.9

0.95

1

Independent variable

Dep

ende

nt v

aria

ble


ODE systems as vector fields

We can do the same for ODEs systems with two variables, at a certain time point

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Variable 1

Var

iabl

e 2

0.65 0.7 0.75 0.8 0.85 0.9 0.950.6

0.65

0.7

0.75

0.8

Variable 1

Var

iabl

e 2

d1 0.45 0.45

dd

0.45 0.33d

xxy x

ty

xy yt


Example: A 1-D First Order ODE

Consider the following initial value problem

This ODE and its solution follow a general form(Johann Bernoulli, Basel, 17th – 18th century)

( )2

(0) 1

dy ty t

dty

( )d ( )d

d ( )( ) ( ) ( )

d

( ) e ( ) e dp s s p s s

y tr t p t y t

t

y t k r t t

21( ) 3e 2 1

4ty t t

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

2

3

4

5

6

7

t

y


Some Nomenclature

On the next slide, the following notation will be used

Value of at point

Value of the derivative of at point

Integration step

n n

n n

y y t

y y t

h


Numerical Solution of a 1-D First Order ODE

Since we know the first derivative, Taylor expansion suggests itself as a first approach

This is known as the Euler method, named after Leonhard Euler (Basel, 18th century)

( )2

(0) 1

dy ty t

dty

21( ) 3e 2 1

4ty t t

2

1 1 1

2 2

d( )

d

( , )

n

n n n n n nt

n n n n n

yy y t t O t t

t

y hy O h y h f t y O h

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

2

3

4

5

6

7

t

x

Euler Method

Exact Solution


How does Matlab do it? Non-Stiff Solvers

ode45: Most general solver, best to try first; Uses an explicit Runge-Kutta pair (Dormand-Prince, p = 4 and 5)

ode23: More efficient than ode45 at crude tolerances, better with moderate stiffness; Uses an explicit Runge-Kutta pair (Bogacki-Shampine, p = 2 and 3).

ode113: Better at stringent tolerances and when the ODE function file is very expensive to evaluate; Uses a variable order Adams-Bashforth-Moulton method


How does Matlab do it? Stiff Solvers

ode15s: Use this if ode45 fails or is very inefficient and you suspect that the problem is stiff; Uses multi-step numerical differentiation formulas (approximates the derivative in the next point by extrapolation from the previous points)

ode23s may be more efficient than ode15s at crude tolerances and for some problems (especially if there are largely different time-scales / dynamics involved); Uses a modified Rosenbrock formula of order 2 (one-step method)


Matlab Syntax Hints

All ODE-solvers use the syntax[T, Y] = ode45(ode_fun, tspan, y0, …); ode_fun is a function handle taking two inputs, a scalar t (current

time point) and a vector y (current function values), and returning as output a vector of the derivatives dy/dt in these points

tspan is either a two component vector [tstart, tend] or a vector of time points where the solution is needed [tstart, t1, t2, ...]

y0 are the values of y at tstart

Use parametrizing functions to pass more arguments to ode_fun if neededf_new = @(t,y)ode_fun(t, y, k, p);

This can be done directly in the solver call [T, Y] = ode45(@(t,y)ode_fun(t, y, k, p), tspan, y0);


Exercise 1: Damped harmonic oscillator

A mass on a string is subject to a force when out of equilibrium

According to Newton’s second law (including friction), the resulting movement follows the solution to an ODE:

m

y

F

2

2

d d

d d

y ym ky at t

Spring force

Friction


Assignment 1

1. Rewrite the motion equation for the damped harmonic oscillator as an explicit, first order ODE system

2. Use ode45 to solve the problem Set up a function file to calculate the derivatives, its header should

read something likefunction dy = harm_osc(t, y, m, k, a)

Set m = 1; k = 10; a = 0.5; y0 = 1; dy/dt0 = 0; Use tSpan = [0, 20];

3. Plot the position of the mass y against time

4. Plot the speed of the mass dy/dt against time


Exercise 2: Radioactive decay

A decaying radioactive element changes its concentration according to the ODE

where k is the decay constant The analytical solution to this problem reads

d

d

yky

t

0 expy t y kt


Assignment 21. Plot the behavior of the radioactive decay problem as a

vector field in the y vs t plane Find online the function vector_field.m. It plots the derivatives of

first order ODEs in different space and time points Plot the vector field for y values between 0 and 1, time between 0

and 2 and k = 1. Can you see what a solver has to do? Is it possible to switch from one trajectory in the vector field to

another? What follows for the uniqueness of the solutions?

2. Implement the forward Euler method in a function file The file header should read something likefunction [t,y] = eulerForward(f, t0, tend, y0, h)

Use y0 = 1, k = 1 and h = 0.1 to solve the radioactive decay problem from t0 = 0 to tEnd = 10 and plot the solution together with the analytical solution.

What happens if you increase the step size h? What happens if you increase the step size above 2?


Exercise 3: Lotka-Volterra equations

Predator-prey dynamics can be modeled with a simple ODE system:

where x is the number of prey, y is the number of predators and a, b, c, d are parameters

d

dd

d

xx a by

ty

y c dxt


Assignment 3

1. Plot the behavior of the predator-prey system as a vector field in the x vs y plane Find online the function vector_field2D.m which plots this vector

field Plot x and y between 0 and 1, and use a = c = 0.5 and b = d = 1 Without solving the ODEs, what can you say about the solution

behavior just by looking at the trajectories?

2. Solve the system using ode45, and x0 = y0 = 0.3 as initial values, and tSpan = [0, 50]; Plot x and y against t. Can you observe what you predicted in 1?

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