Ordinary Differential Equations (ODE) in MATLAB
Ordinary Differential Equations (ODE) in MATLAB
What will we learn from the next 5 lectures
Ordinary Differential Equations (ODE) in
MATLAB
Shan He
School for Computational Science
University of Birmingham
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How to solve ODEs using MATLAB.
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How to model biological systems using ODEs in MATLAB.
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How to analyse ODEs using MATLAB.
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Understand bifurcation and chaos using MATLAB.
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Applications of bifurcation and chaos to biological problems.
Module 06-23836: Computational Modelling with MATLAB
Ordinary Differential Equations (ODE) in MATLAB
Outline
Ordinary Differential Equations (ODE) in MATLAB
Concepts about ODE
Outline of Topics
What is ODE
Concepts about ODE
Solving ODE in MATLAB
ODE Solvers in MATLAB
Solving linear ODEs in MATLAB
Solving high order ODEs in MATLAB
Solving ODEs in MATLAB: Advanced topics
Ordinary Differential Equations (ODE) in MATLAB
Concepts about ODE
An Ordinary Differential Equation (ODE) is an equation involving
a function and its derivatives.
Ordinary Differential Equations (ODE) in MATLAB
Concepts about ODE
Definition of ODE
Linear ODE and Homogeneous Linear ODE
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Let y be an unknown function of x.
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Let y ′ =
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Let y (n) =
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Let F be a given function
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Then an ODE of order n is an equation of the form:
dy
dx
A ODE is said to be linear if F can be written as a linear
combination of the derivatives of y together with a constant
term, all possibly depending on x:
be the first derivative with respect to x.
dny
dx n
an (x)y n + an−1 (x)y n−1 + · · · + a1 (x)y ′ + a0 y = r (x)
be the nth derivative with respect to x.
or more concisely,
yn =
F (x, y , y ′ , . . . , y (n) ) = 0
n−1
X
ai (x)y (i) + r (x)
i=0
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where ai (x) and r (x) are continuous functions in x and the
function r (x) is called the source term.
A linear ODE is said to be homogeneous if r (x) = 0,
otherwise it is called non-homogeneous or inhomogeneous.
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
ODE Solvers in MATLAB
Solution to ODE
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
ODE Solvers in MATLAB
ODE Solvers in MATLAB
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If an ODE is linear, it can be solved by analytical methods.
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In general, an nth-order ODE has n linearly independent
solutions.
Any linear combination of linearly independent functions
solutions is also a solution.
If an ODE is not linear, most of the case it cannot be solved
exactly: we will use MATLAB to obtain approximate solutions
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
ODE Solvers in MATLAB
How to use MATLAB ODE Solvers
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The MATLAB ODE solvers can be called as a function:
[T,Y] = ode**(@odefun,tspan,y0,options)
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@odefun: Function handle of the ODE function
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tspan: Interval of integration, [t0 ,tfinal ].
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y0: Initial conditions.
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options: Optional parameters.
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The ODE function odefun define the ODEs:
function [dy] = odefun(t, y, parameters)
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving linear ODEs in MATLAB
Solving linear ODE in MATLAB
Steps for solving this equation numerically in MATLAB:
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1
2
3
4
5
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ode45: based on an explicit Runge-Kutta (4, 5) formula and
the Dormand-Prince method.
ode23: based on an explicit Runge-Kutta (2, 3) formula and
the Bogacki and Shampine method.
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We choose according to order of accuracy and the type of
systems (stiff or nonstiff).
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Rule of thumb: Always try ode45 first.
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving linear ODEs in MATLAB
Solving linear ODE in MATLAB
One simple example:
(
u′ =
v′ =
du(x)
dx
dv (x)
dx
= u(x) + v (x)
= u(x)
Analytical solution can be obtained, but how to solve it in
MATLAB numerically?
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving linear ODEs in MATLAB
Solving linear ODE in MATLAB
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Step 1: Create a MATLAB function that defines the ODEs
function dy = simple_ode(t,y)
% function to be integrated
dy = zeros(2,1);
dy(1) = y(1) + y(2);
dy(2) = y(1);
Matlab has several different ODE solvers for the numerical
solution of ODEs:
1
2
3
Step 2: Call a numerical solver provided in MATLAB, e.g.,
[T,Y] = ode45(odefun,tspan,y0)
tspan = [0 10];
y0 = [1 0];
[X,Y] = ode45(@simple_ode,tspan,y0);
Ordinary Differential Equations (ODE) in MATLAB
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving ODE in MATLAB
Solving high order ODEs in MATLAB
Solving high order ODEs in MATLAB
Reduction of ODE order
Reduction of ODE order
Methods:
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Recall an ODE of the general form:
′
F (x, y , y , . . . , y
(n)
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We will use a second order ODE as an example:
(
x ′ = −ye (−t/5) + y ′ e (−t/5) + 1
y ′′ = −2 sin(t)
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Step 1: Introduce a new variable that equals the first
derivative of the free variable in the second order equation:
z = y′
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Step 2: Taking the derivative of each side yields the following:
z ′ = y ′′
)=0
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We said this system is an ODE of order n.
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Any differential equation of order n can be reduced to a
system of n first-order (n = 1) differential equations.
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We do so because high order ODE (n > 1) is difficult to solve.
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MATLAB ODE solvers cannot handle higher order ODE!
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving high order ODEs in MATLAB
Reduction of ODE order
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving high order ODEs in MATLAB
Solve high order ODE in MATLAB
Methods:
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Methods:
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Step 3: Substituting the second order ODE with z and z ′ :

′
(−t/5) + ze (−t/5) + 1

x = −ye
′
z = −2 sin(t)

 ′
y =z
1
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3
4
5
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7
8
9
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving high order ODEs in MATLAB
Solve high order ODE in MATLAB
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving ODEs in MATLAB: Advanced topics
Stiffness of ODE equations
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Methods:
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Step 5: evaluate the system of equations using ODE45:
t0 = 5; % Start time
tf = 20; % Stop time
x0 = [1 -1 3] % Initial conditions
[t,s] = ode45(@ho_ode,[t0,tf],x0);
x = s(:,1);
y = s(:,2);
z = s(:,3);
plot(t,s);
Step 4: Write a MATLAB function ho ode.m to define the
ODE:
function dy = high_order_ode_example(t,x)
% x(1) = x
% x(2) = y
% x(3) = z
dy = [-x(2) * exp(-t/5) + ...
x(3) * exp(-t/5) + 1;
x(3);
-2*sin(t)]
end
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Stiffness is a subtle, difficult, and important concept in the
numerical solution of ordinary differential equations.
“A problem is stiff if the solution being sought varies slowly,
but there are nearby solutions that vary rapidly, so the
numerical method must take small steps to obtain satisfactory
results.”
Example:
y′ = y2 − y3
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y (0) = τ
2
0≤t≤
τ
If we weren’t concerned with efficiency of ODE solver, we
wouldn’t be concerned about stiffness.
Ordinary Differential Equations (ODE) in MATLAB
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving ODE in MATLAB
Solving ODEs in MATLAB: Advanced topics
Solving ODEs in MATLAB: Advanced topics
Passing parameters to ODEs
Passing parameters to ODEs
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By passing the value to ODE function, we can change the
coefficients in ODEs each time we call the ODE solver.
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We need to define the parameters in the ODE function header:
function dy = ode_pv(t,y,A)
% function to be integrated
dy = zeros(2,1);
dy = A*y
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1
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Ordinary Differential Equations (ODE) in MATLAB
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving ODE in MATLAB
Solving ODEs in MATLAB: Advanced topics
Solving ODEs in MATLAB: Advanced topics
Events
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Define events
Recall syntax of the ODE solvers:
[T,Y] = ode**(@odefun,tspan,y0,options)
We generally assume tspan is known, e.g.,
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Define the ODE function:
function dydt = ode_ball(t,y)
dydt = [y(2); -9.8];
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Define event function for this problem:
function [value,isterminal,direction] ...
= events(t,y)
% Locate the time when height passes through
% zero and stop integration.
value = y(1);
% Detect height = 0
isterminal = 1;
% Stop the integration
direction = -1;
% Desired directionality of
% the zero crossings:
1
2
t0 ≤ t ≤ tfinal
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But sometimes it is also important to determine tfinal .
Example: a ball is falling because of gravity, when does it hit
the ground?
Equations:
y ′′ = g = −9.8
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where y (t) is the height of the object at time t
The question: for what t does y (t) = 0? We can use events.
Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving ODEs in MATLAB: Advanced topics
Run ODE Solver with events
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Ordinary Differential Equations (ODE) in MATLAB
Solving ODE in MATLAB
Solving ODEs in MATLAB: Advanced topics
Other ODE solvers
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Define the event function as an option for ODE solver:
opts = odeset(’events’,@ball_events);
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What we have introduced are all for Initial Value Problems for
ODEs.
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Solve it using MATLAB ode45:
y0 = [20; 0];
[t,y,tfinal] = ode45(@ode_ball,[0 Inf]...
,y0,opts);
tfinal
plot(t,y(:,1),’-’,[0 tfinal],[1 0],’o’)
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To solve Boundary Value Problems: bvp4c.
Tutorial.
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To solve Delay ODEs: dde23.
Tutorial.
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To solve Stochastic ODEs: MATLAB SDE Toolbox.
Tutorial.
1
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Call the ODE solver and pass the parameter to the ODE
function:
tspan = [0 10];
initvalue = [1 0];
option = [];
for ii=-10:4
A = [1 ii; 2 -1];
[t,y] = ode45(@ode_pv,...
tspan,initvalue,option,A);
hold on;
plot(t, y);
end