lecture #14 matlab

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Math 211Math 211

Lecture #14

MATLAB’s ODE Solvers

September 27, 2002

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SolversSolvers

MATLAB has several solvers.

Return

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SolversSolvers

MATLAB has several solvers.

• ode45

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SolversSolvers

MATLAB has several solvers.

• ode45

� This is the first choice.

Return

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SolversSolvers

MATLAB has several solvers.

• ode45

� This is the first choice.

• ode23

Return

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SolversSolvers

MATLAB has several solvers.

• ode45

� This is the first choice.

• ode23

• Stiff solvers

Return

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SolversSolvers

MATLAB has several solvers.

• ode45

� This is the first choice.

• ode23

• Stiff solvers

� ode15s

Return Solvers

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ode45ode45

Return Solvers

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ode45ode45

• Uses variable step size.

Return Solvers

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ode45ode45

• Uses variable step size.

� Specify error tolerance instead of step size.

Return Solvers

3

ode45ode45

• Uses variable step size.

� Specify error tolerance instead of step size.

� MATLAB chooses the step size at each step to

achieve the limit on the error.

Return Solvers

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ode45ode45

• Uses variable step size.

� Specify error tolerance instead of step size.

� MATLAB chooses the step size at each step to

achieve the limit on the error.

� The default tolerance is good enough for this course.

Return Solvers

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ode45ode45

• Uses variable step size.

� Specify error tolerance instead of step size.

� MATLAB chooses the step size at each step to

achieve the limit on the error.

� The default tolerance is good enough for this course.

• Syntax

Return Solvers

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ode45ode45

• Uses variable step size.

� Specify error tolerance instead of step size.

� MATLAB chooses the step size at each step to

achieve the limit on the error.

� The default tolerance is good enough for this course.

• Syntax

[t,y] = ode45(derfile,[t0, tf ], y0);

Return Solvers

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ode45ode45

• Uses variable step size.

� Specify error tolerance instead of step size.

� MATLAB chooses the step size at each step to

achieve the limit on the error.

� The default tolerance is good enough for this course.

• Syntax

[t,y] = ode45(derfile,[t0, tf ], y0);

plot(t,y)

Return

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Solving SystemsSolving Systems

Return

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Solving SystemsSolving Systems

• Example:

x′ = v

v′ = −9.8 − 0.04v|v|

Return

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Solving SystemsSolving Systems

• Example:

x′ = v

v′ = −9.8 − 0.04v|v|

• Change to vector notation. (Use MATLAB vector

notation)

Return

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Solving SystemsSolving Systems

• Example:

x′ = v

v′ = −9.8 − 0.04v|v|

• Change to vector notation. (Use MATLAB vector

notation)

� u(1) = x

Return

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Solving SystemsSolving Systems

• Example:

x′ = v

v′ = −9.8 − 0.04v|v|

• Change to vector notation. (Use MATLAB vector

notation)

� u(1) = x

� u(2) = v

Return System

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Derivative m-file ball.mDerivative m-file ball.m

function upr = ball(t,u)

x = u(1);

v = u(2);

xpr = v;

vpr = -9.8 - 0.04*v*abs(v);

upr = [xpr; vpr];

Return System ball.m

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Derivative m-file ballshort.mDerivative m-file ballshort.m

function upr = ballshort(t,u)

upr = zeros(2,1);

upr(1) = u(2);

upr(2) = -9.8 - 0.04*u(2)*abs(u(2));

Return

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Computing and Plotting Solutions to

Systems

Computing and Plotting Solutions to

Systems

Return

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Computing and Plotting Solutions to

Systems

Computing and Plotting Solutions to

Systems• [t,u] = ode45(’ball’,[0,3],[0;50]);

Return

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Computing and Plotting Solutions to

Systems

Computing and Plotting Solutions to

Systems• [t,u] = ode45(’ball’,[0,3],[0;50]);

• plot(t,u) – plots all of the components versus t.

Return

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Computing and Plotting Solutions to

Systems

Computing and Plotting Solutions to

Systems• [t,u] = ode45(’ball’,[0,3],[0;50]);

• plot(t,u) – plots all of the components versus t.

• plot(t,u(:,1)) – first component versus t.

Return

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Computing and Plotting Solutions to

Systems

Computing and Plotting Solutions to

Systems• [t,u] = ode45(’ball’,[0,3],[0;50]);

• plot(t,u) – plots all of the components versus t.

• plot(t,u(:,1)) – first component versus t.

• plot(u(:,1),u(:,2)) – second component versus

the first.

Return

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Computing and Plotting Solutions to

Systems

Computing and Plotting Solutions to

Systems• [t,u] = ode45(’ball’,[0,3],[0;50]);

• plot(t,u) – plots all of the components versus t.

• plot(t,u(:,1)) – first component versus t.

• plot(u(:,1),u(:,2)) – second component versus

the first. This is a phase plane plot.

Return

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Computing and Plotting Solutions to

Systems

Computing and Plotting Solutions to

Systems• [t,u] = ode45(’ball’,[0,3],[0;50]);

• plot(t,u) – plots all of the components versus t.

• plot(t,u(:,1)) – first component versus t.

• plot(u(:,1),u(:,2)) – second component versus

the first. This is a phase plane plot.

• plot3(u(:,1),u(:,2),t) – 3-D plot.

Return

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Solving Higher Order EquationsSolving Higher Order Equations

Return

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Solving Higher Order EquationsSolving Higher Order Equations

• Reduce to a first order system and solve the system.

Return

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Solving Higher Order EquationsSolving Higher Order Equations

• Reduce to a first order system and solve the system.

• Example: The motion of a pendulum is modeled by

θ′′ = − g

Lsin θ − Dθ′.

Return

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Solving Higher Order EquationsSolving Higher Order Equations

• Reduce to a first order system and solve the system.

• Example: The motion of a pendulum is modeled by

θ′′ = − g

Lsin θ − Dθ′.

• Introduce ω = θ′.

Return

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Solving Higher Order EquationsSolving Higher Order Equations

• Reduce to a first order system and solve the system.

• Example: The motion of a pendulum is modeled by

θ′′ = − g

Lsin θ − Dθ′.

• Introduce ω = θ′. Notice

ω′ = − g

Lsin θ − Dθ′.

Return Higher order

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Equivalent First Order SystemEquivalent First Order System

Return Higher order

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Equivalent First Order SystemEquivalent First Order System

θ′ = ω

ω′ = − g

Lsin θ − Dω

Return Higher order

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Equivalent First Order SystemEquivalent First Order System

θ′ = ω

ω′ = − g

Lsin θ − Dω

• Change to vector notation. (Use MATLAB vector

notation)

Return Higher order

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Equivalent First Order SystemEquivalent First Order System

θ′ = ω

ω′ = − g

Lsin θ − Dω

• Change to vector notation. (Use MATLAB vector

notation)

� u(1) = θ

Return Higher order

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Equivalent First Order SystemEquivalent First Order System

θ′ = ω

ω′ = − g

Lsin θ − Dω

• Change to vector notation. (Use MATLAB vector

notation)

� u(1) = θ

� u(2) = ω

Pendulum Short

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Derivative m-file pend.mDerivative m-file pend.m

function upr = pend(t,u)

L= 1;

global D

th = u(1);

om = u(2);

thpr = om;

ompr = -(9.8/L)*sin(th) - D*om;

upr = [thpr; ompr];