Math 211 Math 211 Lecture #14 M ATLAB s ODE Solvers September 26, - - PowerPoint PPT Presentation

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Math 211 Math 211

Lecture #14 MATLAB’s ODE Solvers September 26, 2003

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Matlab Solvers Matlab Solvers

MATLAB has several solvers.

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Matlab Solvers Matlab Solvers

MATLAB has several solvers.

• ode45

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Matlab Solvers Matlab Solvers

MATLAB has several solvers.

• ode45

This is the first choice.

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Matlab Solvers Matlab Solvers

MATLAB has several solvers.

• ode45

This is the first choice.

• ode23

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Matlab Solvers Matlab Solvers

MATLAB has several solvers.

• ode45

This is the first choice.

• ode23

This is a good second choice.

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Matlab Solvers Matlab Solvers

MATLAB has several solvers.

• ode45

This is the first choice.

• ode23

This is a good second choice.

• Stiff solvers for equations/systems with widely different

time scales.

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Matlab Solvers Matlab Solvers

MATLAB has several solvers.

• ode45

This is the first choice.

• ode23

This is a good second choice.

• Stiff solvers for equations/systems with widely different

time scales.

• de15s

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Matlab Solvers Matlab Solvers

MATLAB has several solvers.

• ode45

This is the first choice.

• ode23

This is a good second choice.

• Stiff solvers for equations/systems with widely different

time scales.

• de15s
• All use the same syntax.

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• de45
• de45

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• de45
• de45
• Uses variable step size.

Return Solvers

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• de45
• de45
• Uses variable step size.

Specify error tolerance instead of step size.

Return Solvers

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• de45
• de45
• 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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• de45
• de45
• 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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• de45
• de45
• 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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• de45
• de45
• 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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• de45
• de45
• 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)

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Solving Systems Solving Systems

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Solving Systems Solving Systems

• Example:

x′ = v v′ = −9.8 − 0.04v|v|

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Solving Systems Solving Systems

• Example:

x′ = v v′ = −9.8 − 0.04v|v|

• Change to vector notation. (Use MATLAB vector notation)

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Solving Systems Solving Systems

• Example:

x′ = v v′ = −9.8 − 0.04v|v|

• Change to vector notation. (Use MATLAB vector notation)

u(1)

= x

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Solving Systems Solving 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.m Derivative 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.m Derivative 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));

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Computing and Plotting Solutions to Systems Computing and Plotting Solutions to Systems

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Computing and Plotting Solutions to Systems Computing and Plotting Solutions to Systems

• [t,u] = ode45(’ball’,[0,3],[0;50]);

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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.

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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.

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Solving Higher Order Equations Solving Higher Order Equations

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Solving Higher Order Equations Solving Higher Order Equations

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

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Solving Higher Order Equations Solving Higher Order Equations

• Reduce to a first order system and solve the system.
• Example: The motion of a pendulum is modeled by

θ′′ = − g L sin θ − Dθ′.

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Solving Higher Order Equations Solving Higher Order Equations

• Reduce to a first order system and solve the system.
• Example: The motion of a pendulum is modeled by

θ′′ = − g L sin θ − Dθ′.

• Introduce ω = θ′.

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Solving Higher Order Equations Solving Higher Order Equations

• Reduce to a first order system and solve the system.
• Example: The motion of a pendulum is modeled by

θ′′ = − g L sin θ − Dθ′.

• Introduce ω = θ′. Notice

ω′ = − g L sin θ − Dθ′.

Return Higher order

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Equivalent First Order System Equivalent First Order System

Return Higher order

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Equivalent First Order System Equivalent First Order System

θ′ = ω ω′ = − g L sin θ − Dω

Return Higher order

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Equivalent First Order System Equivalent First Order System

θ′ = ω ω′ = − g L sin θ − Dω

• Change to vector notation. (Use MATLAB vector notation)

Return Higher order

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Equivalent First Order System Equivalent First Order System

θ′ = ω ω′ = − g L sin θ − Dω

• Change to vector notation. (Use MATLAB vector notation)

u(1)

= θ

Return Higher order

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Equivalent First Order System Equivalent First Order System

θ′ = ω ω′ = − g L sin 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.m Derivative m-file pend.m

function upr = pend(t,u) L= 1; global D th = u(1);

• m = u(2);

thpr = om;

• mpr = -(9.8/L)*sin(th) - D*om;

upr = [thpr; ompr];