chapter 13: solving equations

Chapter 13:

Solving Equations

MATLAB for Scientist and Engineers

Using Symbolic Toolbox

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You are going to See that MuPAD solves algebraic equations

and differential equations Plot the solution curve of the differential

equations Experience some chaotic systems described

by a set of nonlinear differential equations.

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Polynomial Equations

solve does it all Solution Target

=0 is the default

Not a simple closed form solution

Numerical values

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Set of Linear Equations

Under-determined equations

Verifying the solutions

evalAt operator

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Set of Nonlinear Equations

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Solving with Assumptions

Give some constraints on solutions

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Exercise 1

Compute the general solution of the system of linear equations

How many free parameters does the solution have?

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Infinite Number of Solutions

General Solutions

No Symbolic Solution? Try Numerical Solution.

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Inequalities

Some Region is the Solution

Checking the Region

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Differential Equations

ode and solve 1st Order

2nd Order

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ODE with Initial Conditions

Initial Conditions

with different Initial Values

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Exercise 2

Solve the following ODE for different values of a=-2, 0,+2 and plot the solutions.

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Set of Differential Equations

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Numerical ODE Solver

Original ODE Matrix Form

Matrix form ODE function

Numeric solution at t=1 initial value

time duration

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Plotting Numerical ODE Solution

plot::Ode2d

with Mapping function

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Ode3d with 3D Mapping

plot::Ode3d

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Chaotic System

Lorenz attractor

Parameters of Lorenz attractor

Initial Condition

Plot Generator

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Ode2d - Plot

Lorenz attractor (cont.)

Initial points are nearly the same.

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Ode3d

Lorenz attractor (cont.)

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Numerical Solution at a Point

Lorenz attractor (cont.)

1t

10t

100t

Chaotic System

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Exercise 3

Compute the general solution y(x) of the differen-tial equation y′ = y2/x .

Determine the solution y(x) for each of the follow-ing initial value problems:

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Exercise 4

Draw the 3-D trajectory of the solution of the fol-lowing system of ordinary differential equations in x(t), y(t), z(t) assuming the initial conditions of x(0)=1, y(0)=0.1, z(0)=-1.

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Difference Equations

Arithmetic Sequence

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Difference Equation - Geometric

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Difference Equations

General Solution

With Initial Conditions

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Exercise 5

The Fibonacci numbers are defined by the recurrence Fn = Fn−1 + Fn−2 with the initial val-ues F0 = 0, F1 = 1. Use solve to find an ex-plicit representation for Fn.

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Key Takeaways

Now, you are able to solve a set of linear and non-linear algebraic

equations, solve a set of ordinary differential equations, solve a set of nonlinear differential equations nu-

merically and plot them in 2D as well as in 3D space

and to solve difference equations.

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Notes

solve(x^2+x=y/4,x) solve({x+y+z=3,x+y=2,x-y-z=1},{x,y,z})

solve(sin(x)=1/2)

ode({y'(x)=y(x),y(0)=1}, y(x))

solve( x^2 < 1, x )

plot::Ode2d(..)solve(ode(y''(t)=-3*y'(t)-2*y(t)+2*t^2, y(t)))

numeric::odesolve(f, 0..1,Y0)

plot::Ode3d(..) eqn := rec(y(n+2)=y(n+1)+2*y(n), y(n))

plot::Curve2d(..)