Differences between revisions 1 and 14 (spanning 13 versions)

Differential Equations

First order DEs

IVPs, Direction Fields, Isoclines

Direction Fields, Autonomous DEs

Separable DEs, Exact DEs, Linear 1st order DEs

Numerical method: Euler (or Constant Slope)

Applications (Growth/Cooling/Circuits/Falling body)

Higher order DEs

IVPs/General solutions, Basic theory

Numerical methods for higher order DEs

Constant coefficient case: Undetermined Coefficients

Application: springs (free, damped, forced, pure resonance)

Application: Electrical Circuits

Laplace Transform (LT) methods

Inverse Laplace & Derivatives

1st Translation Thrm

Partial Fractions, completing the square

Unit Step Functions

SAGE can define piecewise functions like

x |→ sin(2π·x)

on (0,1),

x |→ 1−(x−1)2

on (1,3),

x |→ −x

on (3,5), as follows:

sage: f(x) = sin(x*pi/2)
sage: g(x) = 1-(x-1)^2
sage: h(x) = -x
sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]])
sage: latex(P)

However, at the moment only Laplace transforms of "piecewise polynomial" functions are implemented:

sage: f(x) = x^2+1      
sage: g(x) = 1-(x-1)^3
sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]])
sage: P.laplace(x,s)
(s^3 - 6)*e^(-s)/s^4 - ((2*s^2 + 2*s + 2)*e^(-s)/s^3) + (7*s^3 + 12*s^2 + 12*s + 6)*e^(-3*s)/s^4 + (-3*s - 1)*e^(-3*s)/s^2 + (5*s + 1)*e^(-5*s)/s^2 + (s^2 + 2)/s^3

2nd Translation Theorem

Derivative thrms, Solving DEs

Convolution theorem

You can find the convolution of any piecewise defined function with another (off the domain of definition, they are assumed to be zero). Here is f , f*f , and f*f*f , where f(x)=1 , 0<x<1 :

sage: x = PolynomialRing(QQ, 'x').gen()
sage: f = Piecewise([[(0,1),1*x^0]])
sage: g = f.convolution(f)
sage: h = f.convolution(g)
sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1))

The command show(P+Q+R) displays this:

http://sage.math.washington.edu/home/wdj/art/convolutions.png

Though SAGE doesn't simplify very well, you can see that the LT(f*f) is equal to LT(f)2:

sage: f.laplace(x,s)
1/s - e^(-s)/s
sage: g.laplace(x,s)
-(s + 1)*e^(-s)/s^2 + (s - 1)*e^(-s)/s^2 + e^(-(2*s))/s^2 + 1/s^2
sage: (f.laplace(x,s)^2).expand()
-2*e^(-s)/s^2 + e^(-(2*s))/s^2 + 1/s^2

Dirac Delta Function

Application: Lanchester's equations

Application: Electrical networks

PDEs

Separation of Variables

Heat Equation., Fourier's solution

Fourier Series

If f(x) is a piecewise-defined polynomial function on −L<x<L then the Fourier series

f(x)~2a0+∑∞n=1[ancos(Lnπx)+bnsin(Lnπx)],

converges. In addition to computing the coefficients an,bn , it will also compute the partial sums (as a string), plot the partial sums (as a function of x over (−L,L) , for comparison with the plot of f(x) itself), compute the value of the FS at a point, and similar computations for the cosine series (if f(x) is even) and the sine series (if f(x) is odd). Also, it will plot the partial F.S. Cesaro mean sums (a smoother" partial sum illustrating how the Gibbs phenomenon is mollified).

sage: f1 = lambda x:-1
sage: f2 = lambda x:2
sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_cosine_coefficient(5,pi)
-3/(5*pi)
sage: f.fourier_series_sine_coefficient(2,pi)
-3/pi
sage: f.fourier_series_partial_sum(3,pi)
-3*sin(2*x)/pi + sin(x)/pi - 3*cos(x)/pi + 1/4

Plotting the partial sums is implemented: Typing f.plot_fourier_series_partial_sum(15,pi,-5,5) yields

http://sage.math.washington.edu/home/wdj/art/fourier-partial-sum1.png

and typing f.plot_fourier_series_partial_sum_cesaro(15,pi,-5,5) yields the much smoother version:

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+== First order DEs ==

=== IVPs, Direction Fields, Isoclines ===

=== Direction Fields, Autonomous DEs ===

=== Separable DEs, Exact DEs, Linear 1st order DEs ===

=== Numerical method: Euler (or Constant Slope) ===

=== Applications (Growth/Cooling/Circuits/Falling body) ===

== Higher order DEs ==

=== IVPs/General solutions, Basic theory ===

=== Numerical methods for higher order DEs ===

=== Constant coefficient case: Undetermined Coefficients ===

=== Application: springs (free, damped, forced, pure resonance) ===

=== Application: Electrical Circuits ===

== Laplace Transform (LT) methods ==

=== Inverse Laplace & Derivatives ===

=== 1st Translation Thrm ===

=== Partial Fractions, completing the square ===

=== Unit Step Functions ===

SAGE can define piecewise functions like

 $x \mapsto \sin (\frac{π \cdot x}{2})$ 
on  $(0, 1)$ ,
 $x \mapsto 1 - (x - 1)^{2}$ 
on  $(1, 3)$ ,
 $x \mapsto - x$ 
on   $(3, 5)$ , as follows:

{{{
sage: f(x) = sin(x*pi/2)
sage: g(x) = 1-(x-1)^2
sage: h(x) = -x
sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]])
sage: latex(P)
}}}

However, at the moment only Laplace transforms of "piecewise polynomial" functions are implemented:

{{{
sage: f(x) = x^2+1      
sage: g(x) = 1-(x-1)^3
sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]])
sage: P.laplace(x,s)
(s^3 - 6)*e^(-s)/s^4 - ((2*s^2 + 2*s + 2)*e^(-s)/s^3) + (7*s^3 + 12*s^2 + 12*s + 6)*e^(-3*s)/s^4 + (-3*s - 1)*e^(-3*s)/s^2 + (5*s + 1)*e^(-5*s)/s^2 + (s^2 + 2)/s^3
}}}

=== 2nd Translation Theorem ===

=== Derivative thrms, Solving DEs ===

=== Convolution theorem ===

You can find the convolution of any piecewise defined function with another (off the domain of definition, they are assumed to be zero). Here is  $f$  ,  $f * f$  , and  $f * f * f$  , where  $f (x) = 1$  ,  $0 < x < 1$  :

{{{
sage: x = PolynomialRing(QQ, 'x').gen()
sage: f = Piecewise([[(0,1),1*x^0]])
sage: g = f.convolution(f)
sage: h = f.convolution(g)
sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1))
}}}
The command show(P+Q+R) displays this:

http://sage.math.washington.edu/home/wdj/art/convolutions.png

Though SAGE doesn't simplify very well, you can see that the  $L T (f * f)$  is equal to 
 $L T (f)^{2}$ :

{{{
sage: f.laplace(x,s)
1/s - e^(-s)/s
sage: g.laplace(x,s)
-(s + 1)*e^(-s)/s^2 + (s - 1)*e^(-s)/s^2 + e^(-(2*s))/s^2 + 1/s^2
sage: (f.laplace(x,s)^2).expand()
-2*e^(-s)/s^2 + e^(-(2*s))/s^2 + 1/s^2
}}}

=== Dirac Delta Function ===

=== Application: Lanchester's equations ===

=== Application: Electrical networks ===

== PDEs ==

=== Separation of Variables ===

=== Heat Equation., Fourier's solution ===

=== Fourier Series ===

If  $f (x)$   is a piecewise-defined polynomial function on  $- L < x < L$   then the Fourier series

 $f (x) \sim \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} [a_{n} \cos (\frac{n π x}{L}) + b_{n} \sin (\frac{n π x}{L})],$ 

converges. In addition to computing the coefficients  $a_{n}, b_{n}$  , it will also compute the partial sums (as a string), plot the partial sums (as a function of  $x$  over  $(- L, L)$  , for comparison with the plot of  $f (x)$  itself), compute the value of the FS at a point, and similar computations for the cosine series (if  $f (x)$  is even) and the sine series (if  $f (x)$  is odd). Also, it will plot the partial F.S. Cesaro mean sums (a ``smoother" partial sum illustrating how the Gibbs phenomenon is mollified).

{{{
sage: f1 = lambda x:-1
sage: f2 = lambda x:2
sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_cosine_coefficient(5,pi)
-3/(5*pi)
sage: f.fourier_series_sine_coefficient(2,pi)
-3/pi
sage: f.fourier_series_partial_sum(3,pi)
-3*sin(2*x)/pi + sin(x)/pi - 3*cos(x)/pi + 1/4
}}}
Plotting the partial sums is implemented: Typing `f.plot_fourier_series_partial_sum(15,pi,-5,5)` yields

http://sage.math.washington.edu/home/wdj/art/fourier-partial-sum1.png

and typing `f.plot_fourier_series_partial_sum_cesaro(15,pi,-5,5)` yields the much smoother version:


http://sage.math.washington.edu/home/wdj/art/fourier-partial-sum2.png

=== Convergence, Dirichlet's theorem ===

=== Fourier Sine Series, Fourier Cosine Series ===

=== Heat Eqn. Ends at Zero ===

=== Heat Eqn. Both Ends Insulated ===

Diff for "Differential_Equations"