Differences between revisions 1 and 15 (spanning 14 versions)
 ⇤ ← Revision 1 as of 2007-06-24 23:27:22 → Size: 28 Editor: DavidJoyner Comment: ← Revision 15 as of 2008-11-14 13:42:08 → ⇥ Size: 4387 Editor: anonymous Comment: converted to 1.6 markup Deletions are marked like this. Additions are marked like this. Line 2: Line 2: == 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$$x \ {\mapsto}\ \sin ( \frac{\pi \cdot x}{2} )$$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)^2sage: h(x) = -xsage: 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)^3sage: 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

# Differential Equations

## Laplace Transform (LT) methods

### Unit Step Functions

SAGE can define piecewise functions like

x \ {\mapsto}\ \sin ( \frac{\pi \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

### 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:

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

## PDEs

### Fourier Series

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

\displaystyle f(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi 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

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

### Heat Eqn. Both Ends Insulated

Differential_Equations (last edited 2008-11-14 13:42:08 by anonymous)