# 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: