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| $$ x \ {\mapsto}\ \sin ( \frac{\pi \cdot x}{2} ) $$ |
$$x \ {\mapsto}\ \sin ( \frac{\pi \cdot x}{2} ) $$ |
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| $$ \ {\mapsto}\ 1 - ( x - 1 )^2 $$ |
$$x \ {\mapsto}\ 1 - ( x - 1 )^2 $$ |
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| 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 }}} |
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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 }}} |
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 \ {\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
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
Convergence, Dirichlet's theorem
Fourier Sine Series, Fourier Cosine Series
Heat Eqn. Ends at Zero