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SandBoxSignum

Edit detail for SandBoxSignum revision 7 of 13

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Editor: Bill Page Time: 2014/07/15 03:32:02 GMT+0
Note: integration

added:
\end{axiom}

Solving sifferential equations, e.g. $L f(x)=c(x)$
\begin{axiom}
-- differential Operator
L(f)==D(f,[x,x])

added:
c(x)

changed:
-c(x-y)*g(y)
integrand:=c(x-y)*g(y)
conv:=integrate(integrand,y=%minusInfinity..%plusInfinity)
conv:=integrate(integrand,y=%minusInfinity..%plusInfinity,"noPole")
conv:=integrate(integrand,y)
-- expected result

added:
-- ramp

added:
test(L conv = c(x))

Ref.

Derivative of abs

fricas

s1:=D(abs(x),x)

$\label{eq1}{abs \left({x}\right)}\over x$

(1)

Type: Expression(Integer)

fricas

D(s1,x)

$\label{eq2}0$

(2)

Type: Expression(Integer)

fricas

limit(abs(x)/x,x=0,"right")

$\label{eq3}1$

(3)

Type: Union(OrderedCompletion?(Expression(Integer)),...)

fricas

limit(abs(x)/x,x=0,"left")

$\label{eq4}- 1$

(4)

Type: Union(OrderedCompletion?(Expression(Integer)),...)

An approximation to signum

fricas

D(tanh(a*x),x)

$\label{eq5}-{a \ {{\tanh \left({a \ x}\right)}^{2}}}+ a$

(5)

Type: Expression(Integer)

fricas

limit(%,a=%plusInfinity)

$\label{eq6}\mbox{\tt "failed"}$

(6)

Type: Union("failed",...)

Patterns for derivatives

fricas

Abs:=operator('Abs)

$\label{eq7}\hbox{\axiomType{Abs}\ }$

(7)

Type: BasicOperator?

fricas

D(abs(x),x)

$\label{eq8}{abs \left({x}\right)}\over x$

(8)

Type: Expression(Integer)

fricas

signum:=operator('signum)

$\label{eq9}signum$

(9)

Type: BasicOperator?

fricas

diracDelta:=operator('diracDelta)

$\label{eq10}diracDelta$

(10)

Type: BasicOperator?

fricas

signum(x)

$\label{eq11}signum \left({x}\right)$

(11)

Type: Expression(Integer)

fricas

D(signum(x),[x,x])

$\label{eq12}{signum_{\ }^{, ,}}\left({x}\right)$

(12)

Type: Expression(Integer)

fricas

diff1:=rule
  signum(x) == Abs(x)/x
  D(Abs(x),x) == signum(x) + 2*x*diracDelta(x)
  D(signum(x),x) == 2*diracDelta(x)

$\label{eq13}\begin{array}{@{}l} \displaystyle \left\{{{signum \left({x}\right)}\mbox{\rm = =}{{{{\tt'}\hbox{\axiomType{Abs}\ }}\left({x}\right)}\over x}}, \: \right. \ \ \displaystyle \left.{{{\hbox{\axiomType{Abs}\ }_{\ }^{,}}\left({x}\right)}\mbox{\rm = =}{{{{\tt'}signum}\left({x}\right)}+{2 \ x \ {{{\tt'}diracDelta}\left({x}\right)}}}}, \: \right. \ \ \displaystyle \left.{{{signum_{\ }^{,}}\left({x}\right)}\mbox{\rm = =}{2 \ {{{\tt'}diracDelta}\left({x}\right)}}}\right\}$

(13)

Type: Ruleset(Integer,Integer,Expression(Integer))

fricas

diff(x,y)==diff1 D(x,y)

Type: Void

fricas

diff(Abs(x),x)

fricas

Compiling function diff with type (Expression(Integer),Variable(x))
       -> Expression(Integer)

$\label{eq14}{{2 \ {{x}^{2}}\ {diracDelta \left({x}\right)}}+{\hbox{\axiomType{Abs}\ } \left({x}\right)}}\over x$

(14)

Type: Expression(Integer)

fricas

diff(Abs(x)/x,x)

$\label{eq15}2 \ {diracDelta \left({x}\right)}$

(15)

Type: Expression(Integer)

Adding the singular part of the derivative

fricas

diff(f,x) == D(f,x)+(limit(f,x=0,"right")-limit(f,x=0,"left"))::EXPR(INT)*diracDelta(x)

   Compiled code for diff has been cleared.
   1 old definition(s) deleted for function or rule diff

Type: Void

fricas

diff(abs(x),x)

fricas

Compiling function diff with type (Expression(Integer),Variable(x))
       -> Expression(Integer)

$\label{eq16}{abs \left({x}\right)}\over x$

(16)

Type: Expression(Integer)

fricas

diff(%,x)

$\label{eq17}2 \ {diracDelta \left({x}\right)}$

(17)

Type: Expression(Integer)

fricas

signum(x)==abs(x)/x

Type: Void

fricas

diff(signum(x),x)

fricas

Compiling function signum with type Variable(x) -> Expression(
      Integer)

$\label{eq18}2 \ {diracDelta \left({x}\right)}$

(18)

Type: Expression(Integer)

fricas

integrate(signum(x),x)

$\label{eq19}\int^{ \displaystyle x}{{{abs \left({\%A}\right)}\over \%A}\ {d \%A}}$

(19)

Type: Union(Expression(Integer),...)

Heaviside

fricas

H(x)==(abs(x)+x)/2/x

Type: Void

fricas

H(-0.3),H(0.3)

fricas

Compiling function H with type Float -> Float

$\label{eq20}\left[{0.0}, \:{1.0}\right]$

(20)

Type: Tuple(Float)

fricas

diff(H(x),x)

fricas

Compiling function H with type Variable(x) -> Expression(Integer)

$\label{eq21}diracDelta \left({x}\right)$

(21)

Type: Expression(Integer)

Solving sifferential equations, e.g.

fricas

-- differential Operator
L(f)==D(f,[x,x])

Type: Void

fricas

-- Green function
g(x)==abs(x)/2

Type: Void

fricas

-- pulse
c(x)==H(x+1/2)-H(x-1/2)

Type: Void

fricas

c(x)

fricas

Compiling function H with type Polynomial(Fraction(Integer)) -> 
      Expression(Integer)

fricas

Compiling function c with type Variable(x) -> Expression(Integer)

$\label{eq22}{{{\left({2 \ x}- 1 \right)}\ {abs \left({{{2 \ x}+ 1}\over 2}\right)}}+{{\left(-{2 \ x}- 1 \right)}\ {abs \left({{{2 \ x}- 1}\over 2}\right)}}}\over{{4 \ {{x}^{2}}}- 1}$

(22)

Type: Expression(Integer)

fricas

-- convolution
integrand:=c(x-y)*g(y)

fricas

Compiling function c with type Polynomial(Integer) -> Expression(
      Integer)

fricas

Compiling function g with type Variable(y) -> Expression(Integer)

$\label{eq23}{\left({\left({ \begin{array}{@{}l} \displaystyle {{\left({2 \ y}-{2 \ x}- 1 \right)}\ {abs \left({{{2 \ y}-{2 \ x}+ 1}\over 2}\right)}}+ \ \ \displaystyle {{\left(-{2 \ y}+{2 \ x}- 1 \right)}\ {abs \left({{{2 \ y}-{2 \ x}- 1}\over 2}\right)}}$

(23)

Type: Expression(Integer)

fricas

conv:=integrate(integrand,y=%minusInfinity..%plusInfinity)

$\label{eq24}\mbox{\tt "potentialPole"}$

(24)

Type: Union(pole: potentialPole,...)

fricas

conv:=integrate(integrand,y=%minusInfinity..%plusInfinity,"noPole")

$\label{eq25}\mbox{\tt "failed"}$

(25)

Type: Union(fail: failed,...)

fricas

conv:=integrate(integrand,y)

$\label{eq26}\int^{ \displaystyle y}{{{{{\left({2 \ x}-{2 \ \%A}- 1 \right)}\ {abs \left({\%A}\right)}\ {abs \left({{{2 \ x}-{2 \ \%A}+ 1}\over 2}\right)}}+{{\left(-{2 \ x}+{2 \ \%A}- 1 \right)}\ {abs \left({\%A}\right)}\ {abs \left({{{2 \ x}-{2 \ \%A}- 1}\over 2}\right)}}}\over{{8 \ {{x}^{2}}}-{{16}\ \%A \ x}+{8 \ {{\%A}^{2}}}- 2}}\ {d \%A}}$

(26)

Type: Union(Expression(Integer),...)

fricas

-- expected result
conv:=1/2*x^2*H(x+1/2)-1/2*x+1/2*x*H(x+1/2)+1/8*H(x+1/2)-1/2*x^2*H(x-1/2)+1/2*x*H(x-1/2)-1/8*H(x-1/2)

$\label{eq27}{{{\left({2 \ x}+ 1 \right)}\ {abs \left({{{2 \ x}+ 1}\over 2}\right)}}+{{\left(-{2 \ x}+ 1 \right)}\ {abs \left({{{2 \ x}- 1}\over 2}\right)}}}\over 8$

(27)

Type: Expression(Integer)

fricas

-- ramp
diff(conv,x)

$\label{eq28}{{abs \left({{{2 \ x}+ 1}\over 2}\right)}-{abs \left({{{2 \ x}- 1}\over 2}\right)}}\over 2$

(28)

Type: Expression(Integer)

fricas

test(L conv = c(x))

fricas

Compiling function L with type Expression(Integer) -> Expression(
      Integer)

$\label{eq29} \mbox{\rm true}$

(29)

Type: Boolean