# 
# 
> restart;
> #This worksheet is intended to give you some feeling of how to use
> convolution to approximately compute derivatives. In the next command
> replace sin(x) with the function of your choice. Use a function for
> which maple can compute its symbolic derivatives. This is the function
> we will study in the rest of the worksheet.
> g := x-> sin(x); #Rplace sin(x) here with other functions.
> g1 := D(g);
> g2 := D(g1);
> #Here we convolve g with a rectangular pulse of width h and plot the
> results for several values of h.
> f0 := (x,h)->(1/(2*h))*evalf(int(g(y),y=x-h..x+h));
> plot([g(x),f0(x,.025),f0(x,.1),f0(x,.5),f0(x,1)],x=-Pi..Pi);
> #This command convolves g with the weak derivative of a tent function.
> This gives an approximation to the derivative of g. We again plot the
> convolution for several values of h along with g itself.
> f1 := (x,h) ->
> (1/(h^2))*(evalf(int(g(y),y=x..x+h))-evalf(int(g(y),y=x-h..x)));
> plot([g1(x),f1(x,.1),f1(x,.25),f1(x,.5)],x=-Pi..Pi);
> #We examine the graphs more carefully near to x=0.
> plot([cos(x),f1(x,.1),f1(x,.25),f1(x,.5)],x=-0.2..0.2);
> #Finally we convolve g with a weight which is the second derivative of
> the function "w" shown in the next plot. This function is supported in
> the interval [-2h,2h]. It has two weak derivatives. We first compute
> and plot w and its first and second weak derivatives.
> w := (x,h)->piecewise(-2*h<= x and x<-h, (x^2+4*h*x+4*h^2), -h<= x and
> x<h, 2*h^2-x^2, h<x and x<2*h, x^2-4*h*x+4*h^2,0)/(4*h^3);
> w1 := (x,h)->piecewise(-2*h<= x and x<-h, 2*x+4*h, -h<= x and x<h,
> -2*x, h<x and x<2*h, 2*x-4*h,0)/(4*h^3);
> w2 := (x,h)->piecewise(-2*h<= x and x<-h, 2, -h<= x and x<h, -2, h<x
> and x<2*h, 2,0)/(4*h^3);
> plot([w(x,.5),w1(x,.5),w2(x,.5)],x=-1..1);#Try this out with different
> h values.
> #Here we convolve g with D^2 w.
> f2 :=
> (x,h)->(1/(2*h^3))*(evalf(int(g(y),y=x-2*h..x-h))+evalf(int(g(y),y=x+h
> ..x+2*h))-evalf(int(g(y),y=x-h..x+h)));
> #These plots show the second derivative of g along with several
> approximations obtained by convolving with D^2 w.
> plot([g2(x),f2(x,.1),f2(x,.25),f2(x,.5)],x=-Pi..Pi);
> #We examine these plots more carefully near to Pi/2.
> plot([g2(x),f2(x,.1),f2(x,.25),f2(x,.5)],x=Pi/2-.2..Pi/2+.2);
>