"Dr.K" <kananipour@gmail.com> wrote in message <dc1cdc58f9104863a9c17df889612857@q12g2000prb.googlegroups.com>...
> Hi everyone, how can i write a first order derivative of Dirac(x)
> respect to x.
> i know that Dirac delta is derivative of Heaviside. in Fact i had to
> solve:
> ?u(x,t))/?x where: u(x,t)=Dirac(xt)
> in MATLAB i wrote:
> if (xt)==0
> k=1;
> else
> k=0;
> end
> u(x,t)=k;
> for Dirac Function, like as impulse force in any x from domain.
> but now what should i do to achieving ?u(x,t))/?x?
          
That is not a correct formulation for the Dirac delta function. For xt equal to zero, it should be k = inf. However, you cannot compute with the Dirac delta function using ordinary numerical quantities. In those terms there is no such function. It would have to have the property that its integral taken over the range [0,0] is one, which in the numerical world is impossible. No such function has this property. It is not a coincidence that this function is contained in the symbolic toolbox where such concepts are symbolically meaningful. Taking its derivative would be all the more meaningless in the numerical world.
In whatever problem you are dealing with if you express it in such a way that the parameters involved approach a limit which represents what the Dirac delta function or its derivative would be, that might give you the kind of solution you are seeking.
For example, suppose you define a function h(x;e) by:
h(x;e) = 1, if x < pi*e
h(x;e) = (1+sin(x/e))/2, if pi*e <= x <= +pi*e
h(x;e) = +1, if x > +pi*e
This looks like a smoothedout version of the Heaviside function. If you allow e to approach zero, this does in fact approach the Heaviside function. If we take the derivative of h(x;e) w.r. to x, multiply it by some arbitrary continuous function f(x), and then integrate that product from inf to +inf, this integral will always approach f(0) as e is allowed to approach zero. This is precisely how the Dirac delta function is supposed to act, but it only makes sense in terms of such a limiting process.
Presumably in the problem you are faced with you could do a similar analysis in terms of what should behave, in the limit, like the derivative of the Dirac delta function.
Roger Stafford
