help on derivative of triangular function (Dirac delta function)

[smsow]

Hope someone can give me some pointers on how to get the 2nd order derivation of the following triangular function.

. . . . .\(\displaystyle T(x)\, =\, \begin{cases}1\, -\, |x|\, (N\, +\, 1),&|x|\, <\, \dfrac{1}{N\, +\, 1} \\ 0,&|x|\, >\, \dfrac{1}{N\, +\, 1} \end{cases}\)

where N is a constant integer.

The solution is as follows:

. . . . .\(\displaystyle \dfrac{d^2\, T(x\, -\, x_n)}{dx^2}\, =\, (N\, +\, 1)\, \Big[\,\delta\, \left(x\, -\, x_{n - 1}\right)\, -\, 2\, \delta\, \left(x\, -\, x_n\right)\, +\, \delta\, \left(x\, -\, x_{n + 1}\right)\, \Big]\)

where \(\displaystyle \, \delta\, (x)\, \) is the Dirac delta function.

Thanks!!

The original problem can be found in the following book page 13
Title: Field Computation by Moment Methods
Author: R.F. Harrington

 

[Deleted member 4993]

Hope someone can give me some pointers on how to get the 2nd order derivation of the following triangular function.

. . . . .\(\displaystyle T(x)\, =\, \begin{cases}1\, -\, |x|\, (N\, +\, 1),&|x|\, <\, \dfrac{1}{N\, +\, 1} \\ 0,&|x|\, >\, \dfrac{1}{N\, +\, 1} \end{cases}\)

where N is a constant integer.

The solution is as follows:

. . . . .\(\displaystyle \dfrac{d^2\, T(x\, -\, x_n)}{dx^2}\, =\, (N\, +\, 1)\, \Big[\,\delta\, \left(x\, -\, x_{n - 1}\right)\, -\, 2\, \delta\, \left(x\, -\, x_n\right)\, +\, \delta\, \left(x\, -\, x_{n + 1}\right)\, \Big]\)

where \(\displaystyle \, \delta\, (x)\, \) is the Dirac delta function.

Thanks!!

The original problem can be found in the following book page 13
Title: Field Computation by Moment Methods
Author: R.F. Harrington

\(\displaystyle \displaystyle{\dfrac{d(|x|)}{dt} \ = \ ??}\)

 

[smsow]

\(\displaystyle \displaystyle{\dfrac{d(|x|)}{dt} \ = \ ??}\)

Thanks for the pointer. Here is some of my work out but it still not unable to get the solution in term of dirac delta function.

\(\displaystyle \mbox{Let }\, |x|, =\, \sqrt{\strut x^2\,}\)

\(\displaystyle \dfrac{d|x|}{dx}\, =\, \dfrac{d(x^2)^{\frac{1}{2}}}{dx}\, =\, \dfrac{1}{2}\, (x^2)^{-\frac{1}{2}}\, \cdot\, 2x\, =\, \dfrac{x}{|x|}\)

and

\(\displaystyle \dfrac{d|x\, -\, a|}{dx}\, =\, \dfrac{d\left(\, (x\, -\, a)^2\, \right)^{\frac{1}{2}}}{dx}\, =\, \dfrac{x\, -\, a}{|x\, -\, a|}\)

and

\(\displaystyle \mbox{So, if }\, T(x\, -\, x_n)\, =\, 1\, -\, |x\, -\, x_n|(N\, +\, 1)\)

\(\displaystyle \mbox{then }\, \dfrac{dT(x\, -\, x_n)}{dx}\, =\, \dfrac{d\left(1\, -\, |x\, -\, x_n|(N\, +\, 1)\right)}{dx}\, =\, \dfrac{-(N\, +\, 1)(x\, -\, x_n)}{|x\, -\, x_n|}\)

\(\displaystyle \mbox{and }\, \dfrac{d^2 T(x\, -\, x_n)}{dx^2}\, =\, \dfrac{d\left(\dfrac{-(N\, +\, 1)(x\, -\, x_n)}{|x\, -\, x_n|}\right)}{dx}\, =\, -(N\, +\, 1)\,\dfrac{d\left(\dfrac{x\, -\, x_n}{|x\, -\, x_n|}\right)}{dx}\)

. . . . .\(\displaystyle =\, -(N\, +\, 1)\left[\, \dfrac{(1)|x\, -\, x_n|\, -\, x\dfrac{x\, -\, x_n}{|x\, -\, x_n|}}{|x\, -\, x_n|^2}\, -\, \dfrac{x_n (x\, -\, x_n)}{|x\, -\, x_n|^3}\, \right]\)

. . . . .\(\displaystyle -(N\, +\, 1)\left[\, \dfrac{1}{|x\, -\, x_n|}\, -\, \dfrac{x(x\, -\, x_n)}{|x\, -\, x_n|^3}\, -\, \dfrac{x_n(x\, -\, x_n)}{|x\, -\, x_n|^3}\, \right]\)

and finally, this will lead to zero..... Not sure where goes wrong.... and the solution still not in term of dirac delta function. Hope to get some help from here. Thanks!