Results 1 to 6 of 6
Like Tree6Thanks
  • 2 Post By romsek
  • 2 Post By HallsofIvy
  • 2 Post By SlipEternal

Thread: Heaviside or dirac delta integration

  1. #1
    Newbie
    Joined
    Nov 2017
    From
    Belgium
    Posts
    3

    Question Heaviside or dirac delta integration

    Hello,

    I basically have this equation (and another very similar) to solve for my biomechanics course:

    Heaviside or dirac delta integration-codecogseqn.gif

    Where H(t) is the heaviside function. I can find the solution ε(t) as the sum of the general solution and the particular solution. The general solution is easy to get, and I can find the particular solution of type:

    Heaviside or dirac delta integration-codecogseqn-3-.gif

    By replacing it into the first equation and determine P(t), however, this lead me to this equation:

    Heaviside or dirac delta integration-codecogseqn-2-.gif

    The integral can be transformed by part into:

    Heaviside or dirac delta integration-codecogseqn-1-.gif

    But I can't solve this, mostly because the integral is undefinite.
    Thank you in advance for your help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,879
    Thanks
    2473

    Re: Heaviside or dirac delta integration

    Are you allowed to use Laplace transforms?

    For the particular solution (slight change in notation so I don't have to type as much)

    let $\beta = \dfrac{\sigma_0}{\eta}$

    $e^\prime(t) + \dfrac{e}{\tau} = \beta H(t)$

    take Laplace transforms

    $sE(s)-e(0) + \dfrac{1}{\tau}E(s) = \dfrac{\beta}{s}$

    $(s^2 + \dfrac s \tau)E(s) = \beta+e(0)$

    $E(s) = \dfrac{\beta+e(0)}{s^2 + \frac s \tau} = \dfrac{\beta+e(0)}{s\left(s+\frac 1 \tau\right)}$

    apply partial fractions

    $E(s) = \dfrac{e(0)-\beta \tau }{s \tau +1}+\dfrac{\beta }{s}$

    and invert the Laplace transform

    $e(t) =\beta H(t) + \dfrac 1 \tau \left(e(0)-\beta \tau\right)e^{-t/\tau}$

    and you would add this to the homogeneous solution to find the solution to the overall diff eq.
    Thanks from topsquark and Kyraz
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,414
    Thanks
    2889

    Re: Heaviside or dirac delta integration

    Quote Originally Posted by Kyraz View Post
    Hello,

    I basically have this equation (and another very similar) to solve for my biomechanics course:

    Click image for larger version. 

Name:	CodeCogsEqn.gif 
Views:	15 
Size:	758 Bytes 
ID:	38252

    Where H(t) is the heaviside function. I can find the solution ε(t) as the sum of the general solution and the particular solution. The general solution is easy to get, and I can find the particular solution of type:

    Click image for larger version. 

Name:	CodeCogsEqn(3).gif 
Views:	15 
Size:	551 Bytes 
ID:	38255

    By replacing it into the first equation and determine P(t), however, this lead me to this equation:

    Click image for larger version. 

Name:	CodeCogsEqn(2).gif 
Views:	1 
Size:	992 Bytes 
ID:	38254

    The integral can be transformed by part into:

    Click image for larger version. 

Name:	CodeCogsEqn(1).gif 
Views:	3 
Size:	987 Bytes 
ID:	38253

    But I can't solve this, mostly because the integral is undefinite.
    Thank you in advance for your help.
    The fact that the integral "is indefinite" just means it can be written as \int_0^t H(t) e^{t/\tau} dt+ C where C is an unknown constant. And, of course, since H(t) is 1 for all x greater than 0, \int H(t)e^{t/\tau}t= \int_0^t e^{t/\tau} dt+ C= \tau e^{t/\tau}+ C.
    Thanks from topsquark and Kyraz
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Nov 2017
    From
    Belgium
    Posts
    3

    Re: Heaviside or dirac delta integration

    Thank you very much for your two solutions Romsek and HallsofIvy (I realized some english mistakes in my previous post, sorry for that, I am used to write in french).
    By continuing my calculation using the result of HallsofIvy in my particular solution, I finally have this for ε(t) =

    Heaviside or dirac delta integration-codecogseqn-3-.gif

    However, the professor has the same solution, with a factor H(t).

    Heaviside or dirac delta integration-codecogseqn-2-.gif

    Given that this model represents the elongation of a tendon, I can guess ε(t) should be =0 for t<0, hence the H(t) in the solution. I don't see from where this additional term comes given that neither my general solution, nor my particular solution introduce the H(t).
    Last edited by Kyraz; Nov 3rd 2017 at 06:41 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2010
    Posts
    2,967
    Thanks
    1140

    Re: Heaviside or dirac delta integration

    Quote Originally Posted by Kyraz View Post
    Thank you very much for your two solutions Romsek and HallsofIvy (I realized some english mistakes in my previous post, sorry for that, I am used to write in french).
    By continuing my calculation using the result of HallsofIvy in my particular solution, I finally have this for ε(t) =

    Click image for larger version. 

Name:	CodeCogsEqn(3).gif 
Views:	7 
Size:	583 Bytes 
ID:	38258

    However, the professor has the same solution, with a factor H(t).

    Click image for larger version. 

Name:	CodeCogsEqn(2).gif 
Views:	1 
Size:	789 Bytes 
ID:	38259

    Given that this model represents the elongation of a tendon, I can guess ε(t) should be =0 for t<0, hence the H(t) in the solution. I don't see from where this additional term comes given that neither my general solution, nor my particular solution introduce the H(t).
    Let $f(x)$ be an integrable function. Define $g(x) = H(x)\cdot f(x)$. It should be obvious that $\displaystyle \int g(x)dx = H(x)\int f(x)dx$. Think about Riemann sums or Lebesgue integrals (depending on what level you are at in your integration theory) to see why this would be true.
    Thanks from topsquark and Kyraz
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Nov 2017
    From
    Belgium
    Posts
    3

    Re: Heaviside or dirac delta integration

    Indeed, I saw Riemann sums and Lebesgue integrals some time ago, but I have never been extremely comfortable with the theory. Thank you again, I finally got the expected solution (but for that, I had to assume the integration constant of my general solution =0, because my initial condition was not enough to determine the two constants in my final solution).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. dirac delta functions...
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: Apr 26th 2011, 07:45 PM
  2. Dirac Delta
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: May 18th 2010, 01:10 AM
  3. Dirac delta function
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Mar 10th 2010, 02:15 PM
  4. integration of dirac delta times exponential
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Apr 28th 2009, 04:37 AM

/mathhelpforum @mathhelpforum