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A Solve the Integral Equation

  1. Nov 29, 2016 #1
    • Member warned that homework questions must be posted in the Homework sections
    Please anyone can help solve this integral equation
    e^t+e^t ∫ (t, 0 ) e^(-τ) x f(τ) dτ
     
  2. jcsd
  3. Nov 29, 2016 #2

    DrClaude

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    Staff: Mentor

    What have you tried? And what is f(τ)?
     
  4. Nov 29, 2016 #3
    Well one can never be sure but I suppose f(t) is the unknown function we wish to find. But first things first, is the equation we trying to solve as follows
    ##e^t+e^t\int_0^t {e^{-\tau}f(\tau)d\tau}=0##?
     
  5. Dec 1, 2016 #4
    Sorry, i haven't tried because i am not understanding the question completely.
     
  6. Dec 1, 2016 #5
    No, it is not equal to 0.
    The question is exactly like this
    "Solve the Integral Equation
    f(t)= same as you have written
     
  7. Dec 1, 2016 #6
    Ok so the equation is
    ##f(t)=e^t+e^t\int_0^t {e^{-\tau}f(\tau)d\tau}##.

    First do some algebraic operations and write the equation in the form ##\int_0^t {e^{-\tau}f(\tau)d\tau}=...##
    Now what operation can you apply at that form (hint: then plan is to convert the integral equation to an ordinary differential equation, there might be other ways to do this problem)
     
  8. Dec 1, 2016 #7
    It is easy to show that the equation can be transformed to the form
    $$g(t)=1+\int_0^t g(\tau)d\tau$$,
    where
    $$g(t)=e^{-t}f(t)$$.

    The equation can as mentioned before be transformed to a differential equation. A more simple way is however to solve it directly by iteration, i.e.
    $$g^{(n)}(t)=1+\int_0^t g^{(n-1)}(\tau)d\tau$$.
    The iteration is started by using ##g^{(0)}=1## and then computing ##g^{(1)}##, ##g^{(2)}##, etc. After a few iterations it should be obvious what ##g(x)## is.
     
  9. Dec 3, 2016 #8

    Dear,
    Could you please solve the complete answer step by step? I am having difficulty in understanding.
    Thanks..!
     
  10. Dec 3, 2016 #9
    As, I understand it is against the rules at this forum to give a complete answer to the problem. You need to show the work you have done.
    But, if you tell me which step you don't understand I will be happy to help. So, which step do you don't understand?
     
  11. Dec 3, 2016 #10
    okay no problem. you just tell me how you write the first step and how you put the 1.?
     
  12. Dec 3, 2016 #11
    You have the equation
    $$f(t)=e^t+e^t\int_0^{t}e^{-\tau}f(\tau)d\tau\quad (1)$$

    Do you understand how to go from Eq. (1) to
    $$g(t)=1+\int_0^{t}g(\tau)d\tau\quad (2) $$,
    where ##g(t)=e^{-t}f(t)## ?
     
  13. Dec 4, 2016 #12

    No, i can't understand how to go from Eq 1. to Eq 2.
    Also, i am bit confusing between "Tau" and "t"
     
  14. Dec 4, 2016 #13
    Multiply both sides of Eq. (1) by ##e^{-t}## and then simplify. What do you get?

    Both ##t## and ##\tau## are variable. For each value of ##t## we have one a left-hand side an integration from ##0## to ##t##. Therefore, to not mess things up we need to introduce an integration variable called ##\tau##. Obviously, the name of the variable is arbitrary.
     
  15. Dec 4, 2016 #14

    yes i have got. now what is after the iteration step? g(x)=?
     
  16. Dec 4, 2016 #15
    As, I understand you now got Eq. (2). Now, we want to solve this by iteration.
    On the left-hand side we start by a "guess" for ##g(\tau)##. As, the inhomogeneous term is 1. We start by putting ##g(\tau)=1## on the left-hand side.
    You now compute a better estimate ##g(t)## from the equation. This should give you ##g(t)=1+t##.
    Now, in the next step we put ##g(\tau)=1+\tau## on the left-hand side and calculate a new ##g(t)##. This procedure should lead to
    $$g(t)=1+t+t^2/2+t^3/6+...$$
    Do you get this?
     
  17. Dec 4, 2016 #16
    Yes, for sure i understand this also..
    Does it ends here or something remaining?
     
  18. Dec 4, 2016 #17
  19. Dec 5, 2016 #18
    Yes i have solved it. And thank you very much for your help.
     
  20. Dec 5, 2016 #19
    You are welcome.
     
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