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Math Help - Fredholm Intergral Equation

  1. #1
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    Fredholm Intergral Equation

    I need to find a general solution,

    f E [0, \infty)

    for the Fredholm intergral equation:

    f(x) = \gamma cos (x) + \delta sin (x) + \lambda (Mf)(x)

    where M is the intergral operator defined by

    (Mf)(x):= \int_0^{\infty} x^{a-1} e^{-bt} f(t)dt

    Where a and b are postive constants and \gamma, \delta and \lambda ∈ ℝ constants.

    I think, that I have to rewrite it:

    f(x)= \gamma cos (x) + \delta sin (x) + \lambda \int_0^{\infty} x^{a-1} e^{-bt} f(t)dt

    And then assuming that λ is a fixed unknown scalar, and so can be incorporated into the equation as follows:

    f(x)= \gamma cos (x) + \delta sin (x) +\int_0^{\infty} x^{a-1} e^{-bt} f(t)dt


    I'm not sure if this is right, and I don't really understand what to do next, can someone help please?

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  2. #2
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    IMHO
    The next step after this:
    <br />
f(x)= \gamma cos x + \delta sin x + \lambda \int_0^{\infty} x^{a-1} e^{-bt} f(t)dt<br />
    is to say: if equation invloved has a solution, its solution can be represented in the following way:
    f(x)=\gamma cos x + \delta sin x+\lambda x^{a-1} C_1,
    where C_1=\int_0^{\infty} e^{-bt}f(t)dt=\int_0^{\infty} e^{-bt}(\gamma cost + \delta sin t + \lambda t^{a-1} C_1)dt
    Knowing C_1 you will find general solution by substituting it into the first equation in this message.
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  3. #3
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    okay

    Thanks, for that, but if I substitute C_1 into the first equation, then won't I just be back where I started?

    I.e.



    Just with x replaced with t's
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  4. #4
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    Did you check this?
    Note that you have equation for C_1 where this stuff takes part in both sides of the equation.
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  5. #5
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    Well basically,

    if I let C_1=\int_0^{\infty} e^{-bt}f(t)dt

    and subsitute it into

    f(x)= \gamma cos x + \delta sin x + \lambda \int_0^{\infty} x^{a-1} e^{-bt} f(t)dt

    Then all I get is

     <br />
f(x)=\gamma cos x + \delta sin x+\lambda x^{a-1} C_1<br />

    And keep going round in circles.

    Maybe at this point I need to solve the integral, but I am not sure how to do this?
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  6. #6
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    No. Look at this:
    You have:
     f(x)= \gamma cos x + \delta sin x + \lambda \int_0^{\infty} x^{a-1} e^{-bt} f(t)dt
    This equation has a solution:
     f(x)=\gamma cos x + \delta sin x+\lambda x^{a-1} C_1
    where C_1=\int_0^{\infty} e^{-bt}f(t)dt
    Lets find C_1 in terms of \lambda:
    C_1=\int_0^{\infty} e^{-bt}f(t)dt=\int_0^{\infty} e^{-bt}(\gamma cost + \delta sin t + \lambda t^{a-1} C_1)dt = I_1 + I_2+ I_3
    I_1=\gamma  \int_0^{\infty} e^{-bt} cost dt = \frac{b \gamma}{b^2+1}
    I_2=\delta \int_0^{\infty} e^{-bt} sint dt= \frac{\delta}{b^2+1}
    I_3=-\lambda C_1 K.
    So:
    C_1=\frac{b \gamma + \delta}{b^2+1}-\lambda C_1 K
    C_1=\frac{b \gamma + \delta}{(b^2+1)(1+\lambda K)}
    if \lambda doesn't equal -\frac{1}{K}.
    Now you should substitute C_1 into
     f(x)=\gamma cos x + \delta sin x+\lambda x^{a-1} C_1
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  7. #7
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    Thanks!

    Thank you very much that was really helpful!!!
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  8. #8
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    Fredholm Integral Equation

    I need to find a general solution,

    f E [0, \infty)

    for the Fredholm intergral equation:

    f(x) = \gamma cos (x) + \delta sin (x) + \lambda (Mf)(x)

    where M is the intergral operator defined by

    (Mf)(x):= \int_0^{\infty} x^{a-1} e^{-bt} f(t)dt

    Where a and b are postive constants and \gamma, \delta and \lambda ∈ ℝ constants.

    Note that the algebra can be reduced using the Gamma function and standard Laplace Transforms.

    I have started a solution to this, which is attached, can anyone tell me if this is all I have to do to get the general solution or is there more, and if so can anyone advise what I do next please?
    Attached Files Attached Files
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  9. #9
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    My Solution??? Is it correct?

    I have the following as the general solution and someone please help me, and advise me if this is correct please?

     <br />
f(x) = \gamma cos (x) + \delta sin (x) + \frac{\lambda x^{a-1}(b \gamma + \delta)}{(b^2+1)(1+\lambda K)}<br />

    I would be really greatful if someone could help so that I could continue on the next part of the question please...
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