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I have an inegral equation like this

$\qquad n(\phi)=\int_0^\sqrt{\phi} f(w)\sqrt{2w+\phi}dw$.

I need to find $f(w)$ analytically. Here $n(\phi)$ is known. Here $n(\phi) = -\frac{2}{\delta ^2 \phi }-\frac{2 e^{-\phi } \sqrt{\phi }}{\sqrt{\pi }}+\frac{e^{-\phi }}{\sqrt{\pi } \sqrt{\phi }}$ Is there any way for this using Mathematica.

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closed as off-topic by MarcoB, m_goldberg, corey979, Edmund, Henrik Schumacher Feb 21 at 21:55

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  • 2
    $\begingroup$ There are uncountably many answers. Which one do you want? The simplest one would be f(w)=const; just find the appropriate constant (depends on n and phi). Probably though you need an f(w) that is zero for w<phi/2 to make the square root real. $\endgroup$ – Roman Feb 20 at 6:41
  • $\begingroup$ You should give some information about n, \[Phi] $\endgroup$ – Ulrich Neumann Feb 20 at 7:33
  • $\begingroup$ n is a function of $\phi$ which I know... Now I have edited the question $\endgroup$ – Hari Krishnan Feb 20 at 9:40
  • $\begingroup$ I think this question is not about Mathematica. $\endgroup$ – Roman Feb 20 at 12:27
  • $\begingroup$ Yes the problem is about mathematica.. I want to know, whether such a problem can be solved using mathematica $\endgroup$ – Hari Krishnan Feb 20 at 12:39
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I'll give it a try, but still think that this kind of question would be better off asked at the math stackexchange.

First, make an educated Ansatz for the shape of the function $f(w)$. As I have no idea what your $n(\phi)$ is, I'll use a Taylor sum with unknown coefficients c[i] to be determined:

f[w_] = Sum[c[i]*w^i, {i, 0, 5}]
(* c[0] + w c[1] + w^2 c[2] + w^3 c[3] + w^4 c[4] + w^5 c[5] *)

(use more terms for actual calculations).

Then, do the integration on the right-hand side to find

R[φ_] = Assuming[φ > 0, Integrate[f[w] Sqrt[2 w + φ], {w, 0, Sqrt[φ]}]];

The series-expansion of this integrated right-hand side $R(\phi)$ around $\phi=0$ is

Assuming[φ > 0, Series[R[φ], {φ, 0, 5}]]
(* lots of output *)

Now you can compare this to the series expansion of your $n(\phi)$:

Assuming[φ > 0, Series[n[φ], {φ, 0, 5}]]
(* some output *)

These series expansions can be matched term-by-term, which allows you to find the coefficients c[i] and thus the function $f(w)$.

If you give a formula for $n(\phi)$ I could maybe give more help; but as the question stands, this abstract discussion is all I can give.

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  • $\begingroup$ Yeah I can give the value of n(\phi). Please see the question, I have editted it $\endgroup$ – Hari Krishnan Feb 21 at 11:28
  • $\begingroup$ In that case, I don't know how to continue. Neither any polynomial w^m with m>-1 nor any Dirac delta function (or its derivatives) gives a component proportional to 1/phi as you need it in n(phi). Again, please ask on the math stackexchange. $\endgroup$ – Roman Feb 21 at 13:01

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