Solving an integral equation analytically [closed]

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.

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

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If this question can be reworded to fit the rules in the help center, please edit the question.

• 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. – Roman Feb 20 at 6:41
• You should give some information about n, \[Phi] – Ulrich Neumann Feb 20 at 7:33
• n is a function of $\phi$ which I know... Now I have edited the question – Hari Krishnan Feb 20 at 9:40
• I think this question is not about Mathematica. – Roman Feb 20 at 12:27
• Yes the problem is about mathematica.. I want to know, whether such a problem can be solved using mathematica – Hari Krishnan Feb 20 at 12:39

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 + w c + w^2 c + w^3 c + w^4 c + w^5 c *)

(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.

• Yeah I can give the value of n(\phi). Please see the question, I have editted it – Hari Krishnan Feb 21 at 11:28
• 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. – Roman Feb 21 at 13:01