How to solve $y=\int_{-w/2}^{w/2} \log|x-x'|\frac{d^2y}{dx'^2}$?

Question

I'm now stucking how to solve

$$f(x)=\int_{-w/2}^{w/2} \log(|x-x'|)\frac{d^2 f(x')}{dx'^2}$$

($w$>0) for $f(x)$.

I get this self-consistent equation solving Maxwell-equation.

From symmetry, $f(x)$ can be devided into 2 region,

$$-w/2≦x≦w/2$$

and

$$|x|>w/2$$

and can be assumed $f(x)=-f(-x)$ from physical constrants.

But this is too hard for me to solve. If you have any idea how to solve this (including numerical method or special case) , please help me.

Answer 1

Making

$$ y_n(x) = \sum_{k=0}^n a_k x^k $$

with $a_0, a_1$ allocated to initial conditions, and developing in series

$$ \ln(x-z) = ((x-z)-1)-\frac{1}{2} ((x-z)-1)^2+\frac{1}{3} ((x-z)-1)^3-\frac{1}{4} ((x-z)-1)^4+\cdots+O\left(((x-z)-1)^{n}\right) $$

we can establish

$$ y_n(x)=\int_{-\frac w2}^{\frac w2}\ln(x-z)y_n''(z)dz $$

and using the $a_k$ coefficient rules we can build a sequence of approximations for $y(x) = \lim_{n\to\infty}y_n(x)$

Follows a MATHEMATICA script which makes those calculations. Attached a plot showing the convergence.

nmax = 12;
parms = {w -> 10, Subscript[a, 0] -> 0, Subscript[a, 1] -> -1};
grs = {};
y[x_, n_] := Sum[Subscript[a, k] x^k, {k, 0, n}]

For[n = 3, n <= nmax, n++,
   A = Table[Subscript[a, k], {k, 0, n}];
   sxz = Normal[Series[Log[u], {u, 1, n}]] /. u -> x - z;
   int = Integrate[sxz D[y[z, n], {z, 2}], z];
   int0 = int /. {z -> -w/2};
   intw = int /. {z -> w/2};
   pol = y[x, n] - (intw - int0);
   coefs = CoefficientList[pol, x];
   coefs0 = Take[coefs, {1, n - 1}];
   coefs0w = coefs0 /. parms;
   soly = Solve[Thread[coefs0w == 0], Take[A, {3, n + 1}]][[1]];
   y0 = y[x, n] /. soly /. parms;
   gr0 = Plot[y0, {x, -w, w} /. parms, PlotStyle -> RGBColor[(nmax - n + 1)/nmax, (nmax - n + 1)/nmax, (nmax - n + 1)/nmax]];
   AppendTo[grs, gr0]
]
Show[grs,AspectRatio -> 1]

This example has the parameters $a_0 = 0,a_1 = -1, w= 10$

NOTE

The series for $\ln(x)$ does not converges uniformly on $\mathbb{R}_+$ and as a consequence the series for $y(x)$ suffers the same problem. Also the convergence for $y_n$ is affected by the $w$ range. According with the observed behavior it seems as

$$ \lim_{n\to\infty}y_n(x) = a_0 + a_1 x $$