I am trying to code the following integral equation to find the solution numerically using Mathematica. In fact the exact solution is `x^2 (1 - x)`

.

First we define the following functions:

`phi[x_]:=Piecewise[{{1, 0 <= x < 1}}, 0] `

`f[x_] := 1/ 1155 (112 (-1 + x)^(3/4) + x (144 (-1 + x)^(3/4) + x (1155 + 256 (-1 + x)^(3/4) - 1280 x^(3/4) - (1155 + 512 (-1 + x)^(3/4)) x + 1024 x^(7/4)))); exactsoln[x_] := x^2 (1 - x); `

I am trying to solve the following integral equation for `u (x)`

(numerically). where

`u[x] - Integrate[(x - t)^(-1/4)*u[t], {t, 0, x}] - Integrate[(x - t)^(-1/4)*u[t], {t, 0, 1}] = f[x]; `

where `f[x]`

is defined as above. Here is the numerical scheme. Our goal is to find the coefficients `c[j, k]`

. We approximate the solution u by the approximated solution `\approx[x]`

which can be written as

`approxsoln[x_, n_] := Sum[c[j, k]*psijk[x, j, k], {j, -n, n}, {k, -2^n, 2^n - 1}] `

If you plug the function `approxsoln[x,n]`

in the integral equation, we will end up by

`Sum[c[j, k]*(psijk[x, j, k] - Integrate[(x - t)^(-1/4)* psijk[t, j, k], {t, 0, x}] - Integrate[(x - t)^(-1/4)* psijk[t, j, k], {t, 0, 1}]), {j, -n, n}, {k, -2^n, 2^n - 1}]; `

Now every thing is known except the coefficients `c[j,k]`

. We need to use a suitable subdivision, may be divide the interval `[0,1]`

into `(2n+1) 2^(n+1)`

points (to meet the size of the truncated sum in `approxsoln[x,n])`

to be used in the equation to construct a system of linear equation, to find these coefficients in order to find the approximated solution (`approxsoln[x,n]`

) defined above. Is there any way to code this problem using Mathematica. I think it is worth to try for `n=10`

first.