Does anyone know how and when and where used Legendre polynomial? I'd like approach to Legendre polynomial.
But i don't make concept with Legendre polynomial.
So if you know where and when does L.P used and how L.P used, would you please let me know about this? To more understand it.

Hmm.. specifically in the application.
Legendre polynomials are orthogonal on the interval $[-1,1]$ and are therefore useful for approximating functions on a finite interval in a least squares sense. I.e., if you are looking for a polynomial $p^*_K(t)$ of degree $K$ which approximates a given function $f(t)$ on the interval $[-1,1]$ - other intervals are possible by shifting and scaling - by minimizing the error

$$\epsilon=\int_{-1}^1[f(t)-p_K(t)]^2dt$$

then the optimal polynomial $p^*_K(t)$ is given by

$$p^*_K(t)=\sum_{k=0}^Ka_k\bar{L}_k(t)\tag{1}$$

where

$$\bar{L}_k(t)=\sqrt{\frac{2k+1}{2}}L_k(t),\quad k\in\mathbf{N}$$

are the normalized Legendre polynomials $L_k(t)$ (i.e., $\bar{L}_k(t)$ are orthonormal, not only orthogonal). Because of the orthonormality of $\bar{L}_k(t)$ the coefficients $a_k$ in (1) can be easily found by computing

$$a_k=\int_{-1}^1\bar{L}_k(t)x(t)dt$$