In polynomial approximation, Legendre polynomials are not better than Chebychev ones in the sense that the interpolation error is minimum only when you select nodes as zeros of Chebychev polynomials. However, error in Legendre polynomial is less. Both them do not exhibit Runge phenomenon which generally happens in case of equispaced points interpolation.The Legendre polynomials are useful in finding solutions of Laplace equation in Spherical coordinates.

Respected Sir

Thanks a lot for your answer but I am not using interpolation I use galerkin method.

Now can you say something which one is better?

Are you using spectral method? Our experience is again chebychev polynomials are better.But Legendre polynomial are easy to apply for computation purpose.

no sir my area of research is singular integral equation, there i am using.polynomial approximation.

Dear @Vaishali, please see my answer at the link:

https://www.researchgate.net/post/What_is_the_most_frequently_used_orthogonal_polynomial_over-1_1#54b004e5cf57d7151e8b4632

I hope this helps.