Legendre Polynomials The Legendre polynomials form an -orthogonal set of polynomials. You will see below why orthogonal polynomials make particularly good choices for approximation. In this section, we are going to write m-files to generate the Legendre polynomials and we are going to confirm that they form an orthogonal set in . Throughout this section, we will be representing polynomials as vectors of coefficients, in the usual way in Matlab. The Legendre polyonomials are a basis for the set of all polynomials, just as the usual monomial powers of are. They are appropriate for use on the interval [-1,1] because they are orthogonal when considered as members of . Polynomials that are orthogonal are discussed by Quarteroni, Sacco, and Saleri in Chapter 10, with Legendre polynomials discussed in Section 10.1.2. The first few Legendre polynomials are:

B. Legendre polynomials method

Legendre series approximation

The most common method of solution of integral equation is by the use of finite differences. In [3] Fox and Goodwin use the Gregory quadrature formula for the evaluation the integral equations. In this research we try to find the numerical solution of non-singular linear integral equations by the direct expansion of the unknown function, f(x)into a series of Chebyshev polynomials of the first kind and into a series of Legendre polynomials (as discussed by Elliott [4]). Then we use given integral equation to obtaining coefficient. In [5] we see the properties of the Chebyshev polynomials together produce an approximating polynomial which minimizes error in its application. This is different from the least squares approximation where the sum of the squares of the errors is minimized; the maximum error itself can be quite large. In the Chebyshev approximation, the average error can be large but the maximum error is minimized. Chebyshev approximations of a function are sometimes said to be mini-max approximations of the function. Chebyshev polynomials form a special class of polynomials especially suited for approximating other functions. They are used in many areas of numerical analysis. It is assumed that expansions ofgiven functions can be found and for functions whose expansions cannot be found in given manners, some curve fitting technique can be used. The Legendre polynomials [7] are one of the important sequences of orthogonal polynomials which has been extensively investigated and applied in interpolation and approximation theory, numerical integration, the solution of the second-and fourth-order elliptic equations, computational fluid dynamics, etc.It is not only powerful tool for the approximation of functions that are difficultto compute, but also essential ingredient of numerical integration and approximate solution of differential and integral equations. The Legendre spectral methods has excellent error properties in the approximation of a smooth function. The orthogonal polynomialexpansion occurs in a wide range of practical problems and applications and plays an important role in many fields of mathematics and physics.