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This note is concerned with the problem of determining approximate solutions of Fred- holm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equa- tions is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail.

1. Introduction

Integral equations can be viewed as equations which are results of transformation of points in a given vector space of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed. Recently, Mandal and Bhattacharya [1] has described a special approximate method of solution of Fredholm integral equations by using Bernstein polynomials which suits the integral equations associated with function spaces spanned by polynomials only.

Varieties of integral equations have been solved numerically in recent times by several workers, utilizing various approximate methods (see Mandal and Bhattacharya [1], Chakrabarti et al. [2], Chakrabarti and Mandal [3], Golberg and Chen [4], Kanwal [5], Mandal and Bera [6] and Polyanin and Manzhirov [7]).

In the present note, we have developed a straightforward method involving expansion of the unknown function of a Fred-holm integral equation of the second kind in terms of polynomials ${x^j}_{j=0}^n$ and obtained an approximate solution of the given integral equation by the use of the method of least-squares. Simple illustrative examples have been dealt with.

We consider here the problem of solving approximately the integral equation of the form

$L\phi = f, \tag{1.1}$

with L being an integral operator of the type

$L\phi(x) = \phi(x) + \int_{\alpha}^{\beta} k(x, t)\phi(t)dt, \quad (\alpha < x < \beta), \tag{1.2}$

where $\phi(t)$ is an unknown square-integrable function to be determined, $k(x, t)$ is the known kernel which is a continuous and square integrable function, and $f(x)$ is a known square-integrable function. We will assume that the integral Eq. (1.1) possesses a unique solution.

Recently, the integral equations of the above type have been solved approximately by Mandal and Bhattacharya [1], by using the expansion of the solution function $\phi(x)$, in the form

Corresponding author. E-mail addresses: alok@math.iisc.ernet.in (A. Chakrabarti), scmartha@math.iisc.ernet.in, scmartha@gmail.com (S.C. Martha).