The Solution of Integral Equations

If the solution of an integral equation can be expanded in the form of a Chebyshev series, the equation can be transformed into an infinite set of algebraic equations in which the unknowns are the coefficients of the Chebyshev series. The algebraic equations are solved by standard iterative procedures, in which it is not necessary to determine beforehand how many coefficients are significant. The method is applicable to equations of either Fredholm or Volterra types. Introduction. The solution of integral equations in Chebyshev series has been the subject of two papers by Elliott [1], [2]. Elliott's method is essentially a collocation method, and it is necessary to decide in advance how many terms in the Chebyshev series are likely to be significant. The method suggested here avoids this difficulty; in many respects it is similar to the method for differential equations given in an earlier paper by the present author [3]. Cons:der first the Fredholm equation (1) y(x) = F(x) + xf K(x,S)y(!;)dS. J —i It is assumed that the variables have been suitably transformed so as to reduce the range of integration to (—1, 1). Let