The relationship of degenerate kernel and projection methods on Fredholm integral equations of the second kind

Abstract

In this paper, we show that the degenerate kernel method for some cases, on the condition that the source function is approximated by the same way of producing degenerate kernel, becomes as a projection method. We consider two ways, including Lagrange interpolation and best approximation methods, of producing degenerate kernel approximations of more general Fredholm integral equation of the second kind. For these two ways, we show that the degenerate kernel method becomes as a Lagrange-collocation method and Galerkin method respectively.

A Unified Formulation of Analytical and Numerical Methods for Solving Linear Fredholm Integral Equations

“…where f r = f r (x). For similar modifications, one can look at [9,19]. Likewise, we may also express the functions {g j (x)} as partial sums of power series, namely…”

“…We formulate the given integral equation as in (19). Specifically, we replace the kernel K(x, t) and the input function f (x) by finite segments of Taylor series of degree n (r = n) about 0 in t and x, respectively, as follows…”

“…For their solution, a variety of methods have been developed; see [5][6][7][8][9] and others. Well-known classical solution techniques are the Direct Computational Method (DCM), the Degenerate Kernel Methods (DKM), the Quadrature Methods (QM), and the Projection Methods (PM); see, for example, the standard treatises [5][6][7], the traditional articles [10][11][12], and the recent papers [13][14][15][16][17][18][19][20][21]. The DCM has the advantage that it delivers the exact solution in closed form, but its application is limited to special cases where the kernel is separable (degenerate) and the integrals involved can be determined analytically.…”