Numerical methods for Fredholm integral equations with singular right-hand sides

Abstract

Fredholm integral equations with the right-hand side having singularities at the endpoints are considered. The singularities are moved into the kernel that is subsequently regularized by a suitable one-to-one map. The Nyström method is applied to the regularized equation. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.

Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules

“…In the next theorem, by exploiting the results introduced in Sect. 3, we extend the well-known stability and convergence results, valid for the Nyström method based on the Gauss rule [6,9,20], to the Nyström method based on the anti-Gauss quadrature formula.…”

“…Conversely, if (37)-(38) are solutions of (32)- (33), then the coefficients a j andã j are solutions of systems (34) and (35), respectively. This is the well-known Nyström method developed for the first time in 1930 [24] and widely analyzed in terms of convergence and stability in different function spaces, according to the smoothness properties of the known functions; see [2,6,9,13,17].…”

“…(1) (collocation methods, projection methods, Galerkin methods, etc.) and have been extensively investigated in terms of stability and convergence in suitable function spaces, also according to the smoothness properties of the kernel k and the right-hand side g; see [2,6,7,9,17,25,[29][30][31][32].…”

On the stability of a modified Nyström method for Mellin convolution equations in weighted spaces

“…and from the hypothesis (12) we can deduce (29). Finally, since the Gauss-Jacobi quadrature rule ( 9) is convergent for any continuous function and the kernel h(x, y) is continuous for 0 ≤ x, y ≤ 1 in virtue of ( 12) and ( 15), the collectively compactness of the sequence {H m } m and (30) follow (see, for instance, [1,13]).…”

“…We study the integral equation in a suitable weighted space of continuous functions. Then, following an idea in [12], we consider an equivalent Mellin integral equation whose unknown is at least a continuous function. Finally, in order to approximate its solution, we apply a modified Nyström method.…”