A Nyström interpolant for some weakly singular nonlinear Volterra integral equations

Analytical and computational methods for a class of nonlinear singular integral equations

“…Then any standard numerical scheme can be applied to the second kind integral equation (36). Let us define a mesh 0 = t 0 < t 1 …”

“…The numerical study of (8) in the case when μ = 1/2, m = 4 was first carried out in [12] and [13]. Other studies, including analytical ones, mainly focused on the direct treatment of (9), for those values of the parameters, were carried out in [7], [9], [26], [8], while in [25], general product integration methods for equation (9) with any μ > 0 were analysed; see also [1], where a Nystrom-type method was considered and comparative results were obtained for some particular examples of (8) and (9).…”

The Jacobi Collocation Method for a Class of Nonlinear Volterra Integral Equations with Weakly Singular Kernel

“…A collocation method with graded meshes was proposed in [10]; the application of a hybrid collocation method, where the basis for the approximating space also includes some fractional powers, was considered in [22]; the use of extrapolation techniques combined with low order methods was investigated in [12]. We also refer to [2], where a Nyström-type method was applied after a smoothing transformation.…”

“…This example has been taken from [2] and the exact solution of (89) is y(z) = √ z. Thus, using the variable transformation x = s 2 and setting z = t 2 , then (89) is transformed into…”

Galerkin and multi-Galerkin methods for weakly singular Volterra–Hammerstein integral equations and their convergence analysis

“…Generally, it is not easy to obtain the exact solution of Volterra-Hammerstein integral equation with weakly singular kernel; hence, we need some numerical methods to find the numerical approximate solutions of these equations. There are various numerical methods to find the numerical approximate solutions of these integral equations, which are documented (see, e.g., Anselone and Davis 1971;Baratella 2013;Brunner 1983Brunner , 1985bBrunner et al 1999;Cao et al 2003;Long et al 2009;Orsi 1996;Rebelo and Diogo 2010;Tao and Yong 2006). The commonly used approximation methods are Galerkin, collocation, Nyström, product integration methods, etc.…”