A Hermite-Type Collocation Method for the Solution of an Integral Equation with a Certain Weakly Singular Kernel

A Comparative Study on Modifications of Decomposition Method

“…He also described a wide range of applications of integral equations with variable boundary, which is one of the most important factors in the development of the theory of integral equations. In recent years, many works have been focusing on the developing and applying of advanced and efficient methods for integral equations such as implicitly linear collocation methods [2], product integration method [3], Hermite-type collocation method [4] and analytical techniques such as Adomian's decomposition method [5,6], homotopy analysis method [7][8][9], homotopy perturbation method [10], the Exp-function method [11], variational iteration method [12] and the Adomian's decomposition method [13]. In this work, we investigate the performance of two modification of Adomian's decomposition method applied to nonlinear Volterra integral equations of the second kind.…”

Collocation Solutions of a Weakly Singular Volterra Integral Equation

“…For an extensive list of references on these and other approaches see [4]. Equation (1.3) has been the subject of several works for the case when µ > 1: certain classes of product integration methods based on Newton-Cotes rules were studied in [10]; Diogo et al [12] considered a fourth order Hermite-type collocation method and Lima and Diogo [17] developed an extrapolation algorithm, based on Euler's method. Recently, it was shown that general collocation methods on uniform meshes based on piecewise polynomials of degree m−1 yield convergence of order m. In [11] it was also proved that if certain derivatives of the exact solution are zero at the origin, then a higher order is attained at the mesh points by a special choice of the collocation points, like the Radau II points.…”

Numerical solvability of a class of Volterra‐Hammerstein integral equations with noncompact kernels

“…Owing to the fact that the kernel k is unbounded in (1.1), the classical arguments of the convergence analysis of the discretization method for weakly singular equations are not applicable. Diogo et al [3] transformed integral equation of the form (1.6) into the linear form of Volterra integral equation of (1.1) with special noncompact kernel…”

“…In [3], a fourth-order Hermite-type collocation method was applied to (1.1) and the analysis and construction of higher-order collocation method has been investigated in [4]. Lima and Diogo in [9] were concerned with the use of a low-order method in conjunction with extrapolation procedures.…”