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

A product integration method for the approximation of the early exercise boundary in the American option pricing problem

“…In this method, the smooth part of the kernel is interpolated in order to manage the weak singularity of the kernel. [21][22][23] In order to present the principles underlying the method, we first choose n + 1 distinct points, {t i } n i=0 in the interval [0, T] and then collocate (12) at these nodes to obtain…”

“…Taking into account the fact that the early exercise boundary has some kind of singularity near the expiry (see, eg, previous studies) and noting that this knowledge must be incorporated in the design of the numerical scheme, we consider here a one‐dimensional reformulation of Kim's integral equation proposed by Hou et al and employ a modified version of the Nyström method, called the product integration method, specifically designed to tackle this singular behavior. More precisely, we employ an approximation of the kernel based on linear barycentric rational interpolation to manage the weakly singular character of the integral equation …”

“…More precisely, we employ an approximation of the kernel based on linear barycentric rational interpolation to manage the weakly singular character of the integral equation. [21][22][23] In this respect, after a brief review of the existing numerical approaches utilized for the approximation of the early exercise boundary, we provide theoretical and numerical evidence that the product integration method based on linear barycentric rational interpolation is an efficient way to approximate the solution. In the sequel, the integral representation of the American option price and its numerical approximation will be considered, and an upper bound for the incurred error will be given.…”

Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels

“…Also, the numerical studies on the Volterra integral equation with discontinuous kernels can be found in [8,9]. Allaei et al in [10] presented an analytical and computational method for a class of nonlinear singular integral equations. Maleknejad et al in [11] proposed a new numerical approach for solving the nonlinear integral equations of Hammerstein and Volterra-Hammerstein.…”

“…In the study of many nonlinear problems in heat conduction, boundary-layer heat transfer, chemical kinetics, and superfluidity, we are often led to singular Volterra integral equations for which real answers are hard to find [10]. In this article, we use efficient functions such as Genocchi polynomials and their operational matrices to solve nonlinear Volterra integral equations with weakly singular kernels of the following form:…”