Extrapolation Method for Non-Linear Weakly Singular Volterra Integral Equation with Time Delay

Abstract

This paper proposes an extrapolation method to solve a class of non-linear weakly singular kernel Volterra integral equations with vanishing delay. After the existence and uniqueness of the solution to the original equation are proved, we combine an improved trapezoidal quadrature formula with an interpolation technique to obtain an approximate equation, and then we enhance the error accuracy of the approximate solution using the Richardson extrapolation, on the basis of the asymptotic error expansion. Simultaneously, a posteriori error estimate for the method is derived. Some illustrative examples demonstrating the efficiency of the method are given.

An Effective Local Radial Basis Function Method for Solving the Delay Volterra Integral Equation of Nonvanishing and Vanishing Types

“…In [12], proposed a sinc-collocation method for the solution of the DVIE. Some numerical methods have been used to approximate the solution of the DVIEs with different types of conditions, such as meshless methods [13], a local meshless method based on a radial basis function-finite difference [4,5], a numerical method based on a radial basis function finite-difference [14], an efficient local meshless collocation algorithm [15], the Legendre spectral collocation method [16], an extrapolation method [17], a well-conditioned Jacobi spectral Galerkin method [18], and the Chebyshev spectral method [19]. e subject of radial basis functions was first studied by Hardy [20] and applied for topographical mapping.…”