Analysis of Generalized Multistep Collocation Solutions for Oscillatory Volterra Integral Equations

Abstract

In this work, we introduce a class of generalized multistep collocation methods for solving oscillatory Volterra integral equations, and study two kinds of convergence analysis. The error estimate with respect to the stepsize is given based on the interpolation remainder, and the nonclassical convergence analysis with respect to oscillation is developed by investigating the asymptotic property of highly oscillatory integrals. Besides, the linear stability is analyzed with the help of generalized Schur polynomials. Several numerical tests are given to show that the numerical results coincide with our theoretical estimates.

Approximate Solution of a Class of Highly Oscillatory Integral Equations Using an Exponential Fitting Collocation Method

“…For VIEs with trigonometric kernels, the theoretical aspects of these equations, such as the existence and uniqueness of solutions and the high-oscillation properties of their solution, have been studied by Brunner et al [31,32]. In addition, several numerical methods have been proposed to approximate the solution of mentioned equations such as collocation method, Filon-type method, generalized multistep method, EF method, meshless collocation methods, and radial basis functions method [33][34][35][36][37][38][39][40][41][42][43]. Te accuracy of the proposed methods depends heavily on the evaluation of the highly oscillatory integrals in the discretization method and the approximation of the solution of the underlying problem by a suitable function.…”

Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels

“…Ma, Fang and Xiang [23] studied the rate of convergence of the direct Filon method for Volterra integral equations with Fourier kernels. Liu and Ma [24] introduced a kind of generalized multi-step collocation method for Volterra integral equations with highly oscillatory Fourier kernels, and studied two convergence analyses. Research into solving the Volterra equation with time and space variables is also quite well established.…”

Preface to “Mathematical Methods, Modelling and Applications”

“…Papers [4][5][6] are linked to step (ii). All the remaining papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] are related to step (iii) covering a wide spectrum of methods, deterministic and random, algebraic and differentials, in different fields Hydrodynamic, Physics, Health Sciences.…”