scholarly journals Analysis of Generalized Multistep Collocation Solutions for Oscillatory Volterra Integral Equations

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2004
Author(s):  
Hao Chen ◽  
Ling Liu ◽  
Junjie Ma
Keyword(s):  
Error Estimate ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Oscillatory Integrals ◽  
Collocation Methods ◽  
Schur Polynomials ◽  
Numerical Tests ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals

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.

scholarly journals Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 168 ◽  
Author(s):  
Chunhua Fang ◽  
Guo He ◽  
Shuhuang Xiang
Keyword(s):  
Integral Equations ◽  
Collocation Method ◽  
Bessel Functions ◽  
Hermite Interpolation ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Fast Computation ◽  
Highly Oscillatory ◽  
Linear Volterra Integral Equations ◽  
Highly Oscillatory Integrals

In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The other one is piecewise Hermite collocation method, which used a two-points Hermite interpolation in each subinterval. These two methods can calculate the approximate value of function value and derivative value simultaneously. Both methods are constructed easily and implemented well by the fast computation of highly oscillatory integrals involving Bessel functions. Under some conditions, the asymptotic convergence order with respect to oscillatory factor of these two methods are established, which are higher than the existing results. Some numerical experiments are included to show efficiency of these two methods.

Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

2017 ◽  
Vol 9 (6) ◽  
pp. 1506-1524
Author(s):  
Xiong Liu ◽  
Yanping Chen
Keyword(s):  
Convergence Analysis ◽  
Integral Equations ◽  
Lebesgue Constant ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Singular Kernel ◽  
Quadrature Formulas ◽  
Weakly Singular Kernel ◽  
Chebyshev Collocation ◽  
Weakly Singular

AbstractIn this paper, a Chebyshev-collocation spectral method is developed for Volterra integral equations (VIEs) of second kind with weakly singular kernel. We first change the equation into an equivalent VIE so that the solution of the new equation possesses better regularity. The integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation points. The convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials, approximation theory for orthogonal polynomials, and the operator theory. The spectral rate of convergence for the proposed method is established in theL∞-norm and weightedL2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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