Discretization of Volterra integral equations of the first kind (II)

SummaryIn the present paper integral equations of the first kind associated with strictly monotone Volterra integral operators are solved by projecting the exact solution of such an equation into the spaceSm(−1)(ZN) of piecewise polynomials of degreem≧0, possessing jump discontinuities on the setZN of knots. Since the majority of “direct” one-step methods (including the higher-order block methods) result from particular discretizations of the moment integrals occuring in the above projection method we obtain a unified convergence analysis for these methods; in addition, the above approach yields the tools to deal with the question of the connection between the location of the collocation points used to determine the projection inSm(−1)(ZN) and the order of convergence of the method.

[1]  H. Brunner.  Global solution of the generalized Abel integral equation by implicit interpolation , 1974 .

[2]  S. P. Nørsett.  C-Polynomials for rational approximation to the exponential function , 1975 .

[3]  Owe Axelsson,et al.  A class ofA-stable methods , 1969 .

[4]  C. Baker,et al.  Stability Regions in the Numerical Treatment of Volterra Integral Equations , 1978 .

[5]  Richard Weiss,et al.  On the solution of volterra integral equations of the first kind , 1973 .

[6]  C. J. Gladwin,et al.  Quadrature rule methods for Volterra integral equations of the first kind , 1979 .

[7]  Robert S. Anderssen,et al.  A product integration method for a class of singular first kind Volterra equations , 1971 .

[8]  Richard Weiss,et al.  High Order Methods for Volterra Integral Equations of the First Kind , 1973 .

[9]  Richard Weiss,et al.  Product Integration for the Generalized Abel Equation , 1972 .

[10]  Hermann Brunner.  Discretization of Volterra integral equations of the first kind , 1977 .

[11]  L. Garey,et al.  Solving Nonlinear Second Kind Volterra Equations by Modified Increment Methods , 1975 .

[12]  H. A. Watts,et al.  A-stable block implicit one-step methods , 1972 .

[13]  H. Brunner.  An approximation property of certain nonlinear Volterra integral operators , 1976 .

[14]  H. Brunner.  The approximate solution of linear and nonlinear first-kind integral equations of Volterra type , 1976 .

[15]  F. Hoog,et al.  Implicit Runge-Kutta methods for second kind Volterra integral equations , 1974 .

[16]  W. Boland,et al.  Product type quadrature formulas , 1971 .

[17]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[18]  K. Wright,et al.  Some relationships between implicit Runge-Kutta, collocation and Lanczosτ methods, and their stability properties , 1970 .

[19]  H. Keller,et al.  Analysis of Numerical Methods , 1969 .

[20]  A. Young,et al.  The application of approximate product-integration to the numerical solution of integral equations , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[21]  Peter Linz.  Product integration methods for Volterra integral equations of the first kind , 1971 .

[22]  S. Venit,et al.  Numerical Analysis: A Second Course. , 1974 .

[23]  Hing-Sum Hung.  The Numerical Solution of Differential and Integral Equations by Spline Functions. , 1970 .

[24]  Peter Linz,et al.  Numerical methods for Volterra integral equations of the first kind , 1969, Comput. J..

[25]  P. M. Prenter.  Splines and variational methods , 1975 .

[26]  J. Ortega.  Numerical Analysis: A Second Course , 1974 .