Algorithm 689: Discretized collocation and iterated collocation for nonlinear Volterra integral equations of the second kind

ABSTRACT

This paper describes a FORTRAN code for calculating approximate solutions to systems of nonlinear Volterra integral equations of the second kind. The algorithm is based on polynomial spline collocation, with the possibility of combination with the corresponding iterated collocation. It exploits certain local superconvergence properties for the error estimation and the stepsize strategy.

REFERENCES

• 1: A~DERSON, R. L. Problem-solving software system for mathematical and statistical FOR- TRAN programming. IMSL user's manual, 1985.
• 2: J. G. Blom , H. Brunner, The numerical solution of nonlinear volterra integral equations of the second kind by collacation and iterated collocation methods, SIAM Journal on Scientific and Statistical Computing, v.8 n.5, p.806-830, September 1, 1987 [doi>10.1137/0908068]
• 3: HETHCOTE, H. W., AND TUDOR, D. W. Integral equation models for endemic infectious diseases. J. Math. Biol. 9 (1980), 37-47.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.9 Integral Equations
          Subjects: Volterra equations

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Numerical algorithms
      G.1.2 Approximation
          Subjects: Spline and piecewise polynomial approximation

General Terms:
Algorithms, Performance, Reliability

Keywords:
(iterated) collocation, error estimations, local superconvergence, nonlinear Volterra integral equations of the second kind, variable stepsize