A parallel iterated correction algorithm for the numerical solution of Volterra equations

Abstract We present an algorithm by which the discrete numerical solution of a Volterra integral equation (with non-singular kernel) obtained from a classical Newton-Cotes-type approximation of its integral part is corrected in an iterative way. The correction mechanism, which is based upon the use of non-purely polynomial quadrature rules, turns out to be parallelizable. A parallel algorithm which is suited for a small number of processors is presented and has been implemented on a ring of transputers. Its efficiency is illustrated and analyzed by means of three test equations.

[1]  Graeme Fairweather,et al.  A Box Method for a Nonlinear Equation of Population Dynamics , 1991 .

[2]  R. Taylor,et al.  The Numerical Treatment of Integral Equations , 1978 .

[3]  L. Delves,et al.  Computational methods for integral equations: Frontmatter , 1985 .

[4]  Ulf Torsten Ehrenmark,et al.  A three-point formula for numerical quadrature of oscillatory integrals with variable frequency , 1988 .

[5]  J. C. López-Marcos,et al.  An explicit extrapolated box scheme for the Gurtin-MacCamy equation , 1994 .

[6]  G. Webb.  Theory of Nonlinear Age-Dependent Population Dynamics , 1985 .

[7]  Numerical solutions to equations modelling nonlinearly interacting age-dependent populations , 1990 .

[8]  H. De Meyer,et al.  On a class of modified Newton-Cotes quadrature formulae based upon mixed-type interpolation , 1990 .

[9]  M. J. Maron,et al.  Numerical Analysis: A Practical Approach , 1982 .

[10]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[11]  Mimmo Iannelli,et al.  Endemic thresholds and stability in a class of age-structured epidemics , 1988 .

[12]  F A Milner,et al.  Rapidly converging numerical algorithms for models of population dynamics , 1992, Journal of mathematical biology.

[13]  Chung-Ki Cho,et al.  Second-order accurate difference methods for a one-sex model of population dynamics , 1993 .

[14]  Carl Erik Fröberg,et al.  Introduction to Numerical Analysis , 1969 .

[15]  H. De Meyer,et al.  On the error estimation for a mixed type of interpolation , 1990 .

[16]  H. De Meyer,et al.  On a new type of mixed interpolation , 1990 .

[17]  Horst R. Thieme,et al.  Global behavior of an age-structured epidemic model , 1991 .

[18]  M. Iannelli,et al.  Analytical and numerical results for the age-structured S-I-S epidemic model with mixed inter-intracohort transmission , 1992 .

[19]  J. C. López-Marcos.  An upwind scheme for a nonlinear hyperbolic integro-differential equation with integral boundary condition , 1991 .

[20]  Jim Douglas,et al.  Numerical methods for a model of population dynamics , 1987 .

[21]  Morton E. Gurtin,et al.  Non-linear age-dependent population dynamics , 1974 .

[22]  Chichia Chiu.  A numerical method for nonlinear age dependent population models , 1990 .