Continuous time collocation methods for Volterra-Fredholm integral equations. (English) Zbl 0662.65116

Second kind integral equations of mixed Volterra-Fredholm type \[ u(x,t)=g(x,t)+\lambda \int^{t}_{0}\int^{b}_{a}K(x,t,\xi,\tau)u(\xi,\tau)d\xi d\tau \] and their nonlinear counterparts arise in various physical and biological problems. We study existence and uniqueness of a solution, continuous time collocation, time discretization and their global and discrete convergence properties.
Reviewer: J.-P.Kauthen


65R20 Numerical methods for integral equations
45B05 Fredholm integral equations
45D05 Volterra integral equations
Full Text: DOI EuDML


[1] Ahlin, A.C.: On error bounds for Gaussian cubature. SIAM Rev.4, 25-39 (1962) · Zbl 0101.34103 · doi:10.1137/1004004
[2] Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations. SIAM J. Numer. Anal.21, 1132-1145 (1984) · Zbl 0575.65134 · doi:10.1137/0721070
[3] Brunner, H., Kauthen, J.-P.: The numerical solution of two-dimensional Volterra integral equations by collocation and iterated collocation. IMA J. Numer. Anal.9, 47-59 (1989) · Zbl 0665.65095 · doi:10.1093/imanum/9.1.47
[4] Brunner, H., Van der Houwen, P.J.: The numerical solution of Volterra equations. Amsterdam: North-Holland 1986 · Zbl 0611.65092
[5] Cerutti, J.H., Parter, S.V.: Collocation methods for parabolic partial differential equations in one space dimension. Numer. Math.26, 227-254 (1976) · Zbl 0362.65094 · doi:10.1007/BF01395944
[6] Davis, P.J., Rabinowitz, P.: Methods of numerical integration, 2nd Ed. New York: Academic Press 1984 · Zbl 0537.65020
[7] Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol.6, 109-130 (1978) · Zbl 0415.92020 · doi:10.1007/BF02450783
[8] Douglas, J. Jr., Dupont, T.: Collocation methods for parabolic equations in a single space variable. Lecture Notes in Math.385. Berlin Heidelberg New York: Springer 1974 · Zbl 0279.65097
[9] Graham, I.G.: Numerical methods for multidimensional integral equations. In: Noye, J., Fletcher, C. (eds.), Computational techniques and applications CTAC-83, pp. 335-351. Amsterdam: North-Holland 1984
[10] Hacia, L.: On approximate solving of the Fourier problems. Demonstratio Math.12, 913-922 (1979) · Zbl 0434.65110
[11] Kauthen, J.-P.: Résolution numérique des équations intégrales de Volterra dans ?2. Diploma Thesis, Univ. of Fribourg 1986
[12] Pachpatte, B.G.: On mixed Volterra-Fredholm type integral equations. Indian J. Pure Appl. Math.17, 488-496 (1986) · Zbl 0597.45012
[13] Sloan, I.H.: Improvement by iteration for compact operator equations. Math. Comput.30, 758-764 (1976) · Zbl 0343.45010 · doi:10.1090/S0025-5718-1976-0474802-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.