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The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations


Authors: Hermann Brunner, Arvet Pedas and Gennadi Vainikko
Journal: Math. Comp. 68 (1999), 1079-1095
MSC (1991): Primary 65R20, 45E10, 45B05
DOI: https://doi.org/10.1090/S0025-5718-99-01073-X
Published electronically: February 8, 1999
MathSciNet review: 1642797
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Abstract: Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.


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  • 1. C. de Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978. MR 80a:65027
  • 2. H. Brunner, Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer. Anal., 20 (1983), 1106-1119. MR 85d:65069
  • 3. H. Brunner, The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comp. 45 (1985), 417-437. MR 87b:65223
  • 4. H. Brunner, The approximate solution of Volterra equations with nonsmooth solutions, Utilitas Math., 27 (1985), 57-95. MR 87b:45027
  • 5. H. Brunner, Collocation methods for one-dimensional Fredholm and Volterra integral equations, in The State of the Art in Numerical Analysis (A. Iserles and M.J.D. Powell, eds.), Clarendon Press, Oxford, 1987: pp. 563-600. MR 89m:65112
  • 6. H. Brunner, and P. J. van der Houwen, The Numerical Solution of Volterra Equations, North-Holland, Amsterdam, 1986. MR 88g:65136
  • 7. G. A. Chandler, Superconvergence of numerical methods to second kind integral equations. PhD Thesis, 1979, Australian National University, Canberra.
  • 8. I. G. Graham, Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels, J. Integral Equations, 4 (1982), 1-30. MR 83e:45006
  • 9. I. G. Graham, Galerkin methods for second kind integral equations with singularities. Math. Comp., 1982, 39, 519-533. MR 84d:65090
  • 10. W. Hackbusch, Integralgleichungen. Theorie und Numerik, Teubner, Stuttgart, 1989. MR 90g:45001
  • 11. F. de Hoog and R. Weiss, On the solution of a Volterra integral equation with a weakly singular kernel, SIAM J. Math Anal., 4 (1973), 561-573. MR 49:1818
  • 12. H. Kaneko, R. Noren and Y. Xu, Regularity of the solution of Hammerstein equations with weakly singular kernel, Integral Equations Operator Theory, 13 (1990), 660-670. MR 92a:45002
  • 13. H. Kaneko, R. Noren and Y. Xu, Numerical solutions for weakly singular Hammerstein equations and their superconvergence, J. Integral Equations Appl. 4, (1992), 391-406. MR 93j:65223
  • 14. R. Kangro, On the smoothness of solutions to an integral equation with a kernel having a singularity on a curve, Acta et comm. Univ. Tartuensis 913 (1990), 24-37. MR 92e:45002
  • 15. U. Kangro, The smoothness of the solution of a two- dimensional integral equation with logarithmic kernel. Proc. Eston. Acad. Sci., Phys., Math 39 (1990), 196-204 (in Russian). MR 92a:45012
  • 16. U. Kangro, The smoothness of the solution to a two-dimensional integral equation with logarithmic kernel, Z. Anal. Anwendungen, 12 (1993), 305-318. MR 94i:45010
  • 17. J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin, 1972. MR 50:2670
  • 18. J. E. Logan, The approximate solution of Volterra integral equations of the second kind, PhD Thesis, 1976, University of Iowa, Iowa City.
  • 19. Ch. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comp., 41 (1983), 87-102. MR 85a:65178
  • 20. R. K. Miller and A. Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal., 2 (1971), 242-258. MR 44:4465
  • 21. A. Pedas, On the solution of integral equations with logarithmic kernel by the first kind spline collocation method, Tartu Riiklik Ülikool, Toimetised (Acta et comm. Univ. Tartuensis), 431 (1977), 130-146 (in Russian). MR 58:3589
  • 22. A. Pedas, On the smoothness of the solution of an integral equation with a weakly singular kernel, Tartu Riiklik Ülikool, Toimetised (Acta et comm. Univ. Tartuensis) 492 (1979), 56-68 (in Russian). MR 82b:45010
  • 23. A. Pedas and G. Vainikko, The smoothness of solutions to nonlinear weakly singular equations, Z. Anal. Anwendungen, 13 (1994), 463-476. MR 95e:45002
  • 24. A. Pedas and G. Vainikko, Superconvergence of piecewise polynomial collocations for nonlinear weakly singular integral equations, J. Integral Equations Appl. 9 (1997), 379-406. CMP 98:11
  • 25. J. Pitkäranta, On the differential properties of solutions to Fredholm equations with weakly singular kernels, J. Inst. Math. Phys., 1979, 24, 109-119. MR 80i:65157
  • 26. J. Pitkäranta, Estimates for derivatives of solutions to weakly singular Fredholm integral equations, SIAM J. Math. Anal., 1980, 11, 952-968. MR 81m:45006
  • 27. J. R. Rice, On the degree of convergence of nonlinear spline approximation, in: Approximation with Special Emphasis on Spline Functions (I. J. Schoenberg, ed.), Academic Press, New York, 1969: pp. 349-365. MR 42:2226
  • 28. G. R. Richter, On weakly singular integral equations with displacement kernels, J. Math. Anal. Appl., 55 (1976), 32-42. MR 53:11322
  • 29. H. J. J. te Riele, Collocation methods for weakly singular second-kind Volterra integral equations with non-smooth solution, IMA J. Numer. Anal., 2 (1982), 437-449. MR 84g:65167
  • 30. C. Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory, 2 (1979), 62-68. MR 80f:45002
  • 31. C. Schneider, Product integration for weakly singular integral equations, Math. Comp., 36, (1981), 207-213. MR 82c:65090
  • 32. L. L. Schumaker, Spline Functions: Basic Theory, Wiley, New York, 1981. MR 82j:41001
  • 33. E. Tamme, Two-grid methods for nonlinear multidimensional weakly singular integral equations, J. Integral Equations Appl., 7, (1995), 99-113. MR 96e:65080
  • 34. P. Uba, The smoothness of solution of weakly singular integral equations with a discontinuous coefficient. Proc. Eston. Acad. Sci. Phys., Math. 37, No.2 (1988), 192-203 (in Russian). MR 89g:45004
  • 35. P. Uba, A collocation method with cubic splines to the solution of a multidimensional weakly singular integral equation. Acta et comm. Univ. Tartuensis 863 (1989), 19-25. MR 91g:65315
  • 36. P. Uba, A collocation method with cubic splines for multidimensional weakly singular nonlinear integral equations, J. Integral Equations Appl., 6 (1994), 257-266. MR 95g:45013
  • 37. G. Vainikko, Piecewise polynomial approximation of a solution to multidimensional weakly singular integral equation. Acta et comm. Univ. Tartuensis 833 (1988), 19-26 (in Russian). MR 90i:65250
  • 38. G. Vainikko, Collocation methods for multidimensional weakly singular integral equations. In: Numer. Anal. and Math. Modelling. Banach Center Publ.: Warsaw, 1990, 91-105 (in Russian). MR 92a:65355
  • 39. G. Vainikko, On the smoothness of the solution of multidimensional weakly singular integral equations. Math. USSR Sbornik 68 (1991), 585-600. (Russian original 1989). MR 91a:45010
  • 40. G. Vainikko, Multidimensional Weakly Singular Integral Equations, Lecture Notes Math., Vol. 1549, Springer-Verlag, Berlin-Heidelberg-New York, 1993. MR 94i:45001
  • 41. G. Vainikko and A. Pedas, The properties of solutions of weakly singular integral equations, J. Austral. Math. Soc. Ser. B, 22 (1981) 419-430 MR 82i:45014
  • 42. G. Vainikko, A. Pedas, P. Uba Methods for Solving Weakly Singular Integral Equations, Univ. of Tartu, Tartu, 1984 (in Russian).
  • 43. G. Vainikko and P. Uba, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, J. Austral. Math. Soc. Ser. B, 22 (1981), 431-438.

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Additional Information

Hermann Brunner
Affiliation: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld., Canada A1C 5S7
Email: hermann@math.mun.ca

Arvet Pedas
Affiliation: Department of Applied Mathematics, University of Tartu, 0000 Liivi 2–206, Tartu EE2400, Estonia
Email: Arvet.Pedas@ut.ee

Gennadi Vainikko
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O.Box 1100, FIN–02015 HUT, Finland
Email: Gennadi.Vainikko@hut.fi

DOI: https://doi.org/10.1090/S0025-5718-99-01073-X
Keywords: Nonlinear weakly singular Volterra and Fredholm integral equations, piecewise polynomial collocation, graded grids, optimal order of convergence
Received by editor(s): September 2, 1997
Published electronically: February 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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