## Journal of Integral Equations and Applications

### On the Regularity of Solutions to Volterra Functional Integro-Differential Equations with Weakly Singular Kernels

#### Article information

Source
J. Integral Equations Applications, Volume 18, Number 2 (2006), 143-167.

Dates
First available in Project Euclid: 5 June 2007

https://projecteuclid.org/euclid.jiea/1181075377

Digital Object Identifier
doi:10.1216/jiea/1181075377

Mathematical Reviews number (MathSciNet)
MR2273346

Zentralblatt MATH identifier
1147.45007

#### Citation

Brunner, Hermann; Ma, Jingtang. On the Regularity of Solutions to Volterra Functional Integro-Differential Equations with Weakly Singular Kernels. J. Integral Equations Applications 18 (2006), no. 2, 143--167. doi:10.1216/jiea/1181075377. https://projecteuclid.org/euclid.jiea/1181075377

#### References

• A. Bellen and M. Zennaro, Numerical methods for delay differential equations, Oxford Univ. Press, Oxford, 2003.
• R. Bellman and K.L. Cooke, Differential-difference equations, Academic Press, New York, 1963.
• H. Brunner, The numerical solution of weakly singular first-kind Volterra integral equations with delay arguments, Proc. Estonian Acad. Sci. Phys. Math. 48 (1999), 90-100.
• --------, The numerical analysis of functional integral and integro-differential equations of Volterra type, Acta Numer. (2004), 55-145.
• --------, Collocation methods for Volterra integral and related functional equations, Cambridge Monogr. Appl. Comput. Math. 15, Cambridge Univ. Press, Cambridge, 2004.
• --------, The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays, Comm. Pure Appl. Anal. 5 (2006), 261-276.
• H. Brunner, A. Pedas and G. Vainikko, Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal. 39 (2001), 957-982.
• --------, Spline collocation method for linear Volterra integro-differential equations with weakly singular kernels, BIT 41 (2001), 891-900.
• H. Brunner and D. Schötzau, $hp$-discontinuous Galerkin time-stepping for Volterra integro-differential equations, SIAM J. Numer. Anal. 44 (2006), 224-245.
• H. Brunner and W. Zhang, Primary discontinuities in solutions for delay integro-differential equations, Methods Appl. Anal. 6 (1999), 525-533.
• J.A. Burns, T.L. Herdman and J. Turi, Neutral functional integro-differential equations with weakly singular kernels, J. Math. Anal. Appl. 145 (1990), 371-401.
• Y. Cao, T.L. Herdman and Y. Xu, A hybrid collocation method for Volterra integral equations with weakly singular kernels, SIAM J. Numer. Anal. 41 (2003), 364-381.
• K. Deimling, Nonlinear Volterra integral equations of the first kind, Nonlinear Anal. 25 (1995), 951-957.
• M. de Gee, Smoothness of solutions of functional differential equations, J. Math. Anal. Appl. 107 (1985), 103-121.
• L.E. El'sgol'ts and S.B. Norkin, Introduction to the theory and application of differential equations with deviating arguments, Academic Press, New York, 1973.
• J.K. Hale, Theory of functional differential equations, Springer-Verlag, New York, 1977.
• J.K. Hale and S.M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993.
• T.L. Herdman and J. Turi, On the solution of a class of integral equations arising in unsteady aerodynamics, in Differential equations: Stability and control (S. Elaydi, ed.), Lecture Notes Pure Appl. Math., vol. 127, Marcel Dekker, New York, 1991, pp. 241-248.
• K. Ito and J. Turi, Numerical methods for a class of singular integro-differential equations based on semigroup approximation, SIAM J. Numer. Anal. 28 (1991), 1698-1722.
• F. Kappel and K.P. Zhang, On neutral functional differential equations with nonatomic difference operator, J. Math. Anal. Appl. 113 (1986), 311-343.
• Ch. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comp. 41 (1983), 87-102.
• K.W. Neves and A. Feldstein, Characterization of jump discontinuities for state dependent delay differential equations, J. Math. Anal. Appl. 56 (1976), 689-707.
• T. Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math. 61 (1992), 373-382.
• --------, A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal. 13 (1993), 93-99.
• D.R. Willé and C.T.H. Baker, The tracking of derivative discontinuities in systems of delay differential equations, Appl. Numer. Math. 9 (1992), 209-222.