Solving volterra integral equations of the second kind by sigmoidal functions approximation

Journal of Integral Equations and Applications

Volumn 25, Issue 2, 2013, Pages 193-222

  Costarelli, Danilo ^a   Spigler, Renato ^a  

^a ROMA TRE UNIVERSITY   (Italy)

Author keywords

Collocation methods; Linear Volterra integral equations; Nonlinear Volterra integral equations; Sigmoidal functions; Unit step functions

Indexed keywords

EID: 84883037302     PISSN: 08973962     EISSN: None     Source Type: Journal    
DOI: 10.1216/JIE-2013-25-2-193     Document Type: Article

References (26)

1. A.V. Adrianov, An analog of the Jackson-Nikol'skij theorem on the approximation by superposition of sigmoidal functions, Math. Notes 59 (1996), 661-664 (transl. of Matem. Z., Russian Academy of Sciences).
2. G. Albano and V. Giorno, On the first exit time problem for a Gompertz-type tumor growth, Lect. Notes Comp. Sci. 5717/2009 (2009), 113-120.
3. K.E. Atkinson, The numerical solution of integral equations of the second kind, CambridgeMono. Appl. Comput. Math. 4, Cambridge University Press, Cambridge, 1997.
4. A.R. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Infor. Theor. 39 (1993), 930-945.
5. K.S. Berenhaut, D.C. Morton and P.T. Fletcher, Bounds for inverses of triangular Toeplitz matrices, SIAM. J. Matrix Anal. Appl. 27 (2005), 212-217.
6. F. Brauer and C. Castillo Chavez, Mathematical models in population biology and epidemiology, Springer, New York, 2001.
7. H. Brunner, Collocation methods for Volterra integral and related functional equations, Cambr. Mono. Appl. Comput. Math. 15, Cambridge University Press, Cambridge, 2004.
8. Debao Chen, Degree of approximation by superpositions of a sigmoidal function, Approx. Theory Appl. 9 (1993), 17-28.
9. H. Chen, T. Chen and R. Liu, A constructive proof of Cybenko's approximation theorem and its extension, in Computing science and statistics, C. Page and R. LePage, eds., 1990.
10. H. Chen, T. Chen and R. Liu, A constructive proof and an extension of Cybenko's approximation theorem, in Computing science and statistics, Springer-Verlag, New York, 1992.
11. E.S. Chumerina, Choice of optimal strategy of tumor chemotherapy in Gompertz model, J. Comput. Systems Sci. Inter. 48 (2009), 325-331.
12. D. Costarelli and R. Spigler, Constructive approximation by superposition of sigmoidal functions, Anal. Theor. Appl. 29 (2013), 169-196.
13. G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Contr. Sig. Syst. 2 (1989), 303-314.
14. B. Gao and Y. Xu, Univariant approximation by superpositions of a sigmoidal function, J. Math. Anal. Appl. 178 (1993), 221-226.
15. N. Hahm and B. Hong, Approximation order to a function in C(R) by superposition of a sigmoidal function, Appl. Math. Lett. 15 (2002), 591-597.
16. N. Hritonenko and Y. Yatsenko, Mathematical modelling in economics, ecology and the environment, Science Press, Beijing, 2006.
17. A.J. Jerry, Introduction to integral equations with applications, A series of monographs and textbooks, Pure Appl. Math., Marcel Dekker, Inc., New York, 1985.
18. L.K. Jones, Constructive approximations for neural networks by sigmoidal functions, Proc. IEEE 78 (1990), 1586-1589.
19. G. Lewicki and G. Marino, Approximation by superpositions of a sigmoidal function, Z. Anal. Anwend. 22 (2003), 463-470.
20. G. Lewicki and G. Marino, Approximation of functions of finite variation by superpositions of a sigmoidal function, Appl. Math. Lett. 17 (2004), 1147-1152.
21. K. Maleknejad, E. Hashemizadeh and R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernstein's approximation, Comm. Nonlin. Sci. Numer. Simul. 16 (2011), 647-655.
22. K. Maleknejad and E. Najafi, Numerical solution of nonlinear Volterra integral equations using the idea of quasilinearization, Comm. Nonlin. Sci. Numer. Simul. 16 (2011), 93-100.
23. E.A. Mantysaari, M.L. Sevon Aimonen, I. Stranden and K. Vuori, Estimation of non-linear growth models by linearization: A simulation study using a Gompertz function, Genet. Sel. Evol. 38 (2006), 343-358.
24. H.N. Mhaskar and C.A. Micchelli, Approximation by superposition of sigmoidal and radial basis functions, Adv. Appl. Math. 13 (1992), 350-373.
25. A. Vecchio, A bound for the inverse of a lower triangular Toeplitz matrix, SIAM J. Matrix Anal. Appl. 24 (2003), 1167-1174.
26. S.A. Yousefi, Numerical solution of Abel's integral equation by using Legendre wavelets, Appl. Math. Comput. 175 (2006), 574-580.