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Approximating a Retarded-Advanced Differential Equation Using Radial Basis Functions

  • M. Filomena TeodoroEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10408)

Abstract

In last years we have got the approximation of the solution of a linear mixed type functional differential equation, considering the autonomous and non-autonomous case by collocation, least squares and finite element methods considering a polynomial basis. The present work introduces a numerical scheme using collocation and radial basis functions to solve numerically the non-linear mixed type equation with symmetric delay and advance. The results are similar using collocation, B-splines and exponential radial functions. The preliminary results are promising, but more simulations using different basis of radial functions are needed.

Keywords

Mixed type functional differential equation Numerical approach Numerical solution Radial basis function 

Notes

Acknowledgements

This work was supported by Portuguese funds through the Center for Computational and Stochastic Mathematics (CEMAT), The Portuguese Foundation for Science and Technology (FCT), University of Lisbon, Portugal, project UID/Multi-/04621/2013, and Center of Naval Research (CINAV), Naval Academy, Portuguese Navy, Portugal.

References

  1. 1.
    Abell, K., Elmer, C., Humphries, A., Vleck, E.V.: Computation of mixed type functional differential boundary value problems. SIADS - SIAM J. Appl. Dyn. Syst. 4, 755–781 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alvarez-Rodriguez, U., Perez-Leija, A., Egusquiza, I., Grfe, M., Sanz, M., Lamata, L., Szameit, A., Solano, E.: Advanced-retarded differential equations in quantum photonic systems. Sci. Rep. 7 (2017). Art. no. 42933Google Scholar
  3. 3.
    Bell, J.: Behaviour of some models of myelinated axons. IMA J. Math. Appl. Med. Biol. 1, 149–167 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bell, J., Cosner, C.: Threshold conditions for a diffusive model of a myelinated axon. J. Math. Biol. 18, 39–52 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Buhmann, M.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Caruntu, B., Bota, C.: Analytical approximate solutions for a general class of nonlinear delay differential equations. Sci. World J. 2014, 6 (2014). iD 631416CrossRefGoogle Scholar
  7. 7.
    Chi, H., Bell, J., Hassard, B.: Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction. J. Math. Biol. 24, 583–601 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ford, N., Lumb, P.: Mixed-type functional differential equations: a numerical approach. J. Comput. Appl. Math. 229(2), 471–479 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Iakovleva, V., Vanegas, C.: On the solution of differential equations withe delayed and advanced arguments. Electron. J. Differ. Equ. Conf. 13, 57–63 (2005)zbMATHGoogle Scholar
  10. 10.
    Ishizaka, K., Matsudaira, M.: Fluid Mechanical Considerations of Vocal Cord Vibration. Speech Communications Research Laboratory, monography 8 (1972)Google Scholar
  11. 11.
    Ishizaka, K., Matsudaira, M.: Speech Physiology and Acoustic Phonetics. MacMillan, New York (1977)Google Scholar
  12. 12.
    Lima, P., Teodoro, M., Ford, N., Lumb, P.: Analytical and numerical investigation of mixed type functional differential equations. J. Comput. Appl. Math. 234, 2732–2744 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lima, P., Teodoro, M., Ford, N., Lumb, P.: Finite element solution of a linear mixed-type functional differential equation. Numer. Algorithms 55, 301–320 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lima, P., Teodoro, M., Ford, N., Lumb, P.: Analysis and computational approximation of a forward-backward equation arising in nerve conduction. In: Pinelas, S., Chipot, M., Dosla, Z. (eds.) Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics, vol. 47, pp. 475–483. Springer, New York (2013)CrossRefGoogle Scholar
  15. 15.
    Lucero, J.: Advanced-delay equations for aerolastics oscillations in physiology. Biophys. Rev. Lett. 3, 125–133 (2008)CrossRefGoogle Scholar
  16. 16.
    Lucero, J.: A lumped mucosal wave model of vocal folds revisited: recent extensions and oscillation hysteresis. J. Acoust. Soc. Am. 129, 1568–1579 (2011)CrossRefGoogle Scholar
  17. 17.
    Mallet-Paret, J.: The global structure of traveling waves in spatially discrete dynamical systems. J. Dyn. Differ. Equ. 11, 49–128 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: The Mathematical Theory of Optimal Process. Interscience, New York (1962)Google Scholar
  19. 19.
    Rustichini, A.: Functional differential equations of mixed type: the linear autonomous case. J. Dyn. Differ. Equ. 1, 121–143 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rustichini, A.: Hopf bifurcation for functional differential equations of mixed type. J. Dyn. Differ. Equ. 1, 145–177 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shakery, F., Dehghan, M.: Solution of delay differential equations via a homotopy perturbation method. Math. Comput. Model. 48, 486–498 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Teodoro, M.F.: Numerical solution of a forward-backward equation from physiology (accepted for publication in Applied Mathematics and Information Sciences)Google Scholar
  23. 23.
    Teodoro, M.F.: Numerical approximation of a delay-advanced equation from acoustics. In: Vigo-Aguiar, J., et al. (eds.) Mathematical Methods in Science and Engineering, pp. 1086–1089. CMMSE, Spain (2015)Google Scholar
  24. 24.
    Teodoro, M.F.: Numerical approach of a nonlinear forward-backward equation. Int. J. Math. Comput. Methods 1, 75–78 (2016)Google Scholar
  25. 25.
    Teodoro, M.F.: Numerical solution of a delay-advanced equation from acoustics. Int. J. Mech. 11, 107–114 (2017)Google Scholar
  26. 26.
    Teodoro, M.: Approximating a nonlinear advanced-delayed equation from acoustics. In: Sergeyev, Y.D., Kvasov, D.E., Accio, F.D., Mukhametzhanov, M.S. (eds.) AIP Conference Proceedings. Numerical Computations: Theory and Algorithms. vol. 1776. AIP, Melville (2016)Google Scholar
  27. 27.
    Teodoro, M.: Modelling a nonlinear mtfde from acoustics. In: Simos, T., Tsitouras, C. (eds.) AIP Conference Proceedings. Num. Anal. and App. Math., vol. 1738. AIP, Melville (2016)Google Scholar
  28. 28.
    Teodoro, M.: An issue about the existence of solutions for a linear non-autonomous MTFDE. In: Pinelas, S., Došlá, Z., Došý, O., Kloeden, P. (eds.) Differential and Difference Equations with Applications. Proceedings in Mathematics&Statistics, vol. 164. Springer, Cham (2016)Google Scholar
  29. 29.
    Teodoro, M., Lima, P., Ford, N.J., Lumb, P.: Numerical approximation of a nonlinear delay-advance functional differential equations by a finite element method. In: Simos, T., Psihoyios, G., Tsitouras, C., Anastassi, Z. (eds.) AIP Conference Proceedings. Num. Anal. and App. Math., vol. 1479, pp. 406–409. AIP, Melville (2012)Google Scholar
  30. 30.
    Teodoro, M., Lima, P., Ford, N., Lumb, P.: Numerical modelling of a functional differential equation with deviating arguments using a collocation method. In: Simos, T., Psihoyios, G., Tsitouras, C. (eds.) AIP Conference Proceedings. Num. Anal. and App. Math., vol. 1048, pp. 553–557. AIP, Melville (2008)Google Scholar
  31. 31.
    Teodoro, M., Lima, P., Ford, N., Lumb, P.: New approach to the numerical solution of forward-backward equations. Front. Math. China 4, 155–168 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tiago, C., Leitão, V.: Utilização de funções de base radial em problemas unidimensionais de análise estrutural. In: Goicolea, J.M., Soares, C.M., Pastor, M., Bugeda, G. (eds.) Métodos Numéricos em Engenieria, vol. V, SEMNI (2002). (in Portuguese)Google Scholar
  33. 33.
    Titze, I.: The physics of small amplitude oscillation of the vocal folds. J. Acoust. Soc. Am. 83, 1536–1552 (1988)CrossRefGoogle Scholar
  34. 34.
    Titze, I.: Principles of Voice Production. Prentice-Hall, Englewood Cliffs (1994)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CEMAT, Instituto Superior TécnicoLisbon UniversityLisbonPortugal
  2. 2.CINAV, Naval Academy, Base Naval de LisboaAlmadaPortugal

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