• 발행 : 2008.11.30


In this paper, the aim is to solve the neutral delay differential equations in the following form using multiquadric approximation scheme, (1) $$\{_{\;y(t)\;=\;{\phi}(t),\;\;\;\;\;t\;{\leq}\;{t_1},}^{\;y'(t)\;=\;f(t,\;y(t),\;y(t\;-\;{\tau}(t,\;y(t))),\;y'(t\;-\;{\sigma}(t,\;y(t)))),\;{t_1}\;{\leq}\;t\;{\leq}\;{t_f},}$$ where f : $[t_1,\;t_f]\;{\times}\;R\;{\times}\;R\;{\times}\;R\;{\rightarrow}\;R$ is a smooth function, $\tau(t,\;y(t))$ and $\sigma(t,\;y(t))$ are continuous functions on $[t_1,\;t_f]{\times}R$ such that t-$\tau(t,\;y(t))$ < $t_f$ and t - $\sigma(t,\;y(t))$ < $t_f$. Also $\phi(t)$ represents the initial function or the initial data. Hence, we present the advantage of using the multiquadric approximation scheme. In the sequel, presented numerical solutions of some experiments, illustrate the high accuracy and the efficiency of the proposed method even where the data points are scattered.


  1. A. Aminataei and M. M. Mazarei, Numerical solution of elliptic partial differential equarions using direct and indirect radial basis function networks, Euro. J. Scien. Res. 2 (2005), no. 2, p. 5
  2. A. Aminataei and M. Sharan, Using multiquadric method in the numerical solution of ODEs with a singularity point and PDEs in one and two dimensions, Euro. J. Scien. Res. 10 (2005), no. 2, p. 19
  3. R. Bellman, On the computational solution of differential-difference equations, J. Math. Anal. Appl. 2 (1961), 108-110
  4. A. Bellen and M. Zennaro, Adaptive integration of delay differential equations, Advances in time-delay systems, 155-165, Lect. Notes Comput. Sci. Eng., 38, Springer, Berlin, 2004
  5. A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2003
  6. R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, Vol. 20. Springer-Verlag, New York-Heidelberg, 1977
  7. L. E. El'sgol'ts and S. B. Norkin, Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Mathematics in Science and Engineering, Vol. 105. Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973
  8. R. Franke, Scattered data interpolation: Tests of some methods, Math. Comput. 38 (1971), 181-199
  9. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74. Kluwer Academic Publishers Group, Dordrecht, 1992
  10. N. Guglielmi and E. Hairer, Implementing Radau IIA methods for stiff delay differential equations, Computing 67 (2001), no. 1, 1-12
  11. A. Halanay, Differential Equations: Stability, Sscillations, Time Lags, Academic Press, New York-London, 1966
  12. R. L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76 (1971), no. 8, 1705-1915
  13. R. L. Hardy, Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968-1988, Comput. Math. Appl. 19 (1990), no. 8-9, 163-208
  14. E. J. Kansa, Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (1990), no. 8-9, 127-145
  15. E. J. Kansa, Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19 (1990), no. 8-9, 147-161
  16. V. B. Kolmanovskii and A. Myshkis, Applied Theory of Functional-Differential Equations, Mathematics and its Applications (Soviet Series), 85. Kluwer Academic Publishers Group, Dordrecht, 1992
  17. V. B. Kolmanovskii and V. R. Nosov, Stability of Functional-Differential Equations, Mathematics in Science and Engineering, 180. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986
  18. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993
  19. E. Lelarsmee, A. Ruehli, and A. Sangiovanni-Vincentelli, The waveform relaxation method for time domain analysis of large scale integrated circuits, IEEE Transactions on CAD 1 (1982), no. 3, 131-145
  20. W. R. Madych, Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl. 24 (1992), no. 12, 121-138
  21. W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions. II, Math. Comp. 54 (1990), no. 189, 211-230
  22. N. Mai-Duy and T. Tran-Cong, Numerical solution of differential equations using multiquadric radial basis function networks, Neutral Networks 14 (2001), no.2 , 185-199
  23. C. A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986), no. 1, 11-22
  24. M. Zerroukut, H. Power, and C. S. Chen, A numerical method for heat transfer problems using collocation and radial basis functions, Internat. J. Numer. Methods Engrg. 42 (1998), no. 7, 1263-1278<1263::AID-NME431>3.0.CO;2-I

피인용 문헌

  1. Solution of two-dimensional modified anomalous fractional sub-diffusion equation via radial basis functions (RBF) meshless method vol.38, 2014,
  2. Numerical Solution of a Neutral Differential Equation with Infinite Delay vol.20, pp.1, 2012,
  3. The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics vol.37, pp.2, 2013,
  4. The numerical solution of the two–dimensional sinh-Gordon equation via three meshless methods vol.51, 2015,
  5. Laguerre matrix method with the residual error estimation for solutions of a class of delay differential equations vol.37, pp.4, 2014,
  6. An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations vol.50, 2015,
  7. Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives vol.264, 2013,
  8. The meshless method of radial basis functions for the numerical solution of time fractional telegraph equation vol.24, pp.8, 2014,
  9. Optimal Homotopy Asymptotic Method for Solving Delay Differential Equations vol.2013, 2013,
  10. Application of the hybrid functions to solve neutral delay functional differential equations vol.94, pp.3, 2017,
  11. A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model vol.56, 2015,
  12. Differential Transform Method for Some Delay Differential Equations vol.06, pp.03, 2015,
  13. The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions vol.68, pp.3, 2014,