[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.2008.45.4.663

ON THE NUMERICAL SOLUTION OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS USING MULTIQUADRIC APPROXIMATION SCHEME

Vanani, Solat Karimi (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE K. N. TOOSI UNIVERSITY OF TECHNOLOGY)
Aminataei, Azim (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE K. N. TOOSI UNIVERSITY OF TECHNOLOGY)

Publication Information

Bulletin of the Korean Mathematical Society / v.45, no.4, 2008 , pp. 663-670 More about this Journal

Abstract

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$ 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.

Keywords

multiquadric approximation scheme; delay differential equations; neutral delay differential equations;

Citations & Related Records

Times Cited By Web Of Science : 0 (Related Records In Web of Science)
Times Cited By SCOPUS : 4

Reference
Cited By SCOPUS

1	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
2	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
3	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
4	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
5	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
6	R. Franke, Scattered data interpolation: Tests of some methods, Math. Comput. 38 (1971), 181-199
7	K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74. Kluwer Academic Publishers Group, Dordrecht, 1992
8	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 DOI ScienceOn
9	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
10	R. Bellman, On the computational solution of differential-difference equations, J. Math. Anal. Appl. 2 (1961), 108-110 DOI
11	R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, Vol. 20. Springer-Verlag, New York-Heidelberg, 1977
12	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
13	C. A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986), no. 1, 11-22 DOI
14	N. Guglielmi and E. Hairer, Implementing Radau IIA methods for stiff delay differential equations, Computing 67 (2001), no. 1, 1-12 DOI
15	A. Halanay, Differential Equations: Stability, Sscillations, Time Lags, Academic Press, New York-London, 1966
16	R. L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76 (1971), no. 8, 1705-1915
17	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 DOI ScienceOn
18	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 DOI ScienceOn
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 DOI
20	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993
21	W. R. Madych, Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl. 24 (1992), no. 12, 121-138 DOI ScienceOn
22	W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions. II, Math. Comp. 54 (1990), no. 189, 211-230 DOI ScienceOn
23	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 DOI ScienceOn
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 DOI ScienceOn

	Akbar Mohebbi. (2014) Engineering Analysis with Boundary Elements Solution of two-dimensional modified anomalous fractional sub-diffusion equation via radial basis functions (RBF) meshless method / 38 , 72
1	D. Piriadarshani. (2012) Differential Equations and Dynamical Systems Numerical Solution of a Neutral Differential Equation with Infinite Delay / 20 (1) , 17
2	Akbar Mohebbi. (2013) Engineering Analysis with Boundary Elements 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 / 37 (2) , 475
	Mehdi Dehghan. (2015) Engineering Analysis with Boundary Elements The numerical solution of the two–dimensional sinh-Gordon equation via three meshless methods / 51 , 220
4	？uayip Yuzba？ı. (2014) Mathematical Methods in the Applied Sciences Laguerre matrix method with the residual error estimation for solutions of a class of delay differential equations / 37 (4) , 453
	Mehdi Dehghan. (2015) Engineering Analysis with Boundary Elements An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations / 50 , 412
	Akbar Mohebbi. (2013) Computer Methods in Applied Mechanics and Engineering Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives / 264 , 163
8	Akbar Mohebbi. (2014) International Journal of Numerical Methods for Heat & Fluid Flow The meshless method of radial basis functions for the numerical solution of time fractional telegraph equation / 24 (8) , 1636
	N. Ratib Anakira. (2013) Mathematical Problems in Engineering Optimal Homotopy Asymptotic Method for Solving Delay Differential Equations / 2013 , 1
3	S. Sedaghat. (2017) International Journal of Computer Mathematics Application of the hybrid functions to solve neutral delay functional differential equations / 94 (3) , 503
	Mehdi Dehghan. (2015) Engineering Analysis with Boundary Elements A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model / 56 , 129
03	Baoqing Liu. (2015) Applied Mathematics Differential Transform Method for Some Delay Differential Equations / 06 (03) , 585
3	Mehdi Dehghan. (2014) Computers & Mathematics with Applications The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions / 68 (3) , 212