A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation

Applied Mathematical Modelling

Volumn 37, Issue 6, 2013, Pages 4283-4294

  Tohidi, E ^a   Bhrawy, A H ^b,c   Erfani, K ^a  

^a ISLAMIC AZAD UNIVERSITY   (Iran)

^b KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

^c Beni Suef University   (Egypt)

Author keywords

Bernoulli polynomials; Collocation method; Direct method; Functional differential equations; Operational matrix; Pantograph equation

Indexed keywords

BERNOULLI POLYNOMIALS; COLLOCATION METHOD; DIRECT METHOD; FUNCTIONAL DIFFERENTIAL EQUATIONS; OPERATIONAL MATRICES; PANTOGRAPH EQUATION;

PANTOGRAPHS;

NUMERICAL METHODS;

EID: 84872607486     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2012.09.032     Document Type: Article

References (49)

1. Ockendon J.R., Tayler A.B. The dynamics of a current collection system for an electric locomotive Proc. Roy. Soc. London A 1971, 322:447-468.
2. Ajello W.G., Freedman H.I., Wu J. A model of stage structured population growth with density depended time delay. SIAM J. Appl. Math. 1992, 52:855-869.
3. Buhmann M.D., Iserles A. Stability of the discretized pantograph differential equation. Math. Comput. 1993, 60:575-589.
4. Fox L., Mayers D.F., Ockendon J.A., Tayler A.B. On a functional differential equation. J. Inst. Math. Appl. 1971, 8:271-307.
5. Derfel G.A., Vogl F. On the asymptotics of solutions of a class of linear functional-differential equations. Eur. J. Appl. Math. 1996, 7:511-518.
6. Feldstein A., Liu Y. On neutral functional differential equations with variable time delays. Math. Proc. Cambridge Philos. Soc. 1998, 124:371-384.
7. Sezer M. A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials. Int. J. Math. Educ. Sci. Technol. 1996, 27:821-834.
8. Sezer M., Gulsu M. A new polynomial approach for solving difference and Fredholm integro-diffrential equation with mixed argument. Appl. Math. Comput. 2005, 171:332-344.
9. Liu M.Z., Li D. Properties of analytic solution and numerical solution and multi-pantograph equation. Appl. Math. Comput. 2004, 155:853-871.
10. Shakeri F., Dehghan M. Solution of the delay differential equations via homotopy perturbation method. Math. Comput. Model. 2008, 48:486-498.
11. Sezer M., Yalcinbas S., Sahin N. Approximate solution of multi-pantograph equation with variable coeffcients. J. Comput. Appl. Math. 2008, 214:406-416.
12. Li D., Liu M.Z. Runge-Kutta methods for the multi-pantograph delay equation. App. Math. Comput. 2005, 163:383-395.
13. Yu Z.H. Variational iteration method for solving the multi-pantograph delay equation. Phys. Lett. A 2008, 372:6475-6479.
14. Saadatmandi A., Dehghan M. Variational iteration method for solving a generalized pantograph equation. Comput. Math. Appl. 2009, 58:2190-2196.
15. Yuzbasi S., Sahin N., Sezer M. A Bessel collocation method for numerical solution of generalized pantograph equations. Numer. Methods. Partial. Differ. Equ. 2012, 10.1002/num.20660.
16. Yalinbas S., Aynigul M., Sezer M. A collocation method using Hermite polynomials for approximate solution of pantograph equations. J. Franklin Inst. 2011, 348:1128-1139.
17. Koblitz N. p-Adic Numbers, p-Adic Analysis and Zeta-Functions 1984, Springer-Verlag, New York. second ed.
18. Lang S. Introduction to Modular Forms 1976, Springer-Verlag.
19. Boas R.P., Buck R.C. Polynomial Expansions of Analytic Functions 1964, Springer-Verlag, New York.
20. Fairlie D.B., Veselov A.P. Faulhaber and Bernoulli polynomials and solitons. Phys. D 2001, 152-153:47-50.
21. M.P. Grosset, P. Veselov, Elliptic Faulhaber polynomials and Lame densities of states, Int. Math. Res. Not. (2006), Article ID 62120, 31.
22. Doyon B., Lepowsky J., Milas A. Twisted vertex operators and Bernoulli polynomials. Commun. Contemp. Math. 2006, 8:247-307.
23. Derfel G., Iserles A. The pantograph equation in the complex plane. J. Math. Anal. Appl. 1997, 213:117-132.
24. Evans D.J., Raslan K.R. The adomain decomposition method for solving delay differential equation. Int. J. Comput. Math. 2005, 82:49-54.
25. Sezer M., Yalcinbas S., Gulsu M. A taylor polynomial approach for solving generalized pantograph equations with nonhomogenous term. Int. J. Comput. Math. 2008, 85:1055-1063.
26. Saadatmandi A., Dehghan M. A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 2010, 59:1326-1336.
27. Doha E.H., Bhrawy A.H., Ezz-Eldien S.S. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 2011, 35:5662-5672.
28. Doha E.H., Bhrawy A.H., Ezz-Eldien S.S. A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 2011, 62:2364-2373.
29. Doha E.H., Bhrawy A.H., Ezzeldeen S.S. A new jacobi operational matrix: an application for solving fractional differential equation. Appl. Math. Model. 2012, 36:4931-4943.
30. S.A. Yousefi, M. Behroozifar and Mehdi Dehghan, Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials, Appl. Math. Model. (2012) doi:10.1016/j.apm.2011.07.041.
31. Bhrawy A.H., Alofi A.S., Ezzeldeen S.S. A quadrature tau method for fractional differential equations with variable coefficients. Appl. Math. Lett. 2011, 24:2146-2152.
32. Bhrawy A.H., Alofi A.S. The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl. Math. Lett. 2013, 26:25-31.
33. Srivastava H.M., Choi J. Series Associated with the Zeta and Related Functions 2001, Kluwer Academic, Dordreeht.
34. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G. Higher Transcendental Functions 1955, vol. III. McGraw-Hill, New York.
35. Luke Y.L. The Special Functions and Their Approximations 1969, vol. I. Academic Press, New York.
36. Magnus W., Oberhettinger F., Soni R.P. Formulas and Theorems for the Special Functions of Mathematical Physics, Third Enlarged Edition 1966, Springer-Verlag, New York.
37. Mashayekhi S., Ordokhani Y., Razzaghi M. Hybrid functions approach for nonlinear constrained optimal control problems. Commun. Nonlinear. Sci. Numer. Simulat. 2012, 17:1831-1843.
38. Abramowitz M., Stegun I.A. Handbook of mathematical functions with formulas, graphs, and mathematical tables 1972, National Bureau of Standards, Wiley, New York.
39. Rao G.P., Palanisamy K.R. Walsh stretch matrices and functional differential equation. IEEE Trans. Autom. Control 1982, 27:272-276.
40. Hwang C., Shih Y.P. Laguerre series solution of a functional differential equation. Int. J. Syst. Sci. 1982, 13(7):783-788.
41. Sezer M., Akyuz-Das-cioglu A. A Taylor method for numerical solution of generalized pantograph equations with linear functional argument. J. Comput. Appl. Math. 2007, 200:217-225.
42. Ozturk Y., Gulsu M. Approximate solution of linear generalized pantograph equations with variable coefficients on Chebyshev-Gauss grid. J. Adv. Res. Sci. Comput. 2012, 4:36-51.
43. S. Yuzbasi, A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differentialdifference equations, Computers and Mathematics with Applications, doi:10.1016/j.camwa.2012.01.026.
44. Reid W.T. Riccati Differential Equations 1972, Academic Press, New York.
45. Dehghan M., Taleei A. A compact split-step finite difference method for solving the nonlinear Schrdinger equations with constant and variable coefficients. Comput. Phys. Commun. 2010, 181:43-51.
46. Carinena J.F., Marmo G., Perelomov A.M., Ranada M.F. Related operators and exact solutions of Schrdinger equations. Int. J. Modern Phys. A 1998, 13:4913-4929.
47. Scott M.R. Invariant Imbedding and Its Applications to Ordinary Differential Equations 1973, Addison-Wesley.
48. Adomian G. Solving Frontier Problems of Fhysics: Decomposition Method 1994, Kluwer Academic Publishers, Dordrecth.
49. Abbasbandy S. Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method. Appl. Math. Comput. 2006, 172:485-490.