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Bernoulli collocation method for high-order generalized pantograph equations

Year 2015, Volume: 3 Issue: 2, 96 - 109, 19.01.2015

Abstract

In this paper, an approximate method based on Bernoulli polynomials has been presented to obtain the solution ofgeneralized pantograph equations with linear functional arguments. Both initial and boundary value problems have been solved by thiscollocation technique. Approximate solution can also be corrected with the residual function. Some numerical examples have beengiven to illustrate the reliability and efficiency of the method

References

  • J.R. Ockendon, A.B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Roy. Soc. London, Ser.
  • A 322 (1971) 447-468.
  • L. Fox, D.F. Mayers, J.A. Ockendon, A.B. Tayler, On a functional differential equation, J. Inst. Math. Appl. 8 (1971) 271-307.
  • W.G. Ajello, H.I. Freedman, J.Wu, A model of stage structured population growth with density depended time delay, SIAM J.
  • Appl. Math. 52 (1992) 855-869.
  • M.D. Buhmann, A. Iserles, Stability of the discretized pantograph differential equation, Math. Comput. 60 (1993) 575-589.
  • G.R. Morris, A. Feldstein, E.W. Bowen, The Phragmen-Lindel’ of principle and a class of functional-differential equations, in:
  • Proceedings of NRL-MRC Conference on Ordinary Differential Equations, 1972, pp. 513-540.
  • G. Derfel, On compactly supported solutions of a class of functional-differential equations, in: Modern Problems of Functions
  • Theory and Functional Analysis, Karaganda University Press. 1980 (in Russian).
  • G. Derfel, N. Dyn, D. Levin, Generalized refinement equation and subdivision process, J. Approx. Theory 80 (1995) 272-297.
  • G.A. Derfel, F. Vogl, On the asymptotic of solutions of a class of linear functional-differential equations, European J. Appl. Maths. 7 (1996) 511-518.
  • G. Derfel, A.Iserles, The pantograph equation in the complex plane, J. Math. Anal. Appl. 213 (1997) 117-132.
  • A. Feldstein, Y. Liu, On neutral functional-differential equations with variable time delays, Math. Proc. Cambridge Philos. Soc. 124 (1998) 371-384.
  • M.Z. Liu, D. Li, Properties of analytic solution and numerical solution and multi-pantograph equation, Appl. Math. Comput. 155 (2004) 853-871.
  • M. G¨ulsu, M. Sezer, The approximate solution of high-order linear difference equation with variable coefficients in terms of Taylor polynomials, Appl. Math. Comput. 168 (1) (2005) 76-88.
  • M. Sezer, M. G¨ulsu, A new polynomial approach for solving difference and Fredholm integro-difference equation with mixed argument, Appl. Math. Comput. 171 (1) (2005) 332-344.
  • M. G¨ulsu, M. Sezer, A Taylor polynomial approach for solving differential-difference equations, J. Comput. Appl. Math. 186 (2) (2006) 349-364.
  • M. Sezer, A. Aky¨uz-Das¸cıo˘glu, Taylor polynomial solutions of general linear differential–difference equations with variable coefficients, Appl. Math. Comput. 174 (2006) 1526–1538.
  • M. Sezer, S. Yalc¸ınbas¸, N. S¸ahin, Approximate solution of multi-pantograph equation with variable coefficients, J. Comput. Appl. Math. 214 (2008) 406-416.
  • M. Sezer, A. Aky¨uz-Das¸cıo˘glu, A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, J. Comput. Appl. Math. 200 (2007) 217-225.
  • D.J. Evans, K.R. Raslan, The Adomian decomposition method for solving delay differential equation, Int. J. Comput. Math. 82 (1) (2005) 49-54.
  • Y. Keskin, A. Kurnaz, M.E. Kiris, G. Oturanc, Approximate solutions of generalized pantograph equations by the differential transform method, Int. J. Nonlinear Sci. Num. Simulation 8 (2) (2007) 159-164.
  • E. Yusufo˘glu, An efficient algorithm for solving generalized pantograph equations with linear functional argument, Appl. Math. Comput. 217 (2010) 3591–3595.
  • R. Qu, R.P. Agarwal, A subdivision approach to the construction of approximate solutions of boundary-value problems with deviating arguments, Computers Math. Applic. 35 (1998) 121-135.
  • N. M. Temme, Special Functions, an introduction to the classical functions of mathematical physics, John Wiley and Sons Inc., New York, USA, 1996.
  • P. E. Zadunaisky, On the estimation of errors propagated in the numerical integration of ordinary differential equations, Numer. Math. 27 (1976) 21-39.
  • C. H. Hsiao, Wavelet approach to time-varying functional differential equations, Int. J. Comput. Math. 8732 (2010) 528-540.
  • G.P. Rao, K.R. Palanisamy, Walsh stretch matrices and functional differential equations, IEEE Trans. Autom. Control 27 (1982) 272-276.
  • C. Hwang, Y.-P. Shih, Laguerre series solution of a functional differential equation, Int. J. Systems Sci. 13 (7) (1982) 783-788.
  • C. Hwang, Solution of a functional differential equation via delayed unit step functions, Int. J. Systems Sci. 14 (9) (1983) 1065- 1073.
  • Zhan-Hua Yu, Variational iteration method for solving the multi-pantograph delay equation, Physics Letters A 372 (2008) 6475- 6479.
  • A. El-Safty, M. Shadia, On the application of spline function to initial value problems with retarded argument, Int. J. Comput. Math. 32 (1990) 173-179.
  • M. Shadia, Numerical solution of delay differential and neutral differential equations using spline methods, Ph.D. Thesis, Assuit University, 1992.
  • A. El-Safty, M. S. Salim, M. A. El-Khatib, Convergence of the spline function for delay dynamic system, Intern. J. Comput. Math. 80 (4) (2003) 509-518.
  • S¸.Y¨uzbas¸ı, M. Sezer, An exponential approximation for solutions of generalized pantograph-delay differential equations, Applied Mathematical Modelling 37 (2013) 9160–9173.
  • S. Yalc¸inbas¸, M. Aynigul, M. Sezer, A collocation method using Hermite polynomials for approximate solution of pantograph equations, Journal of the Franklin Institute 348 (2011) 1128–1139.
  • E.H. Doha, A.H. Bhrawy, D. Baleanu, R.M. Hafez, A new Jacobi rational–Gauss collocation method solution of generalized pantograph equations, Applied Numerical Mathematics, 77 (2014) 43–54.

Bernoulli collocation method for high-order generalized pantograph equations

Year 2015, Volume: 3 Issue: 2, 96 - 109, 19.01.2015

Abstract

References

  • J.R. Ockendon, A.B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Roy. Soc. London, Ser.
  • A 322 (1971) 447-468.
  • L. Fox, D.F. Mayers, J.A. Ockendon, A.B. Tayler, On a functional differential equation, J. Inst. Math. Appl. 8 (1971) 271-307.
  • W.G. Ajello, H.I. Freedman, J.Wu, A model of stage structured population growth with density depended time delay, SIAM J.
  • Appl. Math. 52 (1992) 855-869.
  • M.D. Buhmann, A. Iserles, Stability of the discretized pantograph differential equation, Math. Comput. 60 (1993) 575-589.
  • G.R. Morris, A. Feldstein, E.W. Bowen, The Phragmen-Lindel’ of principle and a class of functional-differential equations, in:
  • Proceedings of NRL-MRC Conference on Ordinary Differential Equations, 1972, pp. 513-540.
  • G. Derfel, On compactly supported solutions of a class of functional-differential equations, in: Modern Problems of Functions
  • Theory and Functional Analysis, Karaganda University Press. 1980 (in Russian).
  • G. Derfel, N. Dyn, D. Levin, Generalized refinement equation and subdivision process, J. Approx. Theory 80 (1995) 272-297.
  • G.A. Derfel, F. Vogl, On the asymptotic of solutions of a class of linear functional-differential equations, European J. Appl. Maths. 7 (1996) 511-518.
  • G. Derfel, A.Iserles, The pantograph equation in the complex plane, J. Math. Anal. Appl. 213 (1997) 117-132.
  • A. Feldstein, Y. Liu, On neutral functional-differential equations with variable time delays, Math. Proc. Cambridge Philos. Soc. 124 (1998) 371-384.
  • M.Z. Liu, D. Li, Properties of analytic solution and numerical solution and multi-pantograph equation, Appl. Math. Comput. 155 (2004) 853-871.
  • M. G¨ulsu, M. Sezer, The approximate solution of high-order linear difference equation with variable coefficients in terms of Taylor polynomials, Appl. Math. Comput. 168 (1) (2005) 76-88.
  • M. Sezer, M. G¨ulsu, A new polynomial approach for solving difference and Fredholm integro-difference equation with mixed argument, Appl. Math. Comput. 171 (1) (2005) 332-344.
  • M. G¨ulsu, M. Sezer, A Taylor polynomial approach for solving differential-difference equations, J. Comput. Appl. Math. 186 (2) (2006) 349-364.
  • M. Sezer, A. Aky¨uz-Das¸cıo˘glu, Taylor polynomial solutions of general linear differential–difference equations with variable coefficients, Appl. Math. Comput. 174 (2006) 1526–1538.
  • M. Sezer, S. Yalc¸ınbas¸, N. S¸ahin, Approximate solution of multi-pantograph equation with variable coefficients, J. Comput. Appl. Math. 214 (2008) 406-416.
  • M. Sezer, A. Aky¨uz-Das¸cıo˘glu, A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, J. Comput. Appl. Math. 200 (2007) 217-225.
  • D.J. Evans, K.R. Raslan, The Adomian decomposition method for solving delay differential equation, Int. J. Comput. Math. 82 (1) (2005) 49-54.
  • Y. Keskin, A. Kurnaz, M.E. Kiris, G. Oturanc, Approximate solutions of generalized pantograph equations by the differential transform method, Int. J. Nonlinear Sci. Num. Simulation 8 (2) (2007) 159-164.
  • E. Yusufo˘glu, An efficient algorithm for solving generalized pantograph equations with linear functional argument, Appl. Math. Comput. 217 (2010) 3591–3595.
  • R. Qu, R.P. Agarwal, A subdivision approach to the construction of approximate solutions of boundary-value problems with deviating arguments, Computers Math. Applic. 35 (1998) 121-135.
  • N. M. Temme, Special Functions, an introduction to the classical functions of mathematical physics, John Wiley and Sons Inc., New York, USA, 1996.
  • P. E. Zadunaisky, On the estimation of errors propagated in the numerical integration of ordinary differential equations, Numer. Math. 27 (1976) 21-39.
  • C. H. Hsiao, Wavelet approach to time-varying functional differential equations, Int. J. Comput. Math. 8732 (2010) 528-540.
  • G.P. Rao, K.R. Palanisamy, Walsh stretch matrices and functional differential equations, IEEE Trans. Autom. Control 27 (1982) 272-276.
  • C. Hwang, Y.-P. Shih, Laguerre series solution of a functional differential equation, Int. J. Systems Sci. 13 (7) (1982) 783-788.
  • C. Hwang, Solution of a functional differential equation via delayed unit step functions, Int. J. Systems Sci. 14 (9) (1983) 1065- 1073.
  • Zhan-Hua Yu, Variational iteration method for solving the multi-pantograph delay equation, Physics Letters A 372 (2008) 6475- 6479.
  • A. El-Safty, M. Shadia, On the application of spline function to initial value problems with retarded argument, Int. J. Comput. Math. 32 (1990) 173-179.
  • M. Shadia, Numerical solution of delay differential and neutral differential equations using spline methods, Ph.D. Thesis, Assuit University, 1992.
  • A. El-Safty, M. S. Salim, M. A. El-Khatib, Convergence of the spline function for delay dynamic system, Intern. J. Comput. Math. 80 (4) (2003) 509-518.
  • S¸.Y¨uzbas¸ı, M. Sezer, An exponential approximation for solutions of generalized pantograph-delay differential equations, Applied Mathematical Modelling 37 (2013) 9160–9173.
  • S. Yalc¸inbas¸, M. Aynigul, M. Sezer, A collocation method using Hermite polynomials for approximate solution of pantograph equations, Journal of the Franklin Institute 348 (2011) 1128–1139.
  • E.H. Doha, A.H. Bhrawy, D. Baleanu, R.M. Hafez, A new Jacobi rational–Gauss collocation method solution of generalized pantograph equations, Applied Numerical Mathematics, 77 (2014) 43–54.
There are 38 citations in total.

Details

Journal Section Articles
Authors

Ayşegül Daşçıoğlu This is me

Mehmet Sezer This is me

Publication Date January 19, 2015
Published in Issue Year 2015 Volume: 3 Issue: 2

Cite

APA Daşçıoğlu, A., & Sezer, M. (2015). Bernoulli collocation method for high-order generalized pantograph equations. New Trends in Mathematical Sciences, 3(2), 96-109.
AMA Daşçıoğlu A, Sezer M. Bernoulli collocation method for high-order generalized pantograph equations. New Trends in Mathematical Sciences. January 2015;3(2):96-109.
Chicago Daşçıoğlu, Ayşegül, and Mehmet Sezer. “Bernoulli Collocation Method for High-Order Generalized Pantograph Equations”. New Trends in Mathematical Sciences 3, no. 2 (January 2015): 96-109.
EndNote Daşçıoğlu A, Sezer M (January 1, 2015) Bernoulli collocation method for high-order generalized pantograph equations. New Trends in Mathematical Sciences 3 2 96–109.
IEEE A. Daşçıoğlu and M. Sezer, “Bernoulli collocation method for high-order generalized pantograph equations”, New Trends in Mathematical Sciences, vol. 3, no. 2, pp. 96–109, 2015.
ISNAD Daşçıoğlu, Ayşegül - Sezer, Mehmet. “Bernoulli Collocation Method for High-Order Generalized Pantograph Equations”. New Trends in Mathematical Sciences 3/2 (January 2015), 96-109.
JAMA Daşçıoğlu A, Sezer M. Bernoulli collocation method for high-order generalized pantograph equations. New Trends in Mathematical Sciences. 2015;3:96–109.
MLA Daşçıoğlu, Ayşegül and Mehmet Sezer. “Bernoulli Collocation Method for High-Order Generalized Pantograph Equations”. New Trends in Mathematical Sciences, vol. 3, no. 2, 2015, pp. 96-109.
Vancouver Daşçıoğlu A, Sezer M. Bernoulli collocation method for high-order generalized pantograph equations. New Trends in Mathematical Sciences. 2015;3(2):96-109.