Abstract
In this work, a novel numerical method based on generalized Laguerre series is introduced. The numerical technique is applied for the solution of a class of functional differential equations with variable delays. This numerical method is substantially related to generalized Laguerre series also its matrix forms as well as collocation points. By error estimation the pertinent features and applicability of the method are demonstrated.
Keywords
Generalized Laguerre series collocation methods functional differential equations variable delays
Thanks
The author would like to thank the Embassy of France in Turkey for their support her as “2019 Young Visiting Research Fellow”; to the University of Nantes, Jean Leray Mathematics Laboratory to use all the facilities in the department required for completing the work.
References
- Gürbüz, B., Sezer, M., Laguerre polynomial approach for solving Lane-Emden type functional differential equations, Applied Mathematics and Computation, 242, 255-264, (2014).
- Dix, J. G., Asymptotic behavior of solutions to a first-order differential equation with variable delays, Computers & Mathematics with Applications, 50, 10-12, 1791-1800, (2005).
- Graef, J. R., Qian, C., Global attractivity in differential equations with variable delays, The ANZIAM Journal, 41, 4, 568-579, (2000).
- Syski, R., Saaty, T. L., In Modern Nonlinear Equations, McGraw-Hill, New York, (1967).
- Ishiwata, E., Muroya, Y., Brunner, H., A super-attainable order in collocation methods for differential equations with proportional delay, Applied Mathematics and Computation, 198, 1, 227-236, (2008).
- Caraballo, T., Langa, J. A., Robinson, J. C., Attractors for differential equations with variable delays, Journal of Mathematical Analysis and Applications, 260, 2, 421-438, (2001).
- Diblík, J., Svoboda, Z., Šmarda, Z., Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case, Computers & Mathematics with Applications, 56, 2, 556-564, (2008).
- Bellen, A., Zennaro, M., Numerical methods for delay differential equations, Oxford University Press, (2013).
- Reutskiy, S. Y., A new collocation method for approximate solution of the pantograph functional differential equations with proportional delay, Applied Mathematics and Computation, 266, 642-655, (2015).
- Hu, P., Huang, C., Wu, S., Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations, Applied Mathematics and Computation, 211, 1, 95-101, (2009).
- Wang, W., Zhang, Y., Li, S., Stability of continuous Runge-Kutta-type methods for nonlinear neutral delay-differential equations, Applied Mathematical Modelling, 33, 8, 3319-3329, (2009).
- Ishak, F., Suleiman, M. B., Majid, Z. A., Block method for solving pantograph-type functional differential equations, In Proceedings of the World Congress on Engineering, 2, (2013).
- Ishiwata, E., Muroya, Y., Rational approximation method for delay differential equations with proportional delay, Applied Mathematics and Computation, 187, 2, 741-747, (2007).
- Wang, W. S., Li, S. F., On the one-leg θ-methods for solving nonlinear neutral functional differential equations, Applied Mathematics and Computation, 193, 1, 285-301, (2007).
- Wang, W., Qin, T., Li, S., Stability of one-leg θ-methods for nonlinear neutral differential equations with proportional delay, Applied Mathematics and Computation, 213, 1, 177-183, (2009).
- Arfken, G. B., Weber, H. J., Mathematical methods for physicists, Elsevier Inc., (1999).
- Gürbüz, B., Sezer, M., A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, International Journal of Applied Physics and Mathematics, 7, 1, 49, (2017).
- Liu, X. G., Tang, M. L., Martin, R. R., Periodic solutions for a kind of Liénard equation, Journal of Computational and Applied Mathematics, 219, 1, 263-275, (2008).
- Schley, D., Shail, R., Gourley, S. A., Stability criteria for differential equations with variable time delays, International Journal of Education in Mathematics, Science and Technology, 33, 3, 359-375, (2002).
- Yıldızhan, I., Kürkçü, O. K., Sezer, M., A numerical approach for solving pantograph-type functional differential equations with mixed delays using Dickson polynomials of the second kind, Journal of Science and Arts, 18, 3, 667-680, (2018).
- Zhang, B., Fixed points and stability in differential equations with variable delays, Nonlinear Analysis, Theory, Methods and Applications, 63, 5-7, 233-242, (2005).
- Özer, S., An effective numerical technique for the Rosenau-KdV-RLW equation, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20, 3, 1-14, (2018).
- Görgülü, M. Z., Irk, D., Numerical solution of modified regularized long wave equation by using cubic trigonometric B-spline functions, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21, 1, 126-138, (2019).
- Düşünceli F, Çelik E., Numerical solution for high‐order linear complex differential equations with variable coefficients, Numerical Methods for Partial Differential Equations, 34, 5, 1645-58, (2018).
Abstract
Bu çalışmada, genelleştirilmiş Laguerre serisine dayanan yeni bir sayısal yöntem tanıtıldı. Sayısal teknik, fonksiyonel diferansiyel denklemlerin değişken gecikmeli bir sınıfının çözümü için uygulanır. Bu sayısal yöntem, esas olarak genelleştirilmiş Laguerre serileri ile aynı zamanda matris formları ve sıralama noktaları ile ilgilidir. Hata tahmininde, yöntemin ilgili özellikleri ve uygulanabilirliği gösterilmektedir.
Keywords
Genelleştirilmiş Laguerre serileri sıralama yöntemleri fonksiyonel diferansiyel denklemler değişken gecikmeler
References
- Gürbüz, B., Sezer, M., Laguerre polynomial approach for solving Lane-Emden type functional differential equations, Applied Mathematics and Computation, 242, 255-264, (2014).
- Dix, J. G., Asymptotic behavior of solutions to a first-order differential equation with variable delays, Computers & Mathematics with Applications, 50, 10-12, 1791-1800, (2005).
- Graef, J. R., Qian, C., Global attractivity in differential equations with variable delays, The ANZIAM Journal, 41, 4, 568-579, (2000).
- Syski, R., Saaty, T. L., In Modern Nonlinear Equations, McGraw-Hill, New York, (1967).
- Ishiwata, E., Muroya, Y., Brunner, H., A super-attainable order in collocation methods for differential equations with proportional delay, Applied Mathematics and Computation, 198, 1, 227-236, (2008).
- Caraballo, T., Langa, J. A., Robinson, J. C., Attractors for differential equations with variable delays, Journal of Mathematical Analysis and Applications, 260, 2, 421-438, (2001).
- Diblík, J., Svoboda, Z., Šmarda, Z., Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case, Computers & Mathematics with Applications, 56, 2, 556-564, (2008).
- Bellen, A., Zennaro, M., Numerical methods for delay differential equations, Oxford University Press, (2013).
- Reutskiy, S. Y., A new collocation method for approximate solution of the pantograph functional differential equations with proportional delay, Applied Mathematics and Computation, 266, 642-655, (2015).
- Hu, P., Huang, C., Wu, S., Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations, Applied Mathematics and Computation, 211, 1, 95-101, (2009).
- Wang, W., Zhang, Y., Li, S., Stability of continuous Runge-Kutta-type methods for nonlinear neutral delay-differential equations, Applied Mathematical Modelling, 33, 8, 3319-3329, (2009).
- Ishak, F., Suleiman, M. B., Majid, Z. A., Block method for solving pantograph-type functional differential equations, In Proceedings of the World Congress on Engineering, 2, (2013).
- Ishiwata, E., Muroya, Y., Rational approximation method for delay differential equations with proportional delay, Applied Mathematics and Computation, 187, 2, 741-747, (2007).
- Wang, W. S., Li, S. F., On the one-leg θ-methods for solving nonlinear neutral functional differential equations, Applied Mathematics and Computation, 193, 1, 285-301, (2007).
- Wang, W., Qin, T., Li, S., Stability of one-leg θ-methods for nonlinear neutral differential equations with proportional delay, Applied Mathematics and Computation, 213, 1, 177-183, (2009).
- Arfken, G. B., Weber, H. J., Mathematical methods for physicists, Elsevier Inc., (1999).
- Gürbüz, B., Sezer, M., A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, International Journal of Applied Physics and Mathematics, 7, 1, 49, (2017).
- Liu, X. G., Tang, M. L., Martin, R. R., Periodic solutions for a kind of Liénard equation, Journal of Computational and Applied Mathematics, 219, 1, 263-275, (2008).
- Schley, D., Shail, R., Gourley, S. A., Stability criteria for differential equations with variable time delays, International Journal of Education in Mathematics, Science and Technology, 33, 3, 359-375, (2002).
- Yıldızhan, I., Kürkçü, O. K., Sezer, M., A numerical approach for solving pantograph-type functional differential equations with mixed delays using Dickson polynomials of the second kind, Journal of Science and Arts, 18, 3, 667-680, (2018).
- Zhang, B., Fixed points and stability in differential equations with variable delays, Nonlinear Analysis, Theory, Methods and Applications, 63, 5-7, 233-242, (2005).
- Özer, S., An effective numerical technique for the Rosenau-KdV-RLW equation, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20, 3, 1-14, (2018).
- Görgülü, M. Z., Irk, D., Numerical solution of modified regularized long wave equation by using cubic trigonometric B-spline functions, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21, 1, 126-138, (2019).
- Düşünceli F, Çelik E., Numerical solution for high‐order linear complex differential equations with variable coefficients, Numerical Methods for Partial Differential Equations, 34, 5, 1645-58, (2018).
Details
| Primary Language | English |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Publication Date | January 10, 2020 |
| Submission Date | May 27, 2019 |
| Published in Issue | Year 2020 Volume: 22 Issue: 1 |