A MODIFIED TAYLOR COLLOCATION METHOD FOR PANTOGRAPH TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH HYBRID PROPORTIONAL AND VARIABLE DELAYS

Abstract

In this work, high order pantograph type linear functional differential equations with hybrid proportional and variable delays is approximately solved by the modified Taylor matrix method. With this method these functional type differential equations are converted into the matrix form by the Taylor expansion method. The problems are reduced into a set of algebraic equations including Taylor coefficients. By determining the coefficients, the approximate solutions are calculated.  Also, an error analysis technique with residual function is developed for the presented method. Some illustrative examples are given to demonstrate the efficiency and applicability of the method. The computer algebraic system Maple 15 is used for all calculations and graphs. 

Keywords

• Functional differential equations,Proportional and variable delays

References

1. Reference1 Dix, J.G., "Asiymptotic behaviour of solutions to a first-order differential equation with variable delays", Computers and Mathematics with Applications, 50, 1791-1800, 2005.
2. Reference2 Liu, X.G., B., Tang, M.L., Martin, R.R., "Periodic solution for a kind of Lienard Equation", Journal of Computational and Applied Mathematics, 219,1, 263-275, 2008.
3. Reference3 Schley, D., Shail, R. Gourley, S.A., "Stability criteria for differential equations with variable time delays", International Journal of Mathematical Education in Science and Technology, 33, 3, 359-375, 2002.
4. Reference4 Graef, J.R. and Qian, C. "Global attractivity differential equations with variable delays", J. Austral. Math. Soc. Ser. B, 41, 568-579, 2000.
5. Reference5 Caraballo, T. and Langa, J.A., "Attractors for differential equations with variable delays", J. Math. Anal. Appl., 260, 421-438, 2001.
6. Reference6 Diblik, J., Svaboda, Z., Smarda, Z. "Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case", Computers and Mathematics with Applications, 56, 556-564, 2008.
7. Reference7 Zhang, B., "Fixed points and stability in differential equations with variable delays", Nonlinear Analysis, 63, 233-242, 2005.
8. Reference8 Ishawata, E. and Muroya, Y. "Rational approximation method for delay differential equations with proportional delay", Applied Mathematics and Comput., 187,2, 741-747, 2007.

9. Reference9 Ishawata, E., Muroya, Y., Brunner, H. "A super-attainable order in collocation methods for differential equations with proportional delay" Applied Mathematics and Comput., 198,1, 227-236, 2008. Reference10 Hu, P., Huang, C., Wu, S. "Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations", Applied Mathematics and Comput., 211,1, 95-101, 2009.
10. Reference10 Hu, P., Huang, C., Wu, S., "Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations", Applied Mathematics and Comput., 211,1, 95-101, 2009.
11. Reference11 Bellen, A. and Zennaro, M. Numerical methods for delay differential equations in Numerical Mathematics and Scientific Computations, Oxford University Press, New York, 2003.
12. Reference12 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).
13. Reference13 Wang, W.S and Li, S., "On the one-leg h-methods for solving nonlinear neutral functional differential equations", Applied Mathematics and Comput, 193,1,285-301, 2007.
14. Reference14 Wang, W, Qin, T., Li, S.,"Stability of one-leg h-methods for nonlinear neutral differential equations with proportional delay", Applied Mathematics and Comput, 213,1, 177-183,2009.
15. Reference15 Sezer, M. Daşcioglu, A. A., "A Taylor method for numerical solution of generalized pantograph equations with linear functional argument ", Journal of Computational and Applied Mathematics, 200, 217-225,2007.
16. Reference16 Gokmen, E. Sezer, M., " Approximate solution of a model describing biological species living together by Taylor collocation method", New Trends in Math. Sci., 3,2,147-158, 2015.
17. Reference17 Oliveira, F. A., "Collocation and residual correction", Numer Math., 36, 27– 31, 1980.
18. Reference18 Çelik, I., "Approximate calculation ofeigenvalues with the method of weighted residual collocation method", Applied Mathematics and Computation, 160, 2, 401-410, 2005.
19. Reference19 Çelik, I., "Collocation method and residual correction using Chebyshev series", Applied Mathematics and Computation, 174,2, 910-920, 2006.