A class of three-point spline collocation methods for solving delay differential equations

Abstract

In this paper a study of the existence, uniqueness, stability and convergence of a class of C ²-spline collocation methods for solving delay differential equations (DDEs) is introduced. Letting the interior collocation points , j=1(1)3 be dependent on the parameters c ₁, c ₂∈(0, 1) and c ₃=1 it is shown that the proposed methods for DDEs possess a convergence rate of order six if 58−57(c ₁+c ₂)+55c ₁ c ₂=0, and they are unstable if c ₁+c ₂<1. Moreover, the methods are P-stable for 0.8028≤c ₁<c ₂. Numerical results illustrating the behaviour of the methods when faced with some difficult problems are presented and the numerical results are compared to those obtained by other methods.

Keywords:

• Delay differential equations
• Spline collocation methods
• Stability analysis

Acknowledgements

The authors are indebted to Professor S. E. El. Gendi for various valuable suggestions and constructive criticism.

Additional information

Notes on contributors

S. M. Mahmoud

Email: [email protected]