Abstract
This paper considers the numerical solution of delay differential equations. The predictor–corrector scheme based on generalized multistep methods are implemented in variable order variable stepsize techniques. The formulae are represented in divided difference form where the integration coefficients are computed by a simple recurrence relation. This representation produces simpler calculation as compared with the modified divided difference form, but no sacrifice is made in efficiency and accuracy of the method. Numerical results prove that the method is reliable, efficient and accurate. The P- and Q-stability regions for a fixed stepsize of the predictor–corrector scheme are illustrated for various orders.
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References
Al-Mutib, A.N.: Numerical methods for solving delay differential equations. Ph.D.thesis, University of Manchester, UK (1977)
Al-Mutib A.N.: Stability properties of numerical methods for solving delay differential equations. J. Comput. Appl. Math. 10, 71–79 (1984)
Baker, C.T.H., Paul, C.A.H, Wille, D.R.: Issues in the numerical solution of evolutionary delay differential equations. Technical Report No. 248, University of Manchester, UK (1994)
Barwell V.K.: Special stability problems for functional differential equations. BIT 15, 130–135 (1975)
Driver R.D.: Ordinary and Delay Differential Equations. Springer, New York (1977)
Jackiewicz Z.: Asymptotic stability analysis of θ-methods for functional differential equations. Numer. Math. 43, 389–396 (1984)
Jackiewicz Z.: Variable-step variable-order algorithm for the numerical solution of neutral functional differential equations. Appl. Numer. Math. 3, 317–329 (1987)
Jackiewicz Z., Lo E.: The numerical solution of neutral functional differential equations by Adams predictor–corrector methods. Appl. Numer. Math. 8, 477–491 (1991)
Jackiewicz Z., Lo E.: Numerical solution of neutral functional differential equations by Adams methods in divided difference form. J. Comput. Appl. Math. 189, 592–605 (2006)
Kemper G.A.: Linear multistep methods for a class of functional differential equations. Numer. Math. 19, 361–372 (1972)
Lambert J.D.: Computational Methods in Ordinary Differential Equations. Wiley, London (1973)
Shampine L.F., Gordon M.K.: Computer Solution of Ordinary Differential Equations: The Initial Value Problem. W. H. Freeman, San Francisco (1975)
Shampine L.F., Thompson S.: Solving DDEs in MATLAB. Appl. Numer. Math. 37, 441–458 (2001)
Suleiman, M.B.: Generalized multistep Adams and backward differentiation methods for the solution of stiff and non-stiff ordinary differential equations. Ph.D.thesis, University of Manchester, UK (1979)
Suleiman M.B., Ismail F.: Solving delay differential equations using componentwise partitioning by Runge–Kutta method. Appl. Math. Comput. 122, 301–323 (2001)
Wiederholt L.F.: Stability of multistep methods for delay differential equations. Math. Comp. 30, 283–290 (1976)
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Suleiman, M.B., Ishak, F. Numerical solution and stability of multistep method for solving delay differential equations. Japan J. Indust. Appl. Math. 27, 395–410 (2010). https://doi.org/10.1007/s13160-010-0017-6
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DOI: https://doi.org/10.1007/s13160-010-0017-6
Keywords
- Delay differential equations
- Variable order variable stepsize
- Multistep methods
- Predictor–corrector
- Stability regions