Abstract
The Enright’s second derivative formula which is \(A\)-stable up to order four is derived using the multistep collocation method. The continuous schemes obtained are used to generate complementary methods together with the proposed method to solve standard problems via the boundary value techniques such that the numerical solution of a problem is obtained on the domain of integration simultaneously. Implementation on linear and non-linear stiff systems shows that the new algorithm is efficient and error is minimal compared to step-by-step techniques.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amodio P, Mazzia F (1995) Variable-step boundary value methods based on reverse Adams schemes and their grid distribution. Appl Numer Math 18:5–21
Axelsson AOH, Verwer JG (1985) Boundary value techniques for initial value problems in ordinary differential equations. Math Comput 45:153–171
Bickart TA, Rubin WB (1974) Composite multistep methods and stiff stability. In: Willoughby RA (ed) Stiff differential systems. Plenum Press, New York
Brugnano L, Trigiante D (1998) Solving differential problems by multitep initial and boundary value methods. Gordon and Breach Science Publishers, Amsterdam
Butcher JC (2003) Numerical methods for ordinary differential equations. Wiley, England
Cash JR (1981a) On the exponential fitting of composite, multiderivate liinear multistep methods. SIAM J Numer Anal 18(5):808–821
Cash JR (1981b) Second derivative extended backward differentiation formula for the numerical integration of stiff system. SIAM J Numer Anal 18(5):21–36
Dahlquist G (1963) A special stability problem for linear multistep methods. BIT 3:27–43
Ehigie JO, Okunuga SA, Sofoluwe AB (2011) 3-point block methods for direct integration of second order ordinary differential equations. Adv Numer Anal. doi:10.1155/2011/513148
Ehle BL (1969) On pade approximations to the exponential function and A-stable methods for the numerical solution of initial value problems. Research Report CSRR 2010, Dept. AACS, University of Waterloo, Ontario
Enright WH (1974) Second derivative multistep methods for stiff ordinary differential equations. SIAM J Numer Anal 11(2):321–331
Fatunla SO (1992) Parallel methods for second order ODEs. In: Fatunla SO (ed) Computational ordinary differential equations. University Press Plc, Ibadan, pp 87–99
Gear CW (1971) Numerical initial value problems in ordinary differential equations. Prentice-Hall, New Jersey
Hairer E, Wanner G (1996) Solving ordinary differential equations II: stiff and differential algebraic problems. 2nd Revised edn. Springer, Germany
Jackson LW, Kenue SK (1974) A fourth-order exponentially fitted method. SIAM J Numer Anal 11:965–978
Jia-Xiang X, Jiao-Xun K (1988) A class of DBDF methods with the derivative modifying term. J Comput Math 6(1):7–13
Jator SN, Sahi RK (2010) Boundary value technique for initial value problems based on Adams-type second derivative methods. Int J Math Educ Sci Educ iFirst:1–8
Lambert JD (1973) Computational methods in ordinary differential equations. Wiley, New York
Liniger W, Willoughby RA (1970) Efficient integration methods for stiff systems of ordinary differential equations. SIAM J Numer Anal 7(Number, if it exists):47–65
Obrechkoff N (1940) Neue Quadraturformeln. Abh Preuss Akad Wiss Math Nat Kl 4
Okunuga SA (1992) A new composite multiderivative linear multistep methods For solving stiff initial value problems. Ph.D Thesis. University of Lagos, Lagos (submitted)
Okunuga SA, Ehigie JO (2009) A new derivation of continuous collocation multistep methods using power series as basis function. J Mod Math Stat 3(2):43–50
Onumanyi P, Awoyemi DO, Jator SN, Sirisena UW (1994) New linear mutlistep methods with continuous coefficients for first order initial value problems. J Niger Math Soc 13:37–51
Wu X, Xia J (2001) Two low accuracy methods for stiff systems. Appl Math Comput 123:141–153
Acknowledgments
The authors are grateful to the anonymous referee for his useful comments and suggestions in the course of this research work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Antonio Silva Neto.
Rights and permissions
About this article
Cite this article
Ehigie, J.O., Jator, S.N., Sofoluwe, A.B. et al. Boundary value technique for initial value problems with continuous second derivative multistep method of Enright. Comp. Appl. Math. 33, 81–93 (2014). https://doi.org/10.1007/s40314-013-0044-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-013-0044-4