Abstract
In this article, we describe a method to solve differential equations involved in dynamical nonlinear problems with C k spline functions and their algebraic properties. C k spline functions generate k time continuous and derivable approximations. These functions seem to be very efficient in several applied nonlinear differential problems. The computation of functions involved in nonlinear dynamical problem uses the algebraic methods known as Gröbner Bases, which are more robust, more accurate and often more efficient than the purely numerical methods. An example on solving nonlinear differential equations using these methods is presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Rouff (1992) The Computation of Ck spline function, Computers and Mathematics with Applications, 23(1):103–110.
M. Rouff (1996) "Ck spline functions in applied differential problem", Systems Analysis Modelling and Simulations, Gordon & Breach Science Publishers, 26:197–205.
M. Rouff and M. Alaoui (1996) "Computation of dynamical electromagnetic problems using Lagrangian formalism and multidimensional Ck spline functions", Zeitschrift für Angewandte Mathematik und Mechanik, Academie Verlag Berlin, 76(1):513–514.
M. Rouff and W.L. Zhang (1994) "Ck spline functions and linear operators", Computers and Mathematics with Applications, 28(4):51–59.
B. Buchberger (1965) Ein Algorithmus zum Auffnden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. (PhD thesis, Innsbruck).
B. Buchberger (1970) "An Algorithmical Criterion for the Solvability of Algebraic Systems". Aequationes Mathematicae 4(3):374–383. (German).
B. Buchberger (1979) "A Criterion for Detecting Unnecessary Reductions in the Construction of Gröbner Basis". In Proc. EUROSAM 79, vol. 72 of Lect. Notes in Comp. Sci., Springer Verlag, 3–21.
J.C Faugère (1994) Résolution des systèmes d’équations algébriques. (PhD thesis, Université Paris 6).
J.C Faugère (1999) "A new efficient algorithm for computing Grôbner bases (F4)". Journal of Pure and Applied Algebra 139:61–88.
Faugère J.C., Gianni P., Lazard D., Mora T. (1993) "Efficient Computation of Zero-Dimensional Gröbner Basis by Change of Ordering". Journal of Symbolic Computation 16(4):329–344.
D. Lazrad (1992) "Solving zero-dimensional algebraic systems". Journal of Symbolic Computation 13(2):117–132. available by anonymous ftp posso.lip6.fr.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer
About this chapter
Cite this chapter
Lakhdari, Z., Makany, P., Rouff, M. (2006). Computer Algebra and C k Spline Functions: A Combined Tools toSolve Nonlinear Differential Problems. In: Aziz-Alaoui, M., Bertelle, C. (eds) Emergent Properties in Natural and Artificial Dynamical Systems. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-34824-7_13
Download citation
DOI: https://doi.org/10.1007/3-540-34824-7_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34822-1
Online ISBN: 978-3-540-34824-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)