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Linear Complete Differential Resultants and the Implicitization of Linear DPPEs [article]

Sonia L. Rueda, J. Rafael Sendra
2008 arXiv   pre-print
We study the computation by linear complete differential resultants of the implicit equation of a system of n linear differential polynomial parametric equations in n-1 differential parameters.  ...  The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined.  ...  Implicitization of linear DPPEs by linear Complete Differential Resultants Let P(X, U), F i , H i be as in Section 2 and let ID be the implicit ideal of P(X, U).  ... 
arXiv:0712.0785v2 fatcat:2dbtj6vwunbqversab5vhf2j3i

Linear complete differential resultants and the implicitization of linear DPPEs

Sonia L. Rueda, J. Rafael Sendra
2010 Journal of symbolic computation  
We study the computation by linear complete differential resultants of the implicit equation of a system of n linear differential polynomial parametric equations in n − 1 differential parameters.  ...  The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined.  ...  Acknowledgements The authors thank the referees for their questions and comments allowing the achievement of the final version of this paper.  ... 
doi:10.1016/j.jsc.2009.09.003 fatcat:ltyyzb2e5nalhhxd3uc7fhigry

An A- Stable Block Integrator Scheme for the Solution of First Order System of IVP of Ordinary Differential Equations

Muhammad Abdullahi, Shamsuddeen Suleiman, Abdu Masanawa Sagir, Bashir Sule
2022 Asian Journal of Probability and Statistics  
Some test problems are solved with the proposed scheme and the result are compared with some existing method.  ...  Hence, the method is recommended for solving first order system of IVP of ordinary differential equations.  ...  Definition 6 (Explicit and Implicit method) The general linear multi-step method is said to be Explicit if , otherwise it is Implicit .  ... 
doi:10.9734/ajpas/2022/v16i430407 fatcat:k5b7n4ywr5gy7o2vy2dkspxjqi

A perturbed differential resultant based implicitization algorithm for linear DPPEs [article]

Sonia L. Rueda
2010 arXiv   pre-print
A nonzero linear differential polynomial in 𝕀 is obtained from the lowest degree term of _ϕ and used to provide an implicitization algorithm for .  ...  We prove the existence of a linear perturbation _ϕ of so that the linear complete differential resultant _ϕ associated to _ϕ is nonzero.  ...  Rafael Sendra for kindly reading and commenting on some parts of this work.  ... 
arXiv:1003.4375v1 fatcat:a3cmwxduo5dgtcj27eg4h6q45m

A perturbed differential resultant based implicitization algorithm for linear DPPEs

Sonia L. Rueda
2011 Journal of symbolic computation  
A nonzero linear differential polynomial in ID is obtained, from the lowest degree term of ∂CRes φ , and used to provide an implicitization for P .  ...  We prove the existence of a linear perturbation P φ of P , so that the linear complete differential resultant ∂CRes φ associated to P φ is nonzero.  ...  DPPEs, and use linear complete differential resultants to give a candidate for the implicit equation of the system.  ... 
doi:10.1016/j.jsc.2011.05.001 fatcat:yqcxqqz2gneu7ls2u4za7jaoiy

Page 6923 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews  
Both negative and positive results are presented.  ...  These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in  ... 

Page 3740 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
As a result, we produce a new efficient procedure to solve linear systems arising in an application of implicit Runge-Kutta methods to large-scale differential-algebraic equations of index 1.  ...  Linear and non-linear illustrations are given to show the practical usefulness of the approach developed.  ... 

Transactions of the chemical engineering congress

2010 Journal of the Society of Chemical Industry  
This ODE involves a reduced number of dependent variables, and some evaluations of implicit functions defined, either from the original algebraic constraints, or from the hidden ones.  ...  After two deflation steps, this DAE is converted to an equivalent first-order explicit ordinary differential equation (ODE).  ...  For such a quasi linear DAE model, the differentiation index is more difficult to define, and to compute, than in the linear case.  ... 
doi:10.1002/jctb.5000562803 fatcat:ytn7bwlfv5cchit5zmm2v3ftue

Converting DAE Models to ODE Models: Application to Reactive Rayleigh Distillation

K. Alloula, F. Monfreda, R. Thery Hetreux, J.P. Belaud
2013 Chemical Engineering Transactions  
This ODE involves a reduced number of dependent variables, and some evaluations of implicit functions defined, either from the original algebraic constraints, or from the hidden ones.  ...  After two deflation steps, this DAE is converted to an equivalent first-order explicit ordinary differential equation (ODE).  ...  For such a quasi linear DAE model, the differentiation index is more difficult to define, and to compute, than in the linear case.  ... 
doi:10.3303/cet1332220 doaj:a5fe247b60374415bfb6f0fbef8bb65b fatcat:lptnkwt2ufcunfgssf4qiepnja

Page 3194 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews  
For the linear autonomous system y’ = Ay, these linearly implicit methods are identical with certain implicit multistep and one-leg methods, respectively.  ...  For the stiff equations implicit backward differentiation formulae are used and for the remaining equations Adams formulae with a variable-order code are used.  ... 

Page 3596 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews  
wave speed and a time constant, from the resulting al- gebraic eigenvalue problem.  ...  ’ of these iterative implicit schemes are revealed and compared.  ... 

Linearization methods for reaction-diffusion equations: 1-D problems

J.I. Ramos
1997 Applied Mathematics and Computation  
The second one requires the solution of two-point, linear, ordinary differential equations and provides either piecewise continuous or piecewise differentiable solutions in space and discrete in time.  ...  The third type is based on the discretization of the spatial coordinate and provides a system of linear, ordinary differential equations in time which can be integrated analytically.  ...  Garc'za-Lbpez for some useful comments on the relation between Rosenbrock's methods and linearized O-methods.  ... 
doi:10.1016/s0096-3003(96)00328-1 fatcat:ikz4iqwdlzcbzgdzqjkg3ydvty

Page 5317 of Mathematical Reviews Vol. , Issue 2004g [page]

2004 Mathematical Reviews  
Differential Equations 19 (2003), no. 2, 170-181. In this paper the author states and proves a result of Jorba-Simo type [A. Jorba and C. Simo, J.  ...  Although there have been a number of articles that approach implicit differential equa- tions algebraically, all such approaches have relied heavily on linear algebra.  ... 

Page 5101 of Mathematical Reviews Vol. , Issue 92i [page]

1992 Mathematical Reviews  
Ordinary differential equations 92i:65118 ods (for example, explicit and implicit Runge-Kutta methods) is discussed.  ...  Other topics treated are the following: the computational complexity involved in implicit or semi-implicit methods and in local error estimation for explicit methods; linear stability theory examined as  ... 

Linear-implicit strong schemes for Itô-Galkerin approximations of stochastic PDEs

P. E. Kloeden, S. Shott
2001 Journal of Applied Mathematics and Stochastic Analysis  
Linear-implicit versions of strong Taylor numerical schemes for finite dimensional It6 stochastic differential equations (SDEs) are shown to have the same order as the original scheme.  ...  The combined truncation and global discretization error of an 7 strong linear-implicit Taylor scheme with time-step A applied to the N dimensional It6-Galerkin SDE for a class of parabolic stochastic partial  ...  thus be of the form KT, UoA' and hence the combined truncation and global discretization error bound will be as asserted in Theorem 2. D  ... 
doi:10.1155/s1048953301000053 fatcat:e7vxpy44zbeh3bizrbkx5ux3iu
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