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Transformation of Differential Algebraic Array Equations to Index One Form

Martin Otter, Hilding Elmqvist
2017 Proceedings of the 12th International Modelica Conference, Prague, Czech Republic, May 15-17, 2017  
Several new algorithms are proposed that in effect transform DAEs (Differential Algebraic Equations) to a special index one form that can be simulated with standard DAE integrators.  ...  The transformation to this form is performed without solving linear and/or nonlinear equation systems, the sparsity of the equations is kept, and array equations remain array equations or differentiated  ...  Acknowledgements The authors would like to thank Martin Arnold from University Halle-Wittenberg for discussions to understand the fine details of multibody integrators.  ... 
doi:10.3384/ecp17132565 dblp:conf/modelica/OtterE17 fatcat:kp5e46dhhbcyhn6ulqga5aszrq

A derivative array approach for linear second order differential-algebraic systems

Lena Scholz
2011 The Electronic Journal of Linear Algebra  
At first, in Section 2, we present the basic results of the analysis of linear second order differential-algebraic equations as derived in [12] , including a new condensed form that allows one to read  ...  We present an index reduction method based on derivative arrays that allows to determine an equivalent second order system of lower index in a numerical computable way.  ...  Therefore, the numerical solution of linear second order differential-algebraic systems of the form (1.1) typically requires the reduction to a first order system on the one hand and an index reduction  ... 
doi:10.13001/1081-3810.1441 fatcat:fltmirbb4nd5rasr35mnidbf6m

Review of applied multidimensional systems theory

Stephen L. Campbell
1984 Linear Algebra and its Applications  
Among the areas of~applied mathematics that heavily use linear algebra, systems theory is currently one of the most active.  ...  The general heading of systems theory includes topics from control theory, circuit theory, and power systems to name a few. For the person with a strong linear algebra background wanting to learn  ...  The emphasis is on the z transform and difference equations.  ... 
doi:10.1016/0024-3795(84)90194-0 fatcat:azsvcaeomrebxolb444ml6lbzi

A Differentiation Index for Partial Differential-Algebraic Equations

Wade S. Martinson, Paul I. Barton
2000 SIAM Journal on Scientific Computing  
This index is thus a generalization of the differentiation index of differential-algebraic equations.  ...  A differentiation index for nonlinear partial differential-algebraic equations is presented.  ...  From here onward the term index will refer to the differentiation index. Algebraic equations may be thought of as constraints on or invariants of the system.  ... 
doi:10.1137/s1064827598332229 fatcat:hwa5qf6kw5dgvivwxocfowvrxy

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink [chapter]

Kumaresan Nallasamy, Kuru Ratnavelu, Bernardine R. Wong
2011 Communications in Computer and Information Science  
Since such differential algebraic systems have a difficulty to solve in their original form, most approaches use some kind of index reduction.  ...  In this paper, the unsteady Navier-Stokes Takagi-Sugeno (T-S) fuzzy equations (UNSTSFEs) are represented as a differential algebraic system of strangeness index one by applying any spatial discretization  ...  Acknowledgements: The funding of this work by the UMRG grant (Account No: RG099/10AFR) is gratefully acknowledged.  ... 
doi:10.1007/978-3-642-19263-0_7 fatcat:7fatu6sc6rcirkmepyiv5rd6xi

Eigenvalue placement in completions of DAES

Stephen L. Campbell, Lise E. Holte
2013 The Electronic Journal of Linear Algebra  
Differential algebraic equations (DAEs) are used to describe many physical processes. A completion of a DAE is an ordinary differential equation whose solutions include those of the DAE.  ...  Algorithms exists for designing stabilized completions of differential algebraic equations.  ...  Differential algebraic equations (DAEs) are mixed systems of differential and algebraic equations F (x ′ , x, t) = 0 for which the Jacobian F x ′ is identically singular.  ... 
doi:10.13001/1081-3810.1667 fatcat:qxposxu4bjbidisi2chwt7sfty

Index reduction for differential-algebraic equations by minimal extension

P. Kunkel, V. Mehrmann
2004 Zeitschrift für angewandte Mathematik und Mechanik  
Based on this information reduced derivative arrays are formed and instead of using expensive subspace computations the index reduction is obtained by introducing new variables.  ...  In this paper a new index reduction technique is discussed for the treatment of differential-algebraic systems for which extra structural information is available.  ...  Acknowledgment We thank Diana Estevez-Schwarz and Caren Tischendorf for many helpful discussions concerning the topological characterization of the high index equations in circuit simulation and Ingo Seufer  ... 
doi:10.1002/zamm.200310127 fatcat:75xlxeia4ve35c3sjg5nccnhjm

Einstein summation for multidimensional arrays

K. Åhlander
2002 Computers and Mathematics with Applications  
one of the most common data structures, at least in scientific computing, is the multidimensional array.  ...  An objectoriented implementation of a C++ library that supports index notation is described.  ...  These quantities may also vary over the computational domain, which requires the data structures to be arrays of arrays, one array referring to the physical quantity and one array referring to the discretization  ... 
doi:10.1016/s0898-1221(02)00210-9 fatcat:vouqlohrlfb5tkmloocgo5lgem

Introduction to maple

2004 Computers and Mathematics with Applications  
The war of attrition. 7.4.1. The discrete war of attrition. 7.4.2. The continuous war of attrition. 7.4.3. The discrete war of aggression. 7.5. The finitely repeated prisoner's dilemma game. 7.5.1.  ...  Centipede games of lengths two and three.  ...  Introduction to computer algebra. 1.1. What is computer algebra? 1.2. Computer algebra systems. 1.3. Some properties of computer algebra systems. 1.4. Advantages of computer algebra. 1.5.  ... 
doi:10.1016/s0898-1221(04)90103-4 fatcat:vqqmhhwhjng4nj4tcbzcobgutm

Matrix tensor notation part I. Rectilinear orthogonal coordinates

W.C. Hassenpflug
1993 Computers and Mathematics with Applications  
This is achieved by the introduction of the base as a tensor quantity in the fundamental equation for the relation between a tensor and its representation as a matrix.  ...  It is also shown that the same notation is equally useful for vectors in abstract higher dimensional space and transformations and transforms in function space.  ...  We apply now the same transformation algebra to the base and to the transformation equation (11) This equation shows that g is the Unit space tensor and the base is merely a mixed form of the unit tensor  ... 
doi:10.1016/0898-1221(93)90110-h fatcat:dtn4dtng3rfptaif3m3ggvog5e

Characterization of classes of singular linear differential-algebraic equations

Peter Kunkel, Volker Mehrmann
2005 The Electronic Journal of Linear Algebra  
In particular, three different characterizations are given for differential-algebraic equations, namely by means of solution spaces, canonical forms, and derivative arrays.  ...  Linear, possibly over-or underdetermined, differential-algebraic equations are studied that have the same solution behavior as linear differential-algebraic equations with well-defined strangeness index  ...  For regular DAEs, it has been shown in [1] that (1.1) has a well-defined differentiation index and that (1.1) can be transformed to a canonical form.  ... 
doi:10.13001/1081-3810.1169 fatcat:veur6fcp4zbfpeszmowapiueui

Page 867 of Mathematical Reviews Vol. , Issue 98B [page]

1998 Mathematical Reviews  
In particular, an index notion called the strangeness index was introduced. The transformation algorithm to the canonical form was not efficiently implementable.  ...  In the present paper, linear index 2 differential-algebraic equations are considered.  ... 

Index and Characteristic Analysis of Linear PDAE Systems

Wade S. Martinson, Paul I. Barton
2003 SIAM Journal on Scientific Computing  
These packages take as input the differential and algebraic model equations that describe the physical behaviors of the process, processing tasks, and discrete events, and provide tools such as model inheritance  ...  Measures of the degeneracy of the system, rather than a differentiation index, bound the order of derivatives of forcing functions that appear in the solution. 1. Introduction.  ...  If the governing PDE has a differentiation index [17] of 2 or higher with respect to the evolution variable, one expects a high-index differential-algebraic equation (DAE) to result from any reasonable  ... 
doi:10.1137/s1064827599363411 fatcat:5x3ag6navjamllgxfpchp24cdy

Mathieu-Bragg photonic lattices [article]

I. Ramos-Prieto, K. Uriostegui, J. Récamier, F. Soto-Eguibar, H. M. Moya-Cessa
2021 arXiv   pre-print
The dynamics is shown to be fully integrable and therefore described in closed form. We examine the trajectories of classical light propagating in such structures.  ...  We show that Bragg diffraction may be modeled by classical light propagation in photonic lattices having a square power law for the refraction index coefficient.  ...  To increase the index of refraction in a lattice, as the one shown in Fig. 1 , may be a difficult task, as the coupling has to grow quadratically; therefore, ways of simulating such waveguide array may  ... 
arXiv:2107.06273v2 fatcat:zxh4plmq25fobdwf5ftmkkw52m

The additional dynamics of least squares completions for linear differential algebraic equations

Irfan Okay, Stephen L. Campbell, Peter Kunkel
2007 Linear Algebra and its Applications  
Several approaches have been proposed for numerically solving lower dimensional, nonlinear, higher index differential algebraic equations (DAEs) for which more classical numerical methods such as backward  ...  One of these approaches is called explicit integration (EI). This approach is based on solving nonlinear DAE derivative arrays using nonlinear singular least squares methods.  ...  Acknowledgments We would like to thank Hongguo Xu for his helpful comments on Lemma 3.  ... 
doi:10.1016/j.laa.2007.02.005 fatcat:nxgy7wwl4jaijm4wrq3ptpiydq
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