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Relevant bibliographies by topics / Differential and difference algebra

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Academic literature on the topic 'Differential and difference algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 February 2022

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Journal articles on the topic "Differential and difference algebra"

1

Mikhalev, A. V., and E. V. Pankrat'ev. "Differential and difference algebra." Journal of Soviet Mathematics 45, no. 1 (April 1989): 912–55. http://dx.doi.org/10.1007/bf01094866.

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Zhang, Yang, and Xiangui Zhao. "Gelfand–Kirillov dimension of differential difference algebras." LMS Journal of Computation and Mathematics 17, no. 1 (2014): 485–95. http://dx.doi.org/10.1112/s1461157014000102.

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AbstractDifferential difference algebras, introduced by Mansfield and Szanto, arose naturally from differential difference equations. In this paper, we investigate the Gelfand–Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand–Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand–Kirillov dimension under some specific conditions and construct an example to show that this upper bound cannot be sharpened any further.

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Zhao, Xiangui, and Yang Zhang. "Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras." Algebra Colloquium 23, no. 04 (September 26, 2016): 701–20. http://dx.doi.org/10.1142/s1005386716000596.

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Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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Li, Wei, and Chun-Ming Yuan. "Elimination Theory in Differential and Difference Algebra." Journal of Systems Science and Complexity 32, no. 1 (February 2019): 287–316. http://dx.doi.org/10.1007/s11424-019-8367-x.

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Hooker, John W. "Some differences between difference equations and differential equations." Journal of Difference Equations and Applications 2, no. 2 (January 1996): 219–25. http://dx.doi.org/10.1080/10236199608808056.

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DOBREV, V. K., H. D. DOEBNER, and C. MRUGALLA. "DIFFERENCE ANALOGUES OF THE FREE SCHRÖDINGER EQUATION." Modern Physics Letters A 14, no. 17 (June 7, 1999): 1113–22. http://dx.doi.org/10.1142/s021773239900119x.

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We propose an infinite family of difference equations, which are derived from the first principle that they are invariant with respect to the Schrödinger algebra. The first member of this family is a difference analogue of the free Schrödinger equation. These equations are obtained via a purely algebraic construction from a corresponding family of singular vectors in Verma modules over the Schrödinger algebra. The crucial moment in the construction is the realization of the Schrödinger algebra through additive difference vector fields, i.e. vector fields with difference operators instead of differential operators. Our method produces also differential-difference equations in which only space- or time-differentiation is replaced with the corresponding difference operators.

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Kakei, Saburo, and Yasuhiro Ohta. "A differential-difference system related to toroidal Lie algebra." Journal of Physics A: Mathematical and General 34, no. 48 (November 28, 2001): 10585–92. http://dx.doi.org/10.1088/0305-4470/34/48/322.

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Levin, Alexander. "Reduced Gröbner Bases, Free Difference–Differential Modules and Difference–Differential Dimension Polynomials." Journal of Symbolic Computation 30, no. 4 (October 2000): 357–82. http://dx.doi.org/10.1006/jsco.1999.0412.

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Dunkl, Charles F. "Computing with Differential-difference Operators." Journal of Symbolic Computation 28, no. 6 (December 1999): 819–26. http://dx.doi.org/10.1006/jsco.1997.0341.

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SMIRNOV, YURI, and ALEXANDER TURBINER. "LIE ALGEBRAIC DISCRETIZATION OF DIFFERENTIAL EQUATIONS." Modern Physics Letters A 10, no. 24 (August 10, 1995): 1795–802. http://dx.doi.org/10.1142/s0217732395001927.

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A certain representation for the Heisenberg algebra in finite difference operators is established. The Lie algebraic procedure of discretization of differential equations with isospectral property is proposed. Using sl 2-algebra based approach, (quasi)-exactly-solvable finite difference equations are described. It is shown that the operators having the Hahn, Charlier and Meissner polynomials as the eigenfunctions are reproduced in the present approach as some particular cases. A discrete version of the classical orthogonal polynomials (like Hermite, Laguerre, Legendre and Jacobi ones) is introduced.

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Dissertations / Theses on the topic "Differential and difference algebra"

1

Hendriks, Peter Anne. "Algebraic aspects of linear differential and difference equations." [S.l. : [Groningen] : s.n.] ; [University Library Groningen] [Host], 1996. http://irs.ub.rug.nl/ppn/153769580.

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Zhao, Xiangui. "Groebner-Shirshov bases in some noncommutative algebras." London Mathematical Society, 2014. http://hdl.handle.net/1993/24315.

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Groebner-Shirshov bases, introduced independently by Shirshov in 1962 and Buchberger in 1965, are powerful computational tools in mathematics, science, engineering, and computer science. This thesis focuses on the theories, algorithms, and applications of Groebner-Shirshov bases for two classes of noncommutative algebras: differential difference algebras and skew solvable polynomial rings. This thesis consists of three manuscripts (Chapters 2--4), an introductory chapter (Chapter 1) and a concluding chapter (Chapter 5). In Chapter 1, we introduce the background and the goals of the thesis. In Chapter 2, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We find lower and upper bounds of the Gelfand-Kirillov dimension of a differential difference algebra under some conditions. We also give examples to demonstrate that our bounds are sharp. In Chapter 3, we generalize the Groebner-Shirshov basis theory to differential difference algebras with respect to any left admissible ordering and develop the Groebner-Shirshov basis theory of finitely generated free modules over differential difference algebras. By using the theory we develop, we present an algorithm to compute the Gelfand-Kirillov dimensions of finitely generated modules over differential difference algebras. In Chapter 4, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Groebner-Shirshov bases in skew solvable polynomial rings over fields. Our algorithm can detect redundant reductions and therefore it is more efficient than the traditional Buchberger algorithm. Finally, in Chapter 5, we summarize our results and propose possible future work.

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El-Nakla, Jehad A. H. "Finite difference methods for solving mildly nonlinear elliptic partial differential equations." Thesis, Loughborough University, 1987. https://dspace.lboro.ac.uk/2134/10417.

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This thesis is concerned with the solution of large systems of linear algebraic equations in which the matrix of coefficients is sparse. Such systems occur in the numerical solution of elliptic partial differential equations by finite-difference methods. By applying some well-known iterative methods, usually used to solve linear PDE systems, the thesis investigates their applicability to solve a set of four mildly nonlinear test problems. In Chapter 4 we study the basic iterative methods and semiiterative methods for linear systems. In particular, we derive and apply the CS, SOR, SSOR methods and the SSOR method extrapolated by the Chebyshev acceleration strategy. In Chapter 5, three ways of accelerating the SOR method are described together with the applications to the test problems. Also the Newton-SOR method and the SOR-Newton method are derived and applied to the same problems. In Chapter 6, the Alternating Directions Implicit methods are described. Two versions are studied in detail, namely, the Peaceman-Rachford and the Douglas-Rachford methods. They have been applied to the test problems for cycles of 1, 2 and 3 parameters. In Chapter 7, the conjugate gradients method and the conjugate gradient acceleration procedure are described together with some preconditioning techniques. Also an approximate LU-decomposition algorithm (ALUBOT algorithm) is given and then applied in conjunction with the Picard and Newton methods. Chapter 8 contains the final conclusions.

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Chen, Shaoshi. "Quelques applications de l'algébre différentielle et aux différences pour le télescopage créatif." Phd thesis, Ecole Polytechnique X, 2011. http://pastel.archives-ouvertes.fr/pastel-00576861.

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Depuis les années 90, la méthode de création télescopique de Zeilberger a joué un rôle important dans la preuve automatique d'identités mettant en jeu des fonctions spéciales. L'objectif de long terme que nous attaquons dans ce travail est l'obtension d'algorithmes et d'implantations rapides pour l'intégration et la sommation définies dans le cadre de cette création télescopique. Nos contributions incluent de nouveaux algorithmes pratiques et des critères théoriques pour tester la terminaison d'algorithmes existants. Sur le plan pratique, nous nous focalisons sur la construction de télescopeurs minimaux pour les fonctions rationnelles en deux variables, laquelle a de nombreuses applications en lien avec les fonctions algébriques et les diagonales de séries génératrices rationnelles. En considérant cette classe d'entrées contraintes, nous parvenons à mâtiner la méthode générale de création télescopique avec réduction bien connue d'Hermite, issue de l'intégration symbolique. En outre, nous avons obtenu pour cette sous-classe quelques améliorations des algorithmes classiques d'Almkvist et Zeilberger. Nos résultats expérimentaux ont montré que les algorithmes à base de réduction d'Hermite battent tous les autres algorithmes connus, à la fois en ce qui concerne la complexité au pire et en ce qui concerne les mesures de temps sur nos implantations. Sur le plan théorique, notre premier résultat est motivé par la conjecture de Wilf et Zeilberger au sujet des fonctions hyperexponentielles-hypergéométriques holonomes. Nous présentons un théorème de structure pour les fonctions hyperexponentielles-hypergéométriques de plusieurs variables, indiquant qu'une telle fonction peut s'écrire comme le produit de fonctions usuelles. Ce théorème étend à la fois le théorème d'Ore et Sato pour les termes hypergéométriques en plusieurs variables et le résultat récent par Feng, Singer et Wu. Notre second résultat est relié au problème de l'existence de télescopeurs. Dans le cas discret à deux variables, Abramov a obtenu un critère qui indique quand un terme hypergéométrique a un télescopeur. Des résultats similaires ont été obtenus pour le $q$-décalage par Chen, Hou et Mu. Ces résultats sont fondamentaux pour la terminaison des algorithmes s'inspirant de celui de Zeilberger. Dans les autres cas mixtes continus/discrets, nous avons obtenu deux critères pour l'existence de télescopeurs pour des fonctions hyperexponentielles-hypergéométriques en deux variables. Nos critères s'appuient sur une représentation standard des fonctions hyperexponentielles-hypergéométriques en deux variables, sur sur deux décompositions additives.

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Masmali, Ibtisam Ali. "Hopf algebra and noncommutative differential structures." Thesis, Swansea University, 2010. https://cronfa.swan.ac.uk/Record/cronfa42676.

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In this thesis I will study noncommutative differential geometry, after the style of Connes and Woronowicz. In particular two examples of differential calculi on Hopf algebras are considered, and their associated covariant derivatives and Riemannian geometry. These are on the Heisenberg group, and on the finite group A4. I consider bimodule connections after the work of Madore. In the last chapter noncommutative fibrations are considerd, with an application to the Leray spectral sequence. NOTATION. In this thesis equations are numbered as round brackets (), where (a.b) denotes equation b in chapter a, and references are indicated by square brackets []. This thesis has been typeset using Latex, and some figures using the Visio program.

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Nagloo, Joel Chris Ronnie. "Model theory, algebra and differential equations." Thesis, University of Leeds, 2014. http://etheses.whiterose.ac.uk/6813/.

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In this thesis, we applied ideas and techniques from model theory, to study the structure of the sets of solutions XII - XV I , in a differentially closed field, of the Painlevé equations. First we show that the generic XII - XV I , that is those with parameters in general positions, are strongly minimal and geometrically trivial. Then, we prove that the generic XII , XIV and XV are strictly disintegrated and that the generic XIII and XV I are ω-categorical. These results, already known for XI , are the culmination of the work started by P. Painlevé (over 100 years ago), the Japanese school and many others on transcendence and the Painlevé equations. We also look at the non generic second Painlevé equations and show that all the strongly minimal ones are geometrically trivial.

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Khalid, Abdul Muqeet. "Hypergeometric equation and differential-difference bispectrality." Thesis, University of Leeds, 2018. http://etheses.whiterose.ac.uk/21411/.

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The bispectral problem was posed by Duistermaat and Grünbaum in 1986. Since then, many interesting links of this problem with nonlinear integrable PDEs, algebraic geometry, orthogonal polynomials and special functions have been found. Bispectral operators of rank one are related to the KP equation and have been completely classified by G. Wilson. For rank greater than 1 some large families related to Bessel functions are known, although the classification problem remains open. If one generalises the bispectral problem by allowing difference operators in the spectral variable, then this has a clear parallel with the three-term recurrence relation in the theory of orthogonal polynomials. This differential-difference version of the bispectral problem has also been studied extensively, more recently in the context of the exceptional orthogonal polynomials. However, the associated special functions have not been treated in such a way, until now. In our work we make a step in that direction by constructing a large family of bispectral operators related to the hypergeometric equation. In this thesis, we will fully explain our construction.

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Khanizadeh, Farbod. "Symmetry structure for differential-difference equations." Thesis, University of Kent, 2014. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.655204.

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Having infinitely many generalised symmetries is one of the definition of integrability for non-linear differential-difference equations. Therefore, it is important to develop tools by which we can produce these quantities and guarantee the integrability. Two different methods of producing generalised symmetries are studied throughout this thesis, namely recursion operators and master symmetries. These are objects that enable one to obtain the hierarchy of symmetries by recursive action on a known symmetry of a given equation. Our first result contains new Hamiltonian, symplectic and recursion operators for several (1 + 1 )-dimensional differential-difference equations both scalar and multicomponent. In fact in chapter 5 we give the factorization of the new recursion operators into composition of compatible Hamiltonian and symplectic operators. For the list of integrable equations we shall also provide the inverse of recursion operators if it exists. As the second result, we have obtained the master symmetry of differentialdifference KP equation. Since for (2+1 )-dimensional differential-difference equations recursion operators take more complicated form, " master symmetries are alternative effective tools to produce infinitely many symmetries. The notion of master symmetry is thoroughly discussed in chapter 6 and as a result of this chapter we obtain the master symmetry for the differential-difference KP (DDKP) equation. Furthermore, we also produce time dependent symmetries through sl(2, C)-representation of the DDKP equation.

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Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.

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Xue, Fei. "Asymptotic solutions of almost diagonal differential and difference systems." Morgantown, W. Va. : [West Virginia University Libraries], 2006. https://eidr.wvu.edu/etd/documentdata.eTD?documentid=4556.

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Books on the topic "Differential and difference algebra"

1

Kondratieva, M. V. Differential and Difference Dimension Polynomials. Dordrecht: Springer Netherlands, 1999.

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Ashyralyev, Allaberen. New Difference Schemes for Partial Differential Equations. Basel: Birkhäuser Basel, 2004.

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Zbigniew, Hajto, ed. Algebraic groups and differential Galois theory. Providence, R.I: American Mathematical Society, 2011.

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Jacques, Sauloy, and Singer Michael F. 1950-, eds. Galois theories of linear difference equations: An introduction. Providence, Rhode Island: American Mathematical Society, 2016.

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Grossman, Robert. Hopf-algebraic structure of combinatorial objects and different operators. [Washington, DC: National Aeronautics and Space Administration, 1989.

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Wolfgang, Kliemann, ed. Dynamical systems and linear algebra. Providence, Rhode Island: American Mathematical Society, 2014.

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1980-, Blazquez-Sanz David, Morales Ruiz, Juan J. (Juan José), 1953-, and Lombardero Jesus Rodriguez 1961-, eds. Symmetries and related topics in differential and difference equations: Jairo Charris Seminar 2009, Escuela de Matematicas, Universidad Sergio Arboleda, Bogotá, Colombia. Providence, R.I: American Mathematical Society, 2011.

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Difference algebra. [New York?]: Springer, 2008.

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Levin, Alexander. Difference Algebra. Dordrecht: Springer Netherlands, 2008. http://dx.doi.org/10.1007/978-1-4020-6947-5.

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Maximon, Leonard C. Differential and Difference Equations. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-29736-1.

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Book chapters on the topic "Differential and difference algebra"

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Kondratieva, M. V., A. B. Levin, A. V. Mikhalev, and E. V. Pankratiev. "Dimension Polynomials in Difference and Difference-Differential Algebra." In Differential and Difference Dimension Polynomials, 281–353. Dordrecht: Springer Netherlands, 1999. http://dx.doi.org/10.1007/978-94-017-1257-6_6.

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Kondratieva, M. V., A. B. Levin, A. V. Mikhalev, and E. V. Pankratiev. "Basic Notions of Differential and Difference Algebra." In Differential and Difference Dimension Polynomials, 123–90. Dordrecht: Springer Netherlands, 1999. http://dx.doi.org/10.1007/978-94-017-1257-6_3.

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Kondratieva, M. V., A. B. Levin, A. V. Mikhalev, and E. V. Pankratiev. "Some Application of Dimension Polynomials in Difference-Differential Algebra." In Differential and Difference Dimension Polynomials, 355–75. Dordrecht: Springer Netherlands, 1999. http://dx.doi.org/10.1007/978-94-017-1257-6_7.

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Zuevsky, Alexander. "Geometric Versus Automorphic Correspondence for Vertex Operator Algebra Modules." In Differential and Difference Equations with Applications, 649–59. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-75647-9_50.

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Slavyanov, Sergey, and Vladimir Papshev. "Differential and Difference Equations for Products of Classical Orthogonal Polynomials." In Computer Algebra in Scientific Computing, 399–404. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/11555964_34.

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Heredero, R., D. Levi, M. Rodríguez, and P. Winternitz. "Symmetries of differential difference equations and Lie algebra contractions." In Bäcklund and Darboux Transformations. The Geometry of Solitons, 233–43. Providence, Rhode Island: American Mathematical Society, 2001. http://dx.doi.org/10.1090/crmp/029/20.

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Abramov, S. A., and M. Petkovšek. "On Polynomial Solutions of Linear Partial Differential and (q-)Difference Equations." In Computer Algebra in Scientific Computing, 1–11. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-32973-9_1.

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Ayryan, E. A., M. D. Malykh, L. A. Sevastianov, and Yu Ying. "On Explicit Difference Schemes for Autonomous Systems of Differential Equations on Manifolds." In Computer Algebra in Scientific Computing, 343–61. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-26831-2_23.

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Bronstein, Manuel. "Computer Algebra Algorithms for Linear Ordinary Differential and Difference equations." In European Congress of Mathematics, 105–19. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8266-8_9.

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Zuevsky, Alexander. "Algebraic Properties of the Semi-direct Product of Kac–Moody and Virasoro Lie Algebras and Associated Bi-Hamiltonian Systems." In Differential and Difference Equations with Applications, 1–7. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-32857-7_1.

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Conference papers on the topic "Differential and difference algebra"

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van Hoeij, Mark. "Closed Form Solutions for Linear Differential and Difference Equations." In ISSAC '17: International Symposium on Symbolic and Algebraic Computation. New York, NY, USA: ACM, 2017. http://dx.doi.org/10.1145/3087604.3087660.

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Levin, Alexander. "Bivariate Dimension Polynomials of Non-Reflexive Prime Difference-Differential Ideals." In ISSAC '18: International Symposium on Symbolic and Algebraic Computation. New York, NY, USA: ACM, 2018. http://dx.doi.org/10.1145/3208976.3209008.

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Bao, Wendi, Yongzhong Song, Theodore E. Simos, George Psihoyios, Ch Tsitouras, and Zacharias Anastassi. "A Local RBF-generated Finite Difference Method for Partial Differential Algebraic Equations." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP, 2011. http://dx.doi.org/10.1063/1.3636968.

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Vanderploeg, Martin J., and Jeff D. Trom. "Automated Linearization of Nonlinear Coupled Differential and Algebraic Equations." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0121.

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Abstract This paper presents a new approach for linearization of large multi-body dynamic systems. The approach uses an analytical differentiation of terms evaluated in a numerical equation formulation. This technique is more efficient than finite difference and eliminates the need to determine finite difference pertubation values. Because the method is based on a relative coordinate formalism, linearizations can be obtained for equilibrium configurations with non-zero Cartesian accelerations. Examples illustrate the accuracy and efficiency of the algorithm, and its ability to compute linearizations for large-scale systems that were previously impossible.

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Zhou, Jianping, and Zhigang Feng. "Transient Response of Distributed Parameter Systems." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-4080.

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Abstract A semi-analytic method is presented for the analysis of transient response of distributed parameter systems which are consist of one dimensional subsystems. The system is first divided into one dimensional sub-systems. Within each subsystem, replacing differentials on time t by finite difference, the governing partial differential equations are reduced to difference-differential equations. The solution of derived ordinary differential equations is obtained in an exact and closed form by distributed transfer function method and represented in nodal displacement parameters. Assemling global equilibrium equations at each nodes according to displacement continuity and force equilibrium requirements gives simutaneous linear algebraic equations. Numerical results are illustrated and compared with that of finite element method.

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Kosasih, Engkos Achmad, and Raldi Artono Koestoer. "Determination of Air Temperature Distribution in the Annular Space Using Finite Difference Method." In ASME 1993 International Computers in Engineering Conference and Exposition. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/cie1993-0057.

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Abstract This papers discusses air temperature distribution in the annular space on forced convection of turbulent air flow, which have been determined using numerical method, and compares the result with experimental data. Partial Differential Equations are presented in the final formulation, whereas turbulent flow model applied the simple algebraic model. These equations are changed into numerical equations by means of Finite Difference Method, in the form of explicit equation systems. The steps for solving these systems will be discussed. The comparison between numerical solutions and experimental data shows a good result especially in the fully developed region of the air flow.

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Siami, A., and M. Farid. "Identification and Defect Detection of Continuous Dynamic Systems." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14364.

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This paper presents a systematic and efficient algorithm using a coupled finite element - finite difference - least square method for identification and defect detection of continuous system using dynamic response of such systems. First the governing partial differential equations of motion of continuous systems such as beams are reduced to a set of ordinary differential equations in time domain using finite elements. Then finite difference method is used to convert these equations into a set of algebraic equations. This set of equations is considered as a set of equality constraints of an optimization problem in which the objective function is the summation of the squares of differences between measured data at specific points and the predicted data obtained by the solution of the governing system of differential of equations. This method has been successfully applied to find mechanical properties of aforementioned systems in an iterative procedure.

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Kime, Katherine A. "Effect of the Spatial Extent of the Control in a Bilinear Control Problem for the Schroedinger Equation." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86440.

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We consider control of the one-dimensional Schroedinger equation through a time-varying potential. Using a finite difference semi-discretization, we consider increasing the extent of the potential from a single central grid-point in space to two or more gridpoints. With the differential geometry package in Maple 8, we compute and compare the corresponding Control Lie Algebras, identifying a trend in the number of elements which span the Control Lie Algebras.

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Saghafi, Mehdi, and Harry Dankowicz. "Nondegenerate Continuation Problems for the Excitation Response of Nonlinear Beam Structures." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-13115.

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This paper investigates the dynamics of a slender beam subjected to transverse periodic excitation. Of particular interest is the formulation of nondegenerate continuation problems that may be analyzed numerically, in order to explore the parameter-dependence of the steady-state excitation response, while accounting for geometric nonlinearities. Several candidate formulations are presented, including finite-difference (FD) and finite-element (FE) discretizations of the governing scalar, integro-partial differential boundary-value problem (BVP), as well as of a corresponding first-order-in-space, mixed formulation. As an example, a periodic BVP — obtained from a Galerkin-type, FE discretization with continuously differentiable, piecewise-polynomial trial and test functions, and an elimination of Lagrange multipliers associated with spatial boundary conditions — is analyzed to determine the beam response via numerical continuation using a MATLAB-based software suite. In the case of an FE discretization of the mixed formulation with continuous, piecewise-polynomial trial and test functions, it is shown that the choice of spatial boundary conditions may render the resultant index-1, differential-algebraic BVP equivariant under a symmetry group of state-space translations. The paper demonstrates several methods for breaking the equivariance in order to obtain a nondegenerate continuation problem, including a projection onto a symmetry-reduced state space or the introduction of an artificial continuation parameter. As is further demonstrated, an orthogonal collocation discretization in time of the BVP gives rise to ghost solutions, corresponding to arbitrary drift in the algebraic variables. This singularity is resolved by using an asymmetric discretization in time.

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Szabó, Zsolt, S. C. Sinha, and Gábor Stépán. "Dynamics of Pipes Containing Pulsative Flow." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-4022.

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Abstract Several mechanical models exist on elastic pipes containing fluid flow. In this paper those models are considered, where the fluid is incompressible, frictionless and its velocity relative to the pipe has the same but time-periodic magnitude along the pipe at a certain time instant. The pipe can be modelled either as a chain of articulated rigid pipes or as a continuum. The dynamic behaviour of the system strongly depends on the different kinds of boundary conditions and on the fact whether the pipe is considered to be inextensible, i.e. the cross-sectional area of the pipe is constant. The equations of motion are derived via Lagrangian equations and Hamilton’s principle. These systems are non-conservative, and the amount of energy carried in and out by the flow appears in the model. It is well-known, that intricate stability problems arise when the flow pulsates and the corresponding mathematical model, a system of ordinary or partial differential equations, becomes time-periodic. There are several standard techniques, like perturbation method, harmonic balance, finite difference, etc., to analyze these models. The method which constructs the state transition matrix used in Floquet theory in terms of the shifted Chebyshev polynomials of the first kind is especially effective for stability analysis of large systems. The implementation of this method using computer algebra enables us to obtain more accurate results and to investigate more complex models. The stability charts are created with respect to three important parameters: the forcing frequency ω, the perturbation amplitude υ and the mean flow velocity U.

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Reports on the topic "Differential and difference algebra"

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Malitsky, N., A. Reshetov, and Y. Yan. ZLIB++: Object-oriented numerical library for differential algebra. Office of Scientific and Technical Information (OSTI), January 1994. http://dx.doi.org/10.2172/10147641.

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Yan, Yiton T. ZLIB++: Object Oriented Numerical Library for Differential Algebra. Office of Scientific and Technical Information (OSTI), July 2003. http://dx.doi.org/10.2172/813295.

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Hyman, J. M., M. Shashkov, M. Staley, S. Kerr, S. Steinberg, and J. Castillo. Mimetic difference approximations of partial differential equations. Office of Scientific and Technical Information (OSTI), August 1997. http://dx.doi.org/10.2172/518902.

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R.A. Axford. Construction of Superconvergent Discretizations with Differential-Difference Invariants. Office of Scientific and Technical Information (OSTI), August 2005. http://dx.doi.org/10.2172/883452.

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Yan, Y. Applications of differential algebra to single-particle dynamics in storage rings. Office of Scientific and Technical Information (OSTI), September 1991. http://dx.doi.org/10.2172/5166998.

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Malitsky, Nikolay. Application of a Differential Algebra approach to a RHIC Helical Dipole. Office of Scientific and Technical Information (OSTI), December 1994. http://dx.doi.org/10.2172/1119445.

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Svetlana G. Shasharina. Final report: Efficient and user friendly C++ library for differential algebra. Office of Scientific and Technical Information (OSTI), September 1998. http://dx.doi.org/10.2172/761041.

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Berz, M., E. Forest, and J. Irwin. Exact computation of derivatives with differential algebra and applications to beam dynamics. Office of Scientific and Technical Information (OSTI), March 1988. http://dx.doi.org/10.2172/7050634.

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Malitsky, N. Application of a Differential Algebra Approach to a RHIC Helical Dipole (12/94). Office of Scientific and Technical Information (OSTI), December 1994. http://dx.doi.org/10.2172/1149789.

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Caspi, S., M. Helm, L. J. Laslett, and V. O. Brady. An approach to 3D magnetic field calculation using numerical and differential algebra methods. Office of Scientific and Technical Information (OSTI), July 1992. http://dx.doi.org/10.2172/7252409.

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