Numerical solutions for implicit differential equations with singularities
DOI:
https://doi.org/10.5433/1679-0375.2022v43n1Espp3Keywords:
Implicit differential equations, Singularities, Differential algebraic equationsAbstract
In this paper we introduce a technique to deal with implicit differential equations exhibiting singularities. Our approach is a geometrical one, we use the concept of contact structure on a manifold associated with the differential equation. In this setting we prove an existence and uniqueness theorem. We also show how it relates to known geometric results for this kind of equation. We also indicate how the method can be implemented by using continuation methods techniques and the BDF (Backward Differentiation Formula).Downloads
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References
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ARNOLD, V. I. Geometrical methods in the theory of ordinary differential equations. Berlin: Springer-Verlag, 1988.
ASCHER, U.; PETZOLD, L. Projected collocation for higher-order higher-index differential-algebraic equations. Journal of Computational and Applied Mathematics, Antwerp, v. 43, p. 243-259, 1992.
ASCHER, U.; PETZOLD, L. Computer methods for ordinary differential equations and differential-algebraic equations. Philadelphia: SIAM, 1998.
BRENAN, K. E.; CAMPBELL, S. L.; PETZOLD L. R. Numerical solution of initial-value problems in differential-algebraic equations. New York: North-Holland, 1989.
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DARA, L. Singularites generiques des equations differentielles multiformes. Bulletin of the Brazilian Mathematical Society, Rio de Janeiro, v. 6, p. 95-128, 1975. DOI: DOI:10.1007/BF02584779.
DARBOUX, M. G. Sur les solutions singulières des équations aux derivées ordinaires du premier ordre. Bulletin des sciences mathématiques et astronomiques, Paris, v. 4, p. 158-176, 1873.
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JACKSON, K. R.; STAKS-DAVIS, R. An alternative implementation of variable step-size multstep formulas for stiff ODEs. ACM Transactions on Mathematical Software, New York, v. 6, n. 3, p. 295-318, 1980.
JEPSON, A.; SPENCE, A. On implicit ordinary differential equations. IMA Journal of Numerical Analysis, Oxford, v. 4, n. 3, p. 253-274, 1984
KUNKEL, P.; MEHRMANN, W. R.; WEICKERT, J. GELDA: a software package for the solution of general linear differential algebraic equations. SIAM Journal Scientific Computing, Philadelphia, v. 18, p. 115-138, 1997.
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PETZOLD, L. Differential/algebraic equations are not ode’s. SIAM Journal on Scientific and Statistical Computing, Philadelphia, v. 3, n. 3, p. 367-384, 1982. DOI: https://doi.org/10.1137/0903023
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RABIER, P. J.; RHEINBOLDT, W. C. Theoretical and numerical analysis of differential-algebraic equations. In: CIARLET, P. G.; LIONS, J. L. (ed.). Handbook of Numerical Analysis. Amsterdam: Elsevier, 2002. v. 8, p. 183–540.
RHEINBOLDT, W. C. Differential-algebraic systems as differential equations on manifolds. Mathematics of Computation, Providence, v. 43, p. 473-482, 1984. DOI: https://doi.org/10.2307/2008288.
RIAZA, R. Differential-algebraic systems: analytical aspects and circuit applications. Singapore: World Scientific, 2008.
SERBAN, R.; HINDMARSH, A. C. CVODES: the sensitivity-enabled ODE solver in SUNDIALS. In: INTERNATIONAL DESIGN ENGINEERING TECHNICAL CONFERENCES AND COMPUTERS AND INFORMATION IN ENGINEERING CONFERENCE, 2005, California. Proceedings […]. California: ASME, 2005. p. 1-13.
SOSEN, H. V. Part I: folds and bifurcations in the solutions of semi-explicit differential-algebraic equations. 1994. Thesis (Doctor) - California Institute of Technology, California, 1994.
THOM, R. Sur les equations différentielles multiformes et leurs intégrales singulières. Bulletin of the Brazilian Mathematical Society, Rio de Janeiro, v. 3, n. 1, p. 1-11, 1971.
TOLSMA, J.; BARTON, P. DAEPACK: An open modeling environment for legacy models. Industrial & Engineering Chemistry Research, Washington, v. 39, n. 6, p. 1826-1839, 2000.
WASOW, W. Asymptotic expansions for ordinary equations. [Paris]: Interscience, 1965.
ARNOLD, V. I. Geometrical methods in the theory of ordinary differential equations. Berlin: Springer-Verlag, 1988.
ASCHER, U.; PETZOLD, L. Projected collocation for higher-order higher-index differential-algebraic equations. Journal of Computational and Applied Mathematics, Antwerp, v. 43, p. 243-259, 1992.
ASCHER, U.; PETZOLD, L. Computer methods for ordinary differential equations and differential-algebraic equations. Philadelphia: SIAM, 1998.
BRENAN, K. E.; CAMPBELL, S. L.; PETZOLD L. R. Numerical solution of initial-value problems in differential-algebraic equations. New York: North-Holland, 1989.
BRENAN, K. E.; CAMPBELL, S. L.; PETZOLD L. R. Numerical solution of initial-value problems in differential-algebraic equations. Philadelphia: SIAM, 1996. Revised and corrected reprint of the 1989 original, with an additional chapter and additional references.
DARA, L. Singularites generiques des equations differentielles multiformes. Bulletin of the Brazilian Mathematical Society, Rio de Janeiro, v. 6, p. 95-128, 1975. DOI: DOI:10.1007/BF02584779.
DARBOUX, M. G. Sur les solutions singulières des équations aux derivées ordinaires du premier ordre. Bulletin des sciences mathématiques et astronomiques, Paris, v. 4, p. 158-176, 1873.
DAVYDOV, A. A. Normal form of a differential equation, not solvable for derivative, in a neighborhood of a singular point. Functional Analysis and its Applications, New York, v. 19, n. 2, p. 81-89, 1985.
DEUFLHARD, P.; HAIRER, E.; ZUGCK, J. Onestep and extrapolation methods for differential-algebraic systems. Numerische Mathematik, Berlin, v. 51, n. 5, p. 501-516, 1987.
FREITAS, S. R. Simplicial aproximation of implicitly defined manifolds and differential-algebraic equations. 1991. Thesis (Doctor) - Catholic University, Rio de Janeiro, 1991.
FREITAS, S. R.; TAVARES, G. Solving implicit ODEs by simplicial methods. In: LAURENT, P. L.; MÉHAUTÉ, A. L.; SCHUMAKER, L. L. (ed.). Curves and Surfaces. Boston: Academic Press, 1991. p. 193- 196.
HAIRER, E.; WANNES, G. Solving ordinary differential equations II. Berlin: Springer-Verlag, 1989.
HAIRER, E.; WANNER, G. Solving ordinary differential equations II: stiff and differential-algebraic problems. 2. ed. Berlin: Springer, 1996.
JACKSON, K. R.; STAKS-DAVIS, R. An alternative implementation of variable step-size multstep formulas for stiff ODEs. ACM Transactions on Mathematical Software, New York, v. 6, n. 3, p. 295-318, 1980.
JEPSON, A.; SPENCE, A. On implicit ordinary differential equations. IMA Journal of Numerical Analysis, Oxford, v. 4, n. 3, p. 253-274, 1984
KUNKEL, P.; MEHRMANN, W. R.; WEICKERT, J. GELDA: a software package for the solution of general linear differential algebraic equations. SIAM Journal Scientific Computing, Philadelphia, v. 18, p. 115-138, 1997.
KUNKEL, P.; MEHRMANN, W. R.; SEUFER, I. GENDA: A software package for the solution of general nonlinear differential-algebraic equations. Institut für Mathematik, Technische Universität Berlin, Berlin, n. 7, p. 30-32, 2002.
KUNKEL, P.; MEHRMANN, V. Differential-algebraic equations analysis and numerical solution. Zurich: EMS Publishing House, 2006.
LUBICH, CH; NOWAK, U.; POHLE, U.; ENGSTLER, CH. MEXX-Numerical software for the integration of constrained mechanical multibody systems. Berlin: ZIB, 1992.
PETZOLD, L. Differential/algebraic equations are not ode’s. SIAM Journal on Scientific and Statistical Computing, Philadelphia, v. 3, n. 3, p. 367-384, 1982. DOI: https://doi.org/10.1137/0903023
RABIER, P. J. Implicit Differential equations near a singular point. Journal of Mathematical Analysis and Applications, Orlando, v. 144, n. 2, p. 425-449, 1989. DOI: https://doi.org/10.1016/0022-247X(89)90344-2
RABIER, P. J.; RHEINBOLDT, W. C. A general existence and uniqueness theory for implicit differential algebraic equations. Differential and Integral Equations, Athens, v. 4, p. 563-582, 1991. DOI: 10.21236/ada217965.
RABIER, P. J.; RHEINBOLDT, W. C. Theoretical and numerical analysis of differential-algebraic equations. In: CIARLET, P. G.; LIONS, J. L. (ed.). Handbook of Numerical Analysis. Amsterdam: Elsevier, 2002. v. 8, p. 183–540.
RHEINBOLDT, W. C. Differential-algebraic systems as differential equations on manifolds. Mathematics of Computation, Providence, v. 43, p. 473-482, 1984. DOI: https://doi.org/10.2307/2008288.
RIAZA, R. Differential-algebraic systems: analytical aspects and circuit applications. Singapore: World Scientific, 2008.
SERBAN, R.; HINDMARSH, A. C. CVODES: the sensitivity-enabled ODE solver in SUNDIALS. In: INTERNATIONAL DESIGN ENGINEERING TECHNICAL CONFERENCES AND COMPUTERS AND INFORMATION IN ENGINEERING CONFERENCE, 2005, California. Proceedings […]. California: ASME, 2005. p. 1-13.
SOSEN, H. V. Part I: folds and bifurcations in the solutions of semi-explicit differential-algebraic equations. 1994. Thesis (Doctor) - California Institute of Technology, California, 1994.
THOM, R. Sur les equations différentielles multiformes et leurs intégrales singulières. Bulletin of the Brazilian Mathematical Society, Rio de Janeiro, v. 3, n. 1, p. 1-11, 1971.
TOLSMA, J.; BARTON, P. DAEPACK: An open modeling environment for legacy models. Industrial & Engineering Chemistry Research, Washington, v. 39, n. 6, p. 1826-1839, 2000.
WASOW, W. Asymptotic expansions for ordinary equations. [Paris]: Interscience, 1965.
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Published
2022-05-17
How to Cite
Castelo, A., Tavares, G., & Bertoco, J. (2022). Numerical solutions for implicit differential equations with singularities. Semina: Ciências Exatas E Tecnológicas, 43(1Esp), 3–16. https://doi.org/10.5433/1679-0375.2022v43n1Espp3
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