Dissertations / Theses on the topic 'Ordinary differential equations'
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MARINO, GISELA DORNELLES. "COMPLEX ORDINARY DIFFERENTIAL EQUATIONS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2007. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=10175@1.
Full textNeste texto estudamos diversos aspectos de singularidades de campos vetoriais holomorfos em dimensão 2. Discutimos detalhadamente o caso particular de uma singularidade sela-nó e o papel desempenhado pelas normalizações setoriais. Isto nos conduz à classificação analítica de difeomorfismos tangentes à identidade. seguir abordamos o Teorema de Seidenberg, tratando da redução de singularidades degeneradas em singularidades simples, através do procedimento de blow-up. Por fim, estudamos a demonstração do Teorema de Mattei-Moussu, acerca da existência de integrais primeiras para folheações holomorfas.
In the present text, we study the different aspects of singularities of holomorphic vector fields in dimension 2. We discuss in detail the particular case of a saddle-node singularity and the role of the sectorial normalizations. This leads us to the analytic classiffication of diffeomorphisms which are tangent to the identity. Next, we approach the Seidenberg Theorem, dealing with the reduction of degenerated singularities into simple ones, by means of the blow-up procedure. Finally, we study the proof of the well-known Mattei-Moussu Theorem concerning the existence of first integrals to holomorphic foliations.
Gehrs, Kai Frederik. "Algorithmic methods for ordinary differential equations." [S.l.] : [s.n.], 2006. http://ubdata.uni-paderborn.de/ediss/17/2007/gehrs.
Full textNg, Chee Loong. "Parameter estimation in ordinary differential equations." Texas A&M University, 2004. http://hdl.handle.net/1969.1/388.
Full textJorba, i. Monte Àngel. "On Quasiperiodic Perturbations of Ordinary Differential Equations." Doctoral thesis, Universitat de Barcelona, 1991. http://hdl.handle.net/10803/2122.
Full textIt is also known that, when the mass parameter "mi" (the mass of the small primary in the normalized units) is less than the Routh critical value "mi"(R) = 1/2(1 - square root (23/27) = 0.03852 ... (this is true in the Earth-Moon case) these points are linearly stable. Applying the KAM theorem to this case we can obtain that there exist invariant tori around these points. Now, if we restrict the motion of the particle to the plane of motion of the primaries we have that, inside each energy level, these tori split the phase space and this allows to prove that the equilateral points are stable (except for two values, "mi" = "mi"2 and "mi"= "mi"3 with low order resonances). In the spatial case, the invariant tori do not split the phase space and, due to the possible Arnold diffusion, these points can be unstable. But Arnold diffusion is a very slow phenomenon and we can have small neighbourhoods of "practical stability", that is, the particle will stay near the equilibrium point for very long time spans.
Unfortunately, the real Earth-Moon system is rather complex. In this case, due to the fact that that the motions of the Earth and the Moon are non circular (even non elliptical) and the strong influence of the Sun, the libration points do not exist as equilibrium points, and we need to define "instantaneous" libration points as the ones forming an equilateral triangle with the Earth and the Moon at each instant. If we perform some numerical integrations starting at (or near) these points we can see that the solutions go away after a short period of time, showing that these regions are unstable.
Two conclusions can be obtained from this fact. First: if we are interested in keeping a spacecraft there, we will need to use some kind of control. Second: the RTBP is not a good model for this problem} because the behaviour displayed by it is different from the one of the real system.
For these reasons, an improved model has been developed in order to study this problem. This model includes the main perturbations (due to the solar effect and to the noncircular motion of the Moon), assuming that they are quasi-periodic. This is a very good approximation for time spans of some thousands of years. It is not clear if this is true for longer time spans, but this matter will not be considered in this work. This model is in good agreement with the vector field of the solar system directly computed by means of the JPL ephemeris, for the time interval for which the JPL model is available.
The study of this kind of models is the main purpose of this work.
First of all, we have focused our attention on linear differential equations with constant coefficients, affected by a small quasi-periodic perturbation. These equations appear as variational equations along a quasi-periodic solution of a general equation and they also serve as an introduction to nonlinear problems.
The purpose is to reduce those systems to constant coefficients ones by means of a quasi-periodic change of variables, as the classical Floquet theorem does for periodic systems. It is also interesting to nave a way to compute this constant matrix, as well as the change of variables. The most interesting case occurs when the unperturbed system is of elliptic type. Other cases, as the hyperbolic one, have already been studied. We have added a parameter ("epsilon") in the system, multiplying the perturbation, such that if "epsilon" is equal to zero we recover the unperturbed system. In this case we have found that, under suitable hypothesis of non-resonance, analyticity and non-degeneracy with respect to "epsilon", it is possible to reduce the system to constant coefficients, for a cantorian set of values of "epsilon". Moreover, the proof is constructive in an iterative way. This means that it is possible to find approximations to the reduced matrix as well as to the change of variables that performs such reduction. These results are given in Chapter 1.
The nonlinear case is now going to be studied. We have then considered an elliptic equilibrium point of an autonomous ordinary differential equation, and we have added a small quasi-periodic perturbation, in such a way that the equilibrium point does not longer exist. As in the linear case, we have put a parameter ("epsilon") multiplying the perturbation. There is some "practical" evidence that there exists a quasi-periodic orbit, having the same basic frequencies that the perturbation, such that, when the perturbation goes to zero, this orbit goes to the equilibrium point. Our results show that, under suitable hypothesis, this orbit exists for a cantorian set of values of "epsilon". We have also found some results related to the stability of this orbit. These results are given in Chapter 2.
A remarkable case occurs when the system is Hamiltonian. Here it is interesting to know what happens to the invariant tori near these points when the perturbation is added. Note that the KAM theorem can not be applied directly due to the fact that the Hamiltonian is degenerated, in the sense that it has some frequencies (the ones of the perturbation) that have fixed values and they do not depend on actions in a diffeomorphic way. In this case, we have found that some tori still exist in the perturbed system. These tori come from the ones of the unperturbed system whose frequencies are non-resonant with those of the perturbation. The perturbed tori add these perturbing frequencies to the ones they already had. This can be described saying that the unperturbed tori are "quasi-periodically dancing" under the "rhythm" of the perturbation. These results can also be found in Chapter 2 and Appendix C.
The final point of this work has been to perform a study of the behaviour near the instantaneous equilateral libration points of the real Earth-Moon system. The purpose of those computations has been to find a way of keeping a spacecraft near these points in an unexpensive way. As it has been mentioned above in the real system these points are not equilibrium points, and their neighbourhood displays unstability. This leads us to use some control to keep the spacecraft there. It would be useful to have an orbit that was always near these points, because the spacecraft could be placed on it. Thus, only a station keeping would be necessary. The simplest orbit of this kind that we can compute is the one that replaces the equilibrium point. In Chapter 3, this computation has been carried out first for a planar simplified model and then for a spatial model. Then, the solution found for this last model has been improved, by means of numerical methods, in order to have a real orbit of the real system (here, by real system we mean the model of solar system provided by the JPL tapes). This improvement has been performed for a given (fixed) time-span. That is sufficient for practical purposes. Finally, an approximation to the linear stability of this refined orbit has been computed, and a very mild unstability has been found, allowing for an unexpensive station keeping. These results are given in Chapter 3 and Appendix A.
Finally, in Appendix B the reader can find the technical details concerning the way of obtaining the models used to study the neighbourhood of the equilateral points. This has been jointly developed with Gerard Gomez, Jaume Llibre, Regina Martinez, Josep Masdemont and Carles Simó.
We study several topics concerning quasi-periodic time-dependent perturbations of ordinary differential equations. This kind of equations appear in many applied problems of Celestial Mechanics, and we have used, as an illustration, the study of the behaviour near the Lagrangian points of the real Earth-Moon system. For this purpose, a model has been developed. It includes the main perturbations (due to the Sun and Moon), assuming that they are quasi-periodic.
Firstly, we deal with linear differential equations with constant coefficients, affected by a small quasi-periodic perturbation, trying to reduce then: to constant coefficients by means of a quasi-periodic change of variables. The most interesting case occurs when the unperturbed system is of elliptic type. We have added a parameter "epsilon" in the system, multiplying the perturbation, such that if "epsilon" is equal to zero we recover the unperturbed system. In this case, under suitable hypothesis of non-resonance, analyicity and non degeneracy with respect to "epsilon", it is possible to reduce the system to constant coefficients, for a cantorian set of values of "epsilon".
In the nonlinear case, we have considered an elliptic equilibrium point of an autonomous differential equation, and we have added a small quasi-periodic perturbation, in such a way that the equilibrium point does not exist. As in the linear case, we have put a parameter ("epsilon") multiplying the perturbation. Then, for a cantorian set of "epsilon", there exists a quasi-periodic orbit having the same basic frequencies as the perturbation, going to the equilibrium point when t: goes to zero. Some results concerning the stability of this orbit are stated. When the system is Hamiltonian, we have found that some tori still exist in the perturbed system. These tori come from the ones of the unperturbed system whose frequencies are non-resonant with those of the perturbation, adding these perturbing frequencies to the ones they already had.
Finally, a study of the behaviour near the Lagrangian points of the real Earth-Moon system is presented. The purpose has been to find the orbit replacing the equilibrium point. This computation has been carried out first for the model mentioned above and then it has been improved numerically, in order to have a real orbit of the real system. Finally, a study of the linear stability of this refined orbit has been done.
Måhl, Anna. "Separation of variables for ordinary differential equations." Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-5620.
Full textIn case of the PDE's the concept of solving by separation of variables
has a well defined meaning. One seeks a solution in a form of a
product or sum and tries to build the general solution out of these
particular solutions. There are also known systems of second order
ODE's describing potential motions and certain rigid bodies that are
considered to be separable. However, in those cases, the concept of
separation of variables is more elusive; no general definition is
given.
In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible.
Lagrange, John. "Power Series Solutions to Ordinary Differential Equations." TopSCHOLAR®, 2001. http://digitalcommons.wku.edu/theses/672.
Full textSharples, Nicholas. "Some problems in irregular ordinary differential equations." Thesis, University of Warwick, 2012. http://wrap.warwick.ac.uk/55877/.
Full textHemmi, Mohamed Ali Carleton University Dissertation Mathematics and statistics. "Series solutions of nonlinear ordinary differential equations." Ottawa, 1994.
Find full textMaclean, John. "Numerical multiscale methods for ordinary differential equations." Thesis, The University of Sydney, 2014. http://hdl.handle.net/2123/12818.
Full textSaravi, Masoud. "Numerical solution of linear ordinary differential equations and differential-algebraic equations by spectral methods." Thesis, Open University, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.446280.
Full textZhang, Quanju. "Ordinary differential equation methods for some optimization problems." HKBU Institutional Repository, 2006. http://repository.hkbu.edu.hk/etd_ra/710.
Full textKunze, Herbert Eduard. "Monotonicity properties of systems of ordinary differential equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/nq21361.pdf.
Full textTam, Kim M. "ODEASY, a query tool for ordinary differential equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ36744.pdf.
Full textCarvalho, Alexandre Nolasco de. "Infinite dimensional dynamics described by ordinary differential equations." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/29585.
Full text蔡澤鍔 and Chak-ngok Choy. "Lie's theory on solvability of ordinary differential equations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1997. http://hub.hku.hk/bib/B3121518X.
Full textKhalaf, Bashir M. S. "Parallel numerical algorithms for solving ordinary differential equations." Thesis, University of Leeds, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.277599.
Full textWoods, Patrick Daniel. "Localisation in reversible fourth-order ordinary differential equations." Thesis, University of Bristol, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299269.
Full textSUN, JIAN. "VISUALIZATIONS OF PERIODIC ORBIT OF ORDINARY DIFFERENTIAL EQUATIONS." University of Cincinnati / OhioLINK, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1012855340.
Full textTung, Michael Ming-Sha. "Spline approximations for systems of ordinary differential equations." Doctoral thesis, Universitat Politècnica de València, 2013. http://hdl.handle.net/10251/31658.
Full textTESIS
Premiado
Sun, Jian. "Visualizations of periodic orbits of ordinary differential equations." Cincinnati, Ohio : University of Cincinnati, 2002. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=ucin1012855340.
Full textChoy, Chak-ngok. "Lie's theory on solvability of ordinary differential equations /." Hong Kong : University of Hong Kong, 1997. http://sunzi.lib.hku.hk/hkuto/record.jsp?B19472675.
Full textJenab, Bita. "Asymptotic theory of second-order nonlinear ordinary differential equations." Thesis, University of British Columbia, 1985. http://hdl.handle.net/2429/24690.
Full textScience, Faculty of
Mathematics, Department of
Graduate
Hynick, Amy Marie. "On pulse detection in initial value ordinary differential equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0021/MQ49375.pdf.
Full textSandberg, Mattias. "Approximation of Optimally Controlled Ordinary and Partial Differential Equations." Doctoral thesis, Stockholm, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4066.
Full textCaberlin, Martin D. "Stiff ordinary and delay differential equations in biological systems." Thesis, McGill University, 2002. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=29416.
Full textBrown, A. A. "Optimisation methods involving the solution of ordinary differential equations." Thesis, University of Hertfordshire, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.374887.
Full textMacdonald, Benn. "Statistical inference for ordinary differential equations using gradient matching." Thesis, University of Glasgow, 2017. http://theses.gla.ac.uk/7987/.
Full textReddinger, Kaitlin Sue. "Numerical Stability & Numerical Smoothness of Ordinary Differential Equations." Bowling Green State University / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1431597407.
Full textOgundare, Babatunde Sunday. "Qualitative and quantitative properties of solutions of ordinary differential equations." Thesis, University of Fort Hare, 2009. http://hdl.handle.net/10353/244.
Full textKhanamiryan, Marianna. "Numerical methods for systems of highly oscillatory ordinary differential equations." Thesis, University of Cambridge, 2010. https://www.repository.cam.ac.uk/handle/1810/226323.
Full textBerman, Peter Hillel. "Computing Galois Groups for Certain Classes of Ordinary Differential Equations." NCSU, 2001. http://www.lib.ncsu.edu/theses/available/etd-20010723-022851.
Full textAs of now, it is an open problem to find an algorithmthat computes the Galois group G of an arbitrary linear ordinary differential operator L in C(x)[D]. We assume thatC is a computable, characteristic-zero,algebraically closed constant field with factorization algorithm.In this dissertation, we present new methods forcomputing differential Galois groups in two special cases.An article by Compoint and Singer presents a decision procedure to compute G in case L is completely reducible or, equivalently, G is reductive. Here, we present the results of an article by Berman and Singerthat reduces the case of a productof two completely reducible operators to thatof a single completely reducible operator;moreover, we give an optimization of that article's core decision procedure.These results rely on results from cohomologydue to Daniel Bertrand.We also give a set of criteria to compute the Galois group of a differential equation of the formy''' + ay' + by = 0, a, b in C[x].Furthermore, we present an algorithm to carry out this computation in case C is the field of algebraic numbers.This algorithm applies the approach used inan article by M. van der Put to study order-two equations with one or two singularpoints. Each step of the algorithm employs a simple, implementable test based on some combination of factorization properties, properties of associated operators,and testing of associated equations for rational solutions. Examples of the algorithm and a Maple implementation writtenby the author are provided.
George, Daniel Pucknell. "Bifurcations and homoclinic orbits in piecewise linear ordinary differential equations." Thesis, University of Cambridge, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.233083.
Full textWoods, Berietta F. "Numerical instabilities in finite-difference models of ordinary differential equations." DigitalCommons@Robert W. Woodruff Library, Atlanta University Center, 1989. http://digitalcommons.auctr.edu/dissertations/389.
Full textSanugi, Bahrom B. "New numerical strategies for initial value type ordinary differential equations." Thesis, Loughborough University, 1986. https://dspace.lboro.ac.uk/2134/14145.
Full textDalal, Nirav. "Applications of stochastic and ordinary differential equations to HIV dynamics." Thesis, University of Strathclyde, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.435132.
Full textMdekazi, Mvuyisi. "Generalized implicit function theorems and applications to ordinary differential equations." Master's thesis, University of Cape Town, 1993. http://hdl.handle.net/11427/14297.
Full textWalker, Matthew Thomas. "Theta Methods For Nonlinear Ordinary Differential Equations and Error Analysis." OpenSIUC, 2014. https://opensiuc.lib.siu.edu/theses/1495.
Full textGarrisi, Daniele. "Ordinary differential equations in Banach spaces and the spectral flow." Doctoral thesis, Scuola Normale Superiore, 2008. http://hdl.handle.net/11384/85668.
Full textYang, Xue-Feng. "Extensions of sturm-liouville theory : nodal sets in both ordinary and partial differential equations." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/28021.
Full textHovhannisyan, A. H., and Bert-Wolfgang Schulze. "On a method for solution of the ordinary differential equations connected with Huygens' equations." Universität Potsdam, 2010. http://opus.kobv.de/ubp/volltexte/2010/4538/.
Full textGuzainuer, Maimaitiyiming. "Boundary Summation Equation Preconditioning for Ordinary Differential Equations with Constant Coefficients on Locally Refined Meshes." Thesis, Linköpings universitet, Matematiska institutionen, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-102573.
Full textDanis, Alexander. "A validated solver for one-dimensional non-autonomous ordinary differential equations." Thesis, Uppsala University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121360.
Full textAriyadasa, A. "Filtering and extrapolation techniques in numerical solution of ordinary differential equations." Thesis, University of Ottawa (Canada), 1985. http://hdl.handle.net/10393/4595.
Full textHayes, Wayne Brian. "Rigorous shadowing of numerical solutions of ordinary differential equations by containment." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/NQ58940.pdf.
Full textLu, Yixia. "Painleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equations." Diss., Tucson, Arizona : University of Arizona, 2005. http://etd.library.arizona.edu/etd/GetFileServlet?file=file:///data1/pdf/etd/azu%5Fetd%5F1103%5F1%5Fm.pdf&type=application/pdf.
Full textNiesen, Jitse. "On the global error of discretization methods for ordinary differential equations." Thesis, University of Cambridge, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.616182.
Full textPatrulescu, Flavius-Olimpiu. "Ordinary differential equations and contact problems : modeling, analysis and numerical methods." Perpignan, 2012. http://www.theses.fr/2012PERP1284.
Full textThe thesis is divided into two parts and eight chapters. The first part contains Chapters 1-3 and presents results concerning the numerical methods for the Cauchy problem associated to ordinary differential equations. The second part refers to the modeling and analysis of some frictionless contact problems for nonlinear elastic or viscoelastic materials. It contains Chapters 4-8. In the first part of the thesis we introduce some Runge-Kutta-type methods for which we obtain new results concerning their consistency, zero-stability, convergence, order of convergence and local truncation error. The second part is devoted to the mathematical study of three contact problems involving deformable bodies. This concerns modeling and the variational analysis of the models, including existence, uniqueness and behavior of the weak solution with respect to the parameters. The study is completed by numerical simulations which validate the theoretical results. The contact processes considered are quasistatic and are treated in the infinitesimal strain theory: the behavior of the material is modeled with elastic and viscoelastic constitutive laws. The contact is frictionless and is modeled with normal compliance and unilateral constraint. The memory effects are also taken into account, both in the constitutive law and in the contact conditions, as well
Rana, Muhammad Sohel. "Analysis and Implementation of Numerical Methods for Solving Ordinary Differential Equations." TopSCHOLAR®, 2017. https://digitalcommons.wku.edu/theses/2053.
Full textHermansyah, Edy. "An investigation of collocation algorithms for solving boundary value problems system of ODEs." Thesis, University of Newcastle Upon Tyne, 2001. http://hdl.handle.net/10443/1976.
Full textMargitus, Michael. "Optimal interpolation grids for accurate numerical solutions of singular ordinary differential equations /." Online version of thesis, 2009. http://hdl.handle.net/1850/10823.
Full text