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Journal articles on the topic 'Second order ODEs'

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Published: 10 March 2023

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1

Kruglikov, Boris. "Symmetries of second order ODEs." Journal of Mathematical Analysis and Applications 461, no. 1 (May 2018): 591–94. http://dx.doi.org/10.1016/j.jmaa.2018.01.026.

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2

Ola Fatunla, Simeon. "Block methods for second order odes." International Journal of Computer Mathematics 41, no. 1-2 (January 1991): 55–63. http://dx.doi.org/10.1080/00207169108804026.

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3

McGrath, Peter. "Bases for Second Order Linear ODEs." American Mathematical Monthly 127, no. 9 (October 20, 2020): 849. http://dx.doi.org/10.1080/00029890.2020.1803626.

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4

Cerda, Patricio, and Pedro Ubilla. "Nonlinear Systems of Second-Order ODEs." Boundary Value Problems 2008 (2008): 1–9. http://dx.doi.org/10.1155/2008/236386.

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5

Cheb-Terrab, E. S., and A. D. Roche. "Integrating Factors for Second-order ODEs." Journal of Symbolic Computation 27, no. 5 (May 1999): 501–19. http://dx.doi.org/10.1006/jsco.1999.0264.

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6

Newman, Ezra T., and Pawel Nurowski. "Projective connections associated with second-order ODEs." Classical and Quantum Gravity 20, no. 11 (May 12, 2003): 2325–35. http://dx.doi.org/10.1088/0264-9381/20/11/324.

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7

Yumaguzhin, Valeriy A. "Differential Invariants of Second Order ODEs, I." Acta Applicandae Mathematicae 109, no. 1 (January 30, 2009): 283–313. http://dx.doi.org/10.1007/s10440-009-9454-0.

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8

Wone, Oumar. "Second order ODEs under area-preserving maps." Analysis and Mathematical Physics 5, no. 1 (July 29, 2014): 87–111. http://dx.doi.org/10.1007/s13324-014-0086-9.

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9

Milson, Robert, and Francis Valiquette. "Point equivalence of second-order ODEs: Maximal invariant classification order." Journal of Symbolic Computation 67 (March 2015): 16–41. http://dx.doi.org/10.1016/j.jsc.2014.08.003.

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10

Reyes, M. A., and H. C. Rosu. "Riccati-parameter solutions of nonlinear second-order ODEs." Journal of Physics A: Mathematical and Theoretical 41, no. 28 (June 19, 2008): 285206. http://dx.doi.org/10.1088/1751-8113/41/28/285206.

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11

Pražák, Dalibor. "Remarks on the uniqueness of second order ODEs." Applications of Mathematics 56, no. 1 (February 2011): 161–72. http://dx.doi.org/10.1007/s10492-011-0014-3.

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12

Mahomed, K. S., and E. Momoniat. "Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/847086.

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Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.
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13

Prince, G. E., J. E. Aldridge, and G. B. Byrnes. "A universal Hamilton-Jacobi equation for second-order ODEs." Journal of Physics A: Mathematical and General 32, no. 5 (January 1, 1999): 827–44. http://dx.doi.org/10.1088/0305-4470/32/5/013.

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14

Sövegjártó, A. "Conservative spline methods for second-order IVPs of ODEs." Computers & Mathematics with Applications 38, no. 9-10 (November 1999): 135–41. http://dx.doi.org/10.1016/s0898-1221(99)00269-2.

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15

Ayub, Muhammad, F. M. Mahomed, Masood Khan, and M. N. Qureshi. "Symmetries of second-order systems of ODEs and integrability." Nonlinear Dynamics 74, no. 4 (September 3, 2013): 969–89. http://dx.doi.org/10.1007/s11071-013-1016-3.

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16

Casey, Stephen, Maciej Dunajski, and Paul Tod. "Twistor Geometry of a Pair of Second Order ODEs." Communications in Mathematical Physics 321, no. 3 (May 21, 2013): 681–701. http://dx.doi.org/10.1007/s00220-013-1729-7.

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17

Garcı́a-Huidobro, M., R. Manásevich, and F. Zanolin. "Strongly Nonlinear Second-Order ODEs with Rapidly Growing Terms." Journal of Mathematical Analysis and Applications 202, no. 1 (August 1996): 1–26. http://dx.doi.org/10.1006/jmaa.1996.0300.

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18

Navarro, Juan F. "Computation of periodic solutions in perturbed second-order ODEs." Applied Mathematics and Computation 202, no. 1 (August 2008): 171–77. http://dx.doi.org/10.1016/j.amc.2007.12.065.

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19

Enguiça, Ricardo, Andrea Gavioli, and Luís Sanchez. "Solutions of second-order and fourth-order ODEs on the half-line." Nonlinear Analysis: Theory, Methods & Applications 73, no. 9 (November 2010): 2968–79. http://dx.doi.org/10.1016/j.na.2010.06.062.

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20

Bulatov, Mikhail V., and Guido Vanden Berghe. "Two-step fourth order methods for linear ODEs of the second order." Numerical Algorithms 51, no. 4 (November 5, 2008): 449–60. http://dx.doi.org/10.1007/s11075-008-9249-9.

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21

YOSHIKAWA, ATSUKO YAMADA. "EQUIVALENCE PROBLEM OF THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS." International Journal of Mathematics 17, no. 09 (October 2006): 1103–25. http://dx.doi.org/10.1142/s0129167x06003837.

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We are concerned with the equivalence problem of third-order ordinary differential equations (ODEs) under bundle diffeomorphisms of three-dimensional real space. For those ODEs, we define canonical structure equations of a bundle on the second jet space, and introduce four torsion parts. We characterize linear third-order ODEs and projective linear third-order ODEs defined on tangent scrolls of space curves by vanishing of some torsion parts.
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22

Ayub, Muhammad, Masood Khan, and F. M. Mahomed. "Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability." Journal of Applied Mathematics 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/147921.

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We present a systematic procedure for the determination of a complete set ofkth-order (k≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of twokth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case ofk= 2 and 31 classes for the case ofk≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of twokth-order (k≥3) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.
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23

Sun, Wenmin, and Jiguang Bao. "New maximum principles for fully nonlinear ODEs of second order." Discrete & Continuous Dynamical Systems - A 19, no. 4 (2007): 813–23. http://dx.doi.org/10.3934/dcds.2007.19.813.

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24

Haarsa, P., and S. Pothat. "The Frobenius method on a second-order homogeneous linear ODEs." Advanced Studies in Theoretical Physics 8 (2014): 1145–48. http://dx.doi.org/10.12988/astp.2014.4798.

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25

Conte, Robert, Tuen-Wai Ng, and Cheng-Fa Wu. "Hayman’s classical conjecture on some nonlinear second-order algebraic ODEs." Complex Variables and Elliptic Equations 60, no. 11 (April 27, 2015): 1539–52. http://dx.doi.org/10.1080/17476933.2015.1033414.

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26

Terracini, Susanna, and Gianmaria Verzini. "Oscillating solutions to second-order ODEs with indefinite superlinear nonlinearities." Nonlinearity 13, no. 5 (June 22, 2000): 1501–14. http://dx.doi.org/10.1088/0951-7715/13/5/305.

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27

Jerie, M., and G. E. Prince. "Jacobi fields and linear connections for arbitrary second-order ODEs." Journal of Geometry and Physics 43, no. 4 (October 2002): 351–70. http://dx.doi.org/10.1016/s0393-0440(02)00030-x.

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28

Cheb-Terrab, E. S., L. G. S. Duarte, and L. A. C. P. da Mota. "Computer algebra solving of second order ODEs using symmetry methods." Computer Physics Communications 108, no. 1 (January 1998): 90–114. http://dx.doi.org/10.1016/s0010-4655(97)00132-x.

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29

URBAN, JAKUB, and JOSEF PREINHAELTER. "Adaptive finite elements for a set of second-order ODEs." Journal of Plasma Physics 72, no. 06 (December 2006): 1041. http://dx.doi.org/10.1017/s0022377806005186.

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30

Blanes, Sergio. "On the construction of symmetric second order methods for ODEs." Applied Mathematics Letters 98 (December 2019): 41–48. http://dx.doi.org/10.1016/j.aml.2019.05.026.

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31

Ramírez, J., J. L. Romero, and C. Muriel. "Reductions of PDEs to second order ODEs and symbolic computation." Applied Mathematics and Computation 291 (December 2016): 122–36. http://dx.doi.org/10.1016/j.amc.2016.06.043.

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32

Gasull, Armengol, Antoni Guillamon, and Jordi Villadelprat. "The period function for second-order quadratic ODEs is monotone." Qualitative Theory of Dynamical Systems 4, no. 2 (September 2004): 329–52. http://dx.doi.org/10.1007/bf02970864.

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33

Dong, Yujun. "On solvability of three point BVPs of second order ODEs." Journal of Mathematical Analysis and Applications 296, no. 1 (August 2004): 131–39. http://dx.doi.org/10.1016/j.jmaa.2004.03.053.

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34

Gidoni, Paolo. "Existence of a periodic solution for superlinear second order ODEs." Journal of Differential Equations 345 (February 2023): 401–17. http://dx.doi.org/10.1016/j.jde.2022.11.054.

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35

Loy Kak Choon and Puteri Nurul Fatihah binti Mohamad Azli. "On Numerical Methods for Second-Order Nonlinear Ordinary Differential Equations (ODEs): A Reduction To A System Of First-Order ODEs." Universiti Malaysia Terengganu Journal of Undergraduate Research 1, no. 4 (October 31, 2019): 1–8. http://dx.doi.org/10.46754/umtjur.v1i4.86.

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2nd-order ODEs can be found in many applications, e.g., motion of pendulum, vibrating springs, etc. We first convert the 2nd-order nonlinear ODEs to a system of 1st-order ODEs which is easier to deal with. Then, Adams-Bashforth (AB) methods are used to solve the resulting system of 1st-order ODE. AB methods are chosen since they are the explicit schemes and more efficient in terms of shorter computational time. However, the step size is more restrictive since these methods are conditionally stable. We find two test cases (one test problem and one manufactured solution) to be used to validate the AB methods. The exact solution for both test cases are available for the error and convergence analysis later on. The implementation of 1st-, 2nd- and 3rd-order AB methods are done using Octave. The error was computed to retrieve the order of convergence numerically and the CPU time was recorded to analyze their efficiency.
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36

Ramazani, Paria, Ali Abdi, Gholamreza Hojjati, and Afsaneh Moradi. "Explicit Nordsieck second derivative general linear methods for ODEs." ANZIAM Journal 64 (June 28, 2022): 69–88. http://dx.doi.org/10.21914/anziamj.v64.16949.

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The paper deals with the construction of explicit Nordsieck second derivative general linear methods with \(s\) stages of order \(p\) with \(p=s\) and high stage order \(q=p\) with inherent Runge–Kutta or quadratic stability properties. Satisfying the order and stage order conditions together with inherent stability conditions leads to methods with some free parameters, which will be used to obtain methods with a large region of absolute stability. Examples of methods with \(r\) external stages and \(p=q=s=r-1\) up to order five are given, and numerical experiments in a fixed stepsize environment are presented. doi: https://doi.org/10.1017/S1446181122000049
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37

Zhang, Yueyang, Zongsheng Gao, and Jilong Zhang. "All admissible meromorphic solutions of certain type of second degree second order algebraic ODEs." Journal of Mathematical Analysis and Applications 452, no. 2 (August 2017): 1182–93. http://dx.doi.org/10.1016/j.jmaa.2017.03.054.

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38

Medveď, Milan. "Generic saddle-node bifurcation for cascade second order ODEs on manifolds." Annales Polonici Mathematici 68, no. 3 (1998): 211–25. http://dx.doi.org/10.4064/ap-68-3-211-225.

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39

Medve\vd, Milan. "A class of vector fields on manifolds containing second order ODEs." Hiroshima Mathematical Journal 26, no. 1 (1996): 127–49. http://dx.doi.org/10.32917/hmj/1206127493.

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40

Abdelli, Mama, and Alain Haraux. "The universal bound property for a class of second order ODEs." Portugaliae Mathematica 76, no. 1 (September 30, 2019): 49–56. http://dx.doi.org/10.4171/pm/2026.

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41

Georgiev, Svetlin, Mohamed Majdoub, and Karima Mebarki. "Multiple nonnegative solutions for a class IVPs for second order ODEs." Filomat 35, no. 14 (2021): 4701–13. http://dx.doi.org/10.2298/fil2114701g.

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We study a class of initial value problems for second order ODEs. The interesting points of our results are that the nonlinearity depends on the solution and its derivative and may change sign. Moreover, it satisfies general polynomial growth conditions. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions. The arguments are based upon a recent theoretical result.
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42

Mgaga, T. C., and K. S. Govinder. "On the linearization of some second-order ODEs via contact transformations." Journal of Physics A: Mathematical and Theoretical 44, no. 1 (November 30, 2010): 015203. http://dx.doi.org/10.1088/1751-8113/44/1/015203.

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43

Evans, D. J., and M. I. Jayes. "New numerical methods for a special class of second order odes." International Journal of Computer Mathematics 48, no. 3-4 (January 1993): 191–201. http://dx.doi.org/10.1080/00207169308804202.

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44

D’Ambrosio, Raffaele, and Beatrice Paternoster. "A general framework for the numerical solution of second order ODEs." Mathematics and Computers in Simulation 110 (April 2015): 113–24. http://dx.doi.org/10.1016/j.matcom.2014.04.007.

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45

Lewicka, Marta, and Marco Spadini. "Branches of forced oscillations in degenerate systems of second-order ODEs." Nonlinear Analysis: Theory, Methods & Applications 68, no. 9 (May 2008): 2623–28. http://dx.doi.org/10.1016/j.na.2007.02.008.

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46

Dalbono, Francesca. "Branches of index-preserving solutions to systems of second order ODEs." Nonlinear Differential Equations and Applications NoDEA 16, no. 5 (May 26, 2009): 569–95. http://dx.doi.org/10.1007/s00030-009-0019-8.

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47

Sharifi, Mohammad, Ali Abdi, Michal Braś, and Gholamreza Hojjati. "HIGH ORDER SECOND DERIVATIVE DIAGONALLY IMPLICIT MULTISTAGE INTEGRATION METHODS FOR ODES." Mathematical Modelling and Analysis 28, no. 1 (January 19, 2023): 53–70. http://dx.doi.org/10.3846/mma.2023.16102.

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Construction of second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods with Runge–Kutta stability property requires to generate the corresponding conditions depending of the parameters of the methods. These conditions which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with Runge–Kutta stability property by using some variant of the Fourier series method which has been already used for the construction of high order general linear methods. Examples of explicit and implicit SDIMSIMs of order five and six are given which respectively are appropriate for both non-stiff and stiff differential systems in a sequential computing environment. Finally, the efficiency of the constructed methods is verified by providing some numerical experiments.
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48

Adeyefa, E., and A. Ibrahim. "A Sixth-order Self-Starting Algorithms for Second Order Initial Value Problems of ODEs." British Journal of Mathematics & Computer Science 15, no. 2 (January 10, 2016): 1–8. http://dx.doi.org/10.9734/bjmcs/2016/24322.

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49

Sȩdziwy, Stanisław. "Boundary value problems for second order differential equations with $$\varphi $$-Laplacians." Archiv der Mathematik 118, no. 1 (December 12, 2021): 101–11. http://dx.doi.org/10.1007/s00013-021-01666-1.

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50

Humane, Pramod. "The Direct Two-Point Block One-Step Method Which is efficient and Suitable for Solving Second-Order Differential Equation." International Journal for Research in Applied Science and Engineering Technology 10, no. 4 (April 30, 2022): 3422–29. http://dx.doi.org/10.22214/ijraset.2022.41779.

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Abstract: A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size.Two point four step direct implicit block method is developed for solving directly the second order system of ordinary differential equations (ODEs) using variable step size. The method will estimate the solutions of Initial Value Problems (IVPs) at two points simultaneously by using four backward steps. The method developed is suitable for the numerical integration of non stiff and mildly stiff differential systems. Numerical results are given to compare the efficiency of the developed method to the existence block method.
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