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Journal articles on the topic 'Second order ODEs'

Author: Grafiati

Published: 10 March 2023

Last updated: 31 July 2025

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1

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

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2

B S, Manjula. "Study of Methods for Solving Second-Order ODEs with Constant Coefficient." International Journal of Science and Research (IJSR) 10, no. 7 (2021): 1550–55. http://dx.doi.org/10.21275/es24928084917.

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3

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

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4

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

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5

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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6

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

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7

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

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8

Pan-Collantes, Antonio J., and José Antonio Álvarez-García. "Autonomous Second-Order ODEs: A Geometric Approach." Axioms 13, no. 11 (2024): 788. http://dx.doi.org/10.3390/axioms13110788.

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Given an autonomous second-order ordinary differential equation (ODE), we define a Riemannian metric on an open subset of the first-order jet bundle. A relationship is established between the solutions of the ODE and the geodesic curves with respect to the defined metric. We introduce the notion of energy foliation for autonomous ODEs and highlight its connection to the classical energy concept. Additionally, we explore the geometry of the leaves of the foliation. Finally, the results are applied to the analysis of Lagrangian mechanical systems. In particular, we provide an autonomous Lagrangi

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9

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

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10

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

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11

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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12

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 (2008): 285206. http://dx.doi.org/10.1088/1751-8113/41/28/285206.

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13

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

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14

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 th

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15

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 (1999): 827–44. http://dx.doi.org/10.1088/0305-4470/32/5/013.

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16

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

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17

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 (2013): 969–89. http://dx.doi.org/10.1007/s11071-013-1016-3.

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18

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

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19

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 (1996): 1–26. http://dx.doi.org/10.1006/jmaa.1996.0300.

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20

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

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21

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 (2010): 2968–79. http://dx.doi.org/10.1016/j.na.2010.06.062.

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22

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

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23

Yahaya, Muritala, Abdulkadir Abubakar, and Yusuf Dauda Jikantoro. "DEVELOPMENT OF A CONTINUOUS BACKWARD DIFFERENTIATION FORMULAE FOR SOLVING FIRST-ORDER AND SECOND-ORDER INITIAL VALUE PROBLEM." FUDMA JOURNAL OF SCIENCES 9, no. 5 (2025): 14–22. https://doi.org/10.33003/fjs-2025-0905-3633.

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Numerical methods for solving ordinary differential equations (ODEs) are essential in modeling dynamical systems across science and engineering. While specialized methods exist for first-order and second-order ODEs, developing a unified approach that efficiently handles both classes remain an active area of research. In this paper, we present a novel two-step hybrid block method based on the backward differentiation formula (BDF), capable of approximating solutions for both first- and second-order ODEs without requiring separate derivations. The method is constructed using interpolation and co

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24

Jama, Mohamed, Kang'ethe Giterere, and Duncan Kioi. "A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients." Applied and Computational Mathematics 14, no. 2 (2025): 78–89. https://doi.org/10.11648/j.acm.20251402.11.

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Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the new analytical method. The new method was derived from the concept of exactness in higher-order ODEs. Specifically, second- and third-order ODEs with variable coefficients are exact if there exist first- and second-order linear ODEs whose derivatives

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25

YOSHIKAWA, ATSUKO YAMADA. "EQUIVALENCE PROBLEM OF THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS." International Journal of Mathematics 17, no. 09 (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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26

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

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27

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 (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 t

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28

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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29

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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30

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 (2015): 1539–52. http://dx.doi.org/10.1080/17476933.2015.1033414.

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31

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

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32

Šremr, Jiří. "On a periodic problem for super-linear second-order ODEs." Mathematica Slovaca 74, no. 6 (2024): 1457–76. https://doi.org/10.1515/ms-2024-0106.

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Abstract The present paper concerns the periodic problem u ″ = p ( t ) u − q ( t , u ) u + f ( t ) ; u ( 0 ) = u ( ω ) , u ′ ( 0 ) = u ′ ( ω ) , $$\begin{array}{} \displaystyle u''=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\, u'(0)=u'(\omega), \end{array}$$ where p, f : [0, ω] → ℝ are Lebesgue integrable functions and q : [0, ω] × ℝ → ℝ is a Carathéodory function. We assume that the anti-maximum principle holds for the corresponding linear problem and provide sufficient conditions guaranteeing the existence and uniqueness of a positive solution to the given non-linear problem. The general result

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33

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

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34

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 (1998): 90–114. http://dx.doi.org/10.1016/s0010-4655(97)00132-x.

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35

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

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36

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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37

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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38

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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39

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 (2004): 329–52. http://dx.doi.org/10.1007/bf02970864.

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40

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

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41

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 environm

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42

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 (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 solutio

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43

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 (2017): 1182–93. http://dx.doi.org/10.1016/j.jmaa.2017.03.054.

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44

Sinkala, Winter. "A New Method for Finding Lie Point Symmetries of First-Order Ordinary Differential Equations." Symmetry 15, no. 12 (2023): 2198. http://dx.doi.org/10.3390/sym15122198.

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The traditional algorithm for finding Lie point symmetries of ordinary differential equations (ODEs) faces challenges when applied to first-order ODEs. This stems from the fact that for first-order ODEs, unlike higher-order ODEs, the determining equation lacks derivatives, rendering it impossible to decompose into simpler PDEs to be solved for the infinitesimals. Consequently, a common technique for determining Lie point symmetries of first-order ODEs involves making speculative assumptions about the form of the infinitesimal generator. In this study, we propose a novel and more efficient appr

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45

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

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46

Abdul Majid, Zanariah, Nur Zahidah Mokhtar, and Mohamed Suleiman. "Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations." Mathematical Problems in Engineering 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/184253.

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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. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in thePE(CE)mmode

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47

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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48

Shah, Said Waqas, F. M. Mahomed, and H. Azad. "A Note on the Integration of Scalar Fourth-Order Ordinary Differential Equations with Four-Dimensional Symmetry Algebras." Mathematical Problems in Engineering 2021 (March 19, 2021): 1–7. http://dx.doi.org/10.1155/2021/6619325.

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The complete integration of scalar fourth-order ODEs with four-dimensional symmetry algebras is performed by utilizing Lie’s method which was invoked to integrate scalar second-order ODEs admitting two-dimensional symmetry algebras. We obtain a complete integration of all scalar fourth-order ODEs that possess four Lie point symmetries.

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49

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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50

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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