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Journal articles on the topic 'Ordinary differential equations'

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Published: 4 June 2021
Last updated: 30 July 2024

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1

Brauer, Fred, Vladimir I. Arnol'd, and Roger Cook. "Ordinary Differential Equations." American Mathematical Monthly 100, no. 8 (October 1993): 810. http://dx.doi.org/10.2307/2324802.

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2

Rawlins, A. D., and M. Sever. "Ordinary Differential Equations." Mathematical Gazette 72, no. 462 (December 1988): 334. http://dx.doi.org/10.2307/3619967.

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3

Kapadia, Devendra A., and V. I. Arnold. "Ordinary Differential Equations." Mathematical Gazette 79, no. 484 (March 1995): 228. http://dx.doi.org/10.2307/3620107.

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4

Dory, Robert A. "Ordinary Differential Equations." Computers in Physics 3, no. 5 (1989): 88. http://dx.doi.org/10.1063/1.4822872.

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5

Li, Haoxuan. "The advance of neural ordinary differential ordinary differential equations." Applied and Computational Engineering 6, no. 1 (June 14, 2023): 1283–87. http://dx.doi.org/10.54254/2755-2721/6/20230709.

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Differential methods are widely used to describe complex continuous processes. The main idea of ordinary differential equations is to treat a specific type of neural network as a discrete equation. Therefore, the differential equation solver can be used to optimize the solution process of the neural network. Compared with the conventional neural network solution, the solution process of the neural ordinary differential equation has the advantages of high storage efficiency and adaptive calculation. This paper first gives a brief review of the residual network (ResNet) and the relationship of ResNet to neural ordinary differential equations. Besides, his paper list three advantages of neural ordinary differential equations compared with ResNet and introduce the class of Deep Neural Network (DNN) models that can be seen as numerical discretization of neural ordinary differential equations (N-ODEs). Furthermore, this paper analyzes a defect of neural ordinary differential equations that do not appear in the traditional deep neural network. Finally, this paper demonstrates how to analyze ResNet with neural ordinary differential equations and shows the main application of neural ordinary differential equations (Neural-ODEs).
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6

Saltas, Vassilios, Vassilios Tsiantos, and Dimitrios Varveris. "Solving Differential Equations and Systems of Differential Equations with Inverse Laplace Transform." European Journal of Mathematics and Statistics 4, no. 3 (June 14, 2023): 1–8. http://dx.doi.org/10.24018/ejmath.2023.4.3.192.

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The inverse Laplace transform enables the solution of ordinary linear differential equations as well as systems of ordinary linear differentials with applications in the physical and engineering sciences. The Laplace transform is essentially an integral transform which is introduced with the help of a suitable generalized integral. The ultimate goal of this work is to introduce the reader to some of the basic ideas and applications for solving initially ordinary differential equations and then systems of ordinary linear differential equations.
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7

Sanchez, David A. "Ordinary Differential Equations Texts." American Mathematical Monthly 105, no. 4 (April 1998): 377. http://dx.doi.org/10.2307/2589736.

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8

Iserles, A., D. W. Jordan, and P. Smith. "Nonlinear Ordinary Differential Equations." Mathematical Gazette 72, no. 460 (June 1988): 155. http://dx.doi.org/10.2307/3618957.

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9

Sánchez, David A. "Ordinary Differential Equations Texts." American Mathematical Monthly 105, no. 4 (April 1998): 377–83. http://dx.doi.org/10.1080/00029890.1998.12004897.

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10

Hadeler, K. P., and S. Walcher. "Reducible Ordinary Differential Equations." Journal of Nonlinear Science 16, no. 6 (June 29, 2006): 583–613. http://dx.doi.org/10.1007/s00332-004-0627-8.

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11

Knorrenschild, Michael. "Differential/Algebraic Equations As Stiff Ordinary Differential Equations." SIAM Journal on Numerical Analysis 29, no. 6 (December 1992): 1694–715. http://dx.doi.org/10.1137/0729096.

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12

MANOFF, S. "GEODESIC AND AUTOPARALLEL EQUATIONS OVER DIFFERENTIABLE MANIFOLDS." International Journal of Modern Physics A 11, no. 21 (August 20, 1996): 3849–74. http://dx.doi.org/10.1142/s0217751x96001814.

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The notions of ordinary, covariant and Lie differentials are considered as operators over differentiable manifolds with different (not only by sign) contravariant and covariant affine connections and metric. The difference between the interpretations of the ordinary differential as a covariant basic vector field and as a component of a contravariant vector field is discussed. By means of the covariant metric and the ordinary differential the notion of the line element is introduced and the geodesic equation is obtained and compared with the autoparallel equation.
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13

Kvinikadze, Giorgi. "On Kneser-type solutions of sublinear ordinary differential equations." Časopis pro pěstování matematiky 115, no. 2 (1990): 118–33. http://dx.doi.org/10.21136/cpm.1990.108371.

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14

Afuwape, Anthony Uyi, and M. O. Omeike. "Ultimate boundedness of some third order ordinary differential equations." Mathematica Bohemica 137, no. 3 (2012): 355–64. http://dx.doi.org/10.21136/mb.2012.142900.

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15

Jankowski, Tadeusz. "On numerical solution of ordinary differential equations with discontinuities." Applications of Mathematics 33, no. 6 (1988): 487–92. http://dx.doi.org/10.21136/am.1988.104326.

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16

Jankowski, Tadeusz. "One-step methods for ordinary differential equations with parameters." Applications of Mathematics 35, no. 1 (1990): 67–83. http://dx.doi.org/10.21136/am.1990.104388.

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17

Vondra, Alexandr. "Geometry of second-order connections and ordinary differential equations." Mathematica Bohemica 120, no. 2 (1995): 145–67. http://dx.doi.org/10.21136/mb.1995.126226.

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18

Crilly, Tony, Robert E. O'Malley, Glenn Fulford, Peter Forrester, Arthur Jones, R. M. Mattheij, and J. Molenaar. "Thinking about Ordinary Differential Equations." Mathematical Gazette 83, no. 497 (July 1999): 367. http://dx.doi.org/10.2307/3619113.

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19

Chambers, Ll G., and Stephen H. Saperstone. "Introduction to Ordinary Differential Equations." Mathematical Gazette 83, no. 497 (July 1999): 370. http://dx.doi.org/10.2307/3619116.

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20

Harte, R. "Exactness in ordinary differential equations." Irish Mathematical Society Bulletin 0024 (1990): 20–40. http://dx.doi.org/10.33232/bims.0024.20.40.

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21

Kim, Suyong, Weiqi Ji, Sili Deng, Yingbo Ma, and Christopher Rackauckas. "Stiff neural ordinary differential equations." Chaos: An Interdisciplinary Journal of Nonlinear Science 31, no. 9 (September 2021): 093122. http://dx.doi.org/10.1063/5.0060697.

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22

Tryhuk, Václav, and Veronika Chrastinová. "Automorphisms of Ordinary Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–32. http://dx.doi.org/10.1155/2014/482963.

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The paper deals with the local theory of internal symmetries of underdetermined systems of ordinary differential equations in full generality. The symmetries need not preserve the choice of the independent variable, the hierarchy of dependent variables, and the order of derivatives. Internal approach to the symmetries of one-dimensional constrained variational integrals is moreover proposed without the use of multipliers.
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23

Haraoka, Yoshishige. "Prolongability of Ordinary Differential Equations." Journal of Nonlinear Mathematical Physics 20, sup1 (November 8, 2013): 70–84. http://dx.doi.org/10.1080/14029251.2013.862435.

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24

Press, William H., and Saul A. Teukolsky. "Integrating Stiff Ordinary Differential Equations." Computers in Physics 3, no. 3 (1989): 88. http://dx.doi.org/10.1063/1.4822847.

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25

Pai, M. "Book reviews - Ordinary differential equations." IEEE Control Systems Magazine 6, no. 1 (February 1986): 50. http://dx.doi.org/10.1109/mcs.1986.1105053.

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26

Weidman, P. "Thinking about ordinary differential equations." European Journal of Mechanics - B/Fluids 18, no. 2 (March 1999): 315–17. http://dx.doi.org/10.1016/s0997-7546(99)80029-6.

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27

Fečkan, Michal. "Singularly perturbed ordinary differential equations." Journal of Mathematical Analysis and Applications 170, no. 1 (October 1992): 214–24. http://dx.doi.org/10.1016/0022-247x(92)90015-6.

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28

Schwarz, Fritz. "Decomposition of ordinary differential equations." Bulletin of Mathematical Sciences 7, no. 3 (November 22, 2017): 575–613. http://dx.doi.org/10.1007/s13373-017-0110-0.

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29

Novick-Cohen, Amy. "Ordinary and partial differential equations." Mathematical Biosciences 94, no. 1 (May 1989): 151–52. http://dx.doi.org/10.1016/0025-5564(89)90075-8.

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30

Tóthová, Mária, and Oleg Palumbíny. "On monotone solutions of the fourth order ordinary differential equations." Czechoslovak Mathematical Journal 45, no. 4 (1995): 737–46. http://dx.doi.org/10.21136/cmj.1995.128553.

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31

Artstein, Zvi. "On singularly perturbed ordinary differential equations with measure-valued limits." Mathematica Bohemica 127, no. 2 (2002): 139–52. http://dx.doi.org/10.21136/mb.2002.134168.

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32

Polášek, Vladimír, and Irena Rachůnková. "Singular Dirichlet problem for ordinary differential equations with $\phi$-Laplacian." Mathematica Bohemica 130, no. 4 (2005): 409–25. http://dx.doi.org/10.21136/mb.2005.134206.

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33

Tsyfra, Ivan. "On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations." Opuscula Mathematica 41, no. 5 (2021): 685–99. http://dx.doi.org/10.7494/opmath.2021.41.5.685.

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We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.
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34

Zhang, Zijia, Yaoming Cai, and Dongfang Zhang. "Solving Ordinary Differential Equations With Adaptive Differential Evolution." IEEE Access 8 (2020): 128908–22. http://dx.doi.org/10.1109/access.2020.3008823.

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35

Goltser, Y., and E. Litsyn. "Volterra integro-differential equations and infinite systems of ordinary differential equations." Mathematical and Computer Modelling 42, no. 1-2 (July 2005): 221–33. http://dx.doi.org/10.1016/j.mcm.2004.01.014.

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36

Dhariwal, Monika, and Nahid Fatima. "RHPF for Solving Ordinary Differential Equations." Journal of Intelligent Systems and Computing 1, no. 1 (December 31, 2020): 37–45. http://dx.doi.org/10.51682/jiscom.00101003.2020.

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The current research paper includes the latest research in the field of science. We have introduced the utilization of the RHPF in the current paper. We have solved ordinary differential equations by using RHPF. We compared the decision of using RHPF with those using HPM to understand the new method's efficacy and benefit. The goal of the analysis is to show that the correct construction of the homotopy determines the solution with less computation than with the current method and produces trustworthy outcomes. We've found three ODE problems. The results show that this method is useful and impressive in explaining the ODE. The RHPF is known as a new idea, new techniques of creation.
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37

HarrarII, D. L., and M. R. Osborne. "Computing eigenvalues of ordinary differential equations." ANZIAM Journal 44 (April 1, 2003): 313. http://dx.doi.org/10.21914/anziamj.v44i0.684.

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38

Skórnik, Krystyna, and Joseph Wloka. "m-Reduction of ordinary differential equations." Colloquium Mathematicum 78, no. 2 (1998): 195–212. http://dx.doi.org/10.4064/cm-78-2-195-212.

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39

Ungar, Abraham. "Addition Theorems in Ordinary Differential Equations." American Mathematical Monthly 94, no. 9 (November 1987): 872. http://dx.doi.org/10.2307/2322823.

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40

Brindley, G., E. L. Ince, and I. N. Sneddon. "The Solution of Ordinary Differential Equations." Mathematical Gazette 72, no. 460 (June 1988): 154. http://dx.doi.org/10.2307/3618956.

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41

Butcher, John C., and Lawrence F. Shampine. "Numerical Solution of Ordinary Differential Equations." Mathematics of Computation 64, no. 211 (July 1995): 1345. http://dx.doi.org/10.2307/2153505.

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42

Anumasa, Srinivas, and P. K. Srijith. "Latent Time Neural Ordinary Differential Equations." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 6 (June 28, 2022): 6010–18. http://dx.doi.org/10.1609/aaai.v36i6.20547.

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Neural ordinary differential equations (NODE) have been proposed as a continuous depth generalization to popular deep learning models such as Residual networks (ResNets). They provide parameter efficiency and automate the model selection process in deep learning models to some extent. However, they lack the much-required uncertainty modelling and robustness capabilities which are crucial for their use in several real-world applications such as autonomous driving and healthcare. We propose a novel and unique approach to model uncertainty in NODE by considering a distribution over the end-time T of the ODE solver. The proposed approach, latent time NODE (LT-NODE), treats T as a latent variable and apply Bayesian learning to obtain a posterior distribution over T from the data. In particular, we use variational inference to learn an approximate posterior and the model parameters. Prediction is done by considering the NODE representations from different samples of the posterior and can be done efficiently using a single forward pass. As T implicitly defines the depth of a NODE, posterior distribution over T would also help in model selection in NODE. We also propose, adaptive latent time NODE (ALT-NODE), which allow each data point to have a distinct posterior distribution over end-times. ALT-NODE uses amortized variational inference to learn an approximate posterior using inference networks. We demonstrate the effectiveness of the proposed approaches in modelling uncertainty and robustness through experiments on synthetic and several real-world image classification data.
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43

Malacka, Zuzana. "Pursuit Curves and Ordinary Differential Equations." Communications - Scientific letters of the University of Zilina 14, no. 1 (March 31, 2012): 66–68. http://dx.doi.org/10.26552/com.c.2012.1.66-68.

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44

Braaksma, Boele. "Multisummability and ordinary meromorphic differential equations." Banach Center Publications 97 (2012): 29–38. http://dx.doi.org/10.4064/bc97-0-2.

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45

POLIASHENKO, MAXIM, and CYRUS K. AIDUN. "COMPUTATIONAL DYNAMICS OF ORDINARY DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 05, no. 01 (February 1995): 159–74. http://dx.doi.org/10.1142/s0218127495000132.

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Discrete schemes, used to perform time integration of ODE’s, are expected to exhibit qualitatively ‘true’ dynamics in terms of the solutions and their stability. In past years, it has been discovered that such discretizations may cause spurious steady states and some explicit schemes may produce ‘computational chaos.’ In this study, we show that implicit time integration schemes, such as the backward Euler method, can also produce computationally chaotic solutions. Furthermore, we show that the opposite phenomenon may also take place both for explicit and for implicit schemes: computationally generated ‘spurious order’ may replace the true chaotic solution before the scheme becomes linearly unstable. The numerical solution may become chaotic again as the discretization step is further increased. The spurious computational order and chaos are discussed by solving low-dimensional dynamical systems, as well as a large system of ODE representing the solution to the Navier-Stokes equation. Our results support the point of view that the deviations in the behavior of the computed solution from the true solution has deterministic character with the time step playing the role of an artificial bifurcation parameter.
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46

Swinnerton-Dyer, Peter, and Thomas Wagenknecht. "Some third-order ordinary differential equations." Bulletin of the London Mathematical Society 40, no. 5 (May 29, 2008): 725–48. http://dx.doi.org/10.1112/blms/bdn046.

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47

Krogh, Fred T. "Stepsize selection for ordinary differential equations." ACM Transactions on Mathematical Software 37, no. 2 (April 2010): 1–11. http://dx.doi.org/10.1145/1731022.1731025.

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48

Kalas, Josef. "Nonuniqueness results for ordinary differential equations." Czechoslovak Mathematical Journal 48, no. 2 (June 1998): 373–84. http://dx.doi.org/10.1023/a:1022853923977.

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49

AM, K. SELV. "Alternative Methods of Ordinary Differential Equations." International Journal of Mathematics Trends and Technology 54, no. 6 (February 25, 2018): 448–53. http://dx.doi.org/10.14445/22315373/ijmtt-v54p554.

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50

Voorhees, Burton, and Alexander Nip. "Ordinary Differential Equations with Star Structure." Journal of Dynamical Systems and Geometric Theories 3, no. 2 (January 2005): 121–52. http://dx.doi.org/10.1080/1726037x.2005.10698495.

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