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Journal articles on the topic 'Methods of ordinary differential equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 28 July 2025

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1

Askarova, A., Ye. Gripp, and G. Yeleussizova. "NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS." Scientific heritage, no. 103 (December 21, 2022): 67–69. https://doi.org/10.5281/zenodo.7467608.

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For many applied problems it is practically impossible to obtain the exact solution of differential equations. In these cases, methods of approximate solution of differential equations are used. This article considers the solution of a differential equation by various numerical methods.

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2

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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3

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

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4

Ramos, J. I. "Linearized methods for ordinary differential equations." Applied Mathematics and Computation 104, no. 2-3 (1999): 109–29. http://dx.doi.org/10.1016/s0096-3003(98)10056-5.

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5

Gear, C. W. "Parallel methods for ordinary differential equations." Calcolo 25, no. 1-2 (1988): 1–20. http://dx.doi.org/10.1007/bf02575744.

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6

Li, Haoxuan. "The advance of neural ordinary differential ordinary differential equations." Applied and Computational Engineering 6, no. 1 (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 R

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7

Shintani, Hisayoshi. "Two-step methods for ordinary differential equations." Hiroshima Mathematical Journal 14, no. 3 (1985): 471–78. http://dx.doi.org/10.32917/hmj/1206132929.

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8

Butcher, John. "General linear methods for ordinary differential equations." Mathematics and Computers in Simulation 79, no. 6 (2009): 1834–45. http://dx.doi.org/10.1016/j.matcom.2007.02.006.

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9

Tranquilli, Paul, and Adrian Sandu. "Exponential-Krylov methods for ordinary differential equations." Journal of Computational Physics 278 (December 2014): 31–46. http://dx.doi.org/10.1016/j.jcp.2014.08.013.

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10

Ramos, J. I., та C. M. García-López. "Linearized Θ-methods I. Ordinary differential equations". Computer Methods in Applied Mechanics and Engineering 129, № 3 (1996): 255–69. http://dx.doi.org/10.1016/0045-7825(95)00915-9.

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11

Jankowski, Tadeusz. "Convergence of multistep methods for systems of ordinary differential equations with parameters." Applications of Mathematics 32, no. 4 (1987): 257–70. http://dx.doi.org/10.21136/am.1987.104257.

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12

März, Roswitha. "Numerical methods for differential algebraic equations." Acta Numerica 1 (January 1992): 141–98. http://dx.doi.org/10.1017/s0962492900002269.

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13

Cardone, Angelamaria, Dajana Conte, Raffaele D’Ambrosio, and Beatrice Paternoster. "Multivalue Collocation Methods for Ordinary and Fractional Differential Equations." Mathematics 10, no. 2 (2022): 185. http://dx.doi.org/10.3390/math10020185.

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The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods for fractional differential equations. The construction of the methods together with the convergence and stability analysis are reported and some numerical experiments are carried out to show the efficiency of the proposed methods.

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14

Jankowski, Tadeusz. "Multistep methods for ordinary differential equations with parameters." Annales Polonici Mathematici 50, no. 1 (1989): 63–69. http://dx.doi.org/10.4064/ap-50-1-63-69.

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15

Göktaş, Ünal, and Devendra Kapadia. "Methods in Mathematica for Solving Ordinary Differential Equations." Mathematical and Computational Applications 16, no. 4 (2011): 784–96. http://dx.doi.org/10.3390/mca16040784.

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16

Pucci, Edvige, and Giuseppe Saccomandi. "On the reduction methods for ordinary differential equations." Journal of Physics A: Mathematical and General 35, no. 29 (2002): 6145–55. http://dx.doi.org/10.1088/0305-4470/35/29/314.

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17

Ahmad, R. R., N. Yaacob, and A. H. Mohd Murid. "Explicit methods in solving stiff ordinary differential equations." International Journal of Computer Mathematics 81, no. 11 (2004): 1407–15. http://dx.doi.org/10.1080/00207160410001661744.

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18

Engquist, Bjorn, and Yen-Hsi Tsai. "Heterogeneous multiscale methods for stiff ordinary differential equations." Mathematics of Computation 74, no. 252 (2005): 1707–43. http://dx.doi.org/10.1090/s0025-5718-05-01745-x.

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19

Bouchereau, Maxime, Philippe Chartier, Mohammed Lemou, and Florian Méhats. "Machine learning methods for autonomous ordinary differential equations." Communications in Mathematical Sciences 22, no. 6 (2024): 1463–82. http://dx.doi.org/10.4310/cms.2024.v22.n6.a1.

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20

Faltinsen, Stig, Arne Marthinsen, and Hans Z. Munthe-Kaas. "Multistep methods integrating ordinary differential equations on manifolds." Applied Numerical Mathematics 39, no. 3-4 (2001): 349–65. http://dx.doi.org/10.1016/s0168-9274(01)00103-9.

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21

Zingg, D. W., and T. T. Chisholm. "Runge–Kutta methods for linear ordinary differential equations." Applied Numerical Mathematics 31, no. 2 (1999): 227–38. http://dx.doi.org/10.1016/s0168-9274(98)00129-9.

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22

Wu, Jingwen, and Hongjiong Tian. "Functionally-fitted block methods for ordinary differential equations." Journal of Computational and Applied Mathematics 271 (December 2014): 356–68. http://dx.doi.org/10.1016/j.cam.2014.04.013.

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23

Butcher, J. C. "Trees and numerical methods for ordinary differential equations." Numerical Algorithms 53, no. 2-3 (2009): 153–70. http://dx.doi.org/10.1007/s11075-009-9285-0.

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24

Izzo, Giuseppe, and Zdzislaw Jackiewicz. "Generalized linear multistep methods for ordinary differential equations." Applied Numerical Mathematics 114 (April 2017): 165–78. http://dx.doi.org/10.1016/j.apnum.2016.04.009.

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25

Iserles, Arieh, Geetha Ramaswami, and Mark Sofroniou. "Runge-Kutta methods for quadratic ordinary differential equations." BIT Numerical Mathematics 38, no. 2 (1998): 315–46. http://dx.doi.org/10.1007/bf02512370.

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26

Guo, Ben-yu, and Zhong-qing Wang. "Legendre–Gauss collocation methods for ordinary differential equations." Advances in Computational Mathematics 30, no. 3 (2008): 249–80. http://dx.doi.org/10.1007/s10444-008-9067-6.

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27

Muriel, C. "New methods of reduction for ordinary differential equations." IMA Journal of Applied Mathematics 66, no. 2 (2001): 111–25. http://dx.doi.org/10.1093/imamat/66.2.111.

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28

Suleiman, Mohamed B., Fudziah Bt Ismail, and Kamel Ariffin B. M. Atan. "Partitioning ordinary differential equations using Runge-Kutta methods." Applied Mathematics and Computation 79, no. 2-3 (1996): 289–309. http://dx.doi.org/10.1016/0096-3003(95)00247-2.

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29

Ramos, J. I. "Piecewise homotopy methods for nonlinear ordinary differential equations." Applied Mathematics and Computation 198, no. 1 (2008): 92–116. http://dx.doi.org/10.1016/j.amc.2007.08.030.

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30

Bensebah, A., F. Dubeau, and J. Gélinas. "Projection methods and approximations for ordinary differential equations." Approximation Theory and its Applications 13, no. 3 (1997): 78–90. http://dx.doi.org/10.1007/bf02837013.

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31

Zhang, Hong, Ying Liu, and Romit Maulik. "Semi-Implicit Neural Ordinary Differential Equations." Proceedings of the AAAI Conference on Artificial Intelligence 39, no. 21 (2025): 22416–24. https://doi.org/10.1609/aaai.v39i21.34398.

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Classical neural ODEs trained with explicit methods are intrinsically limited by stability, crippling their efficiency and robustness for stiff learning problems that are common in graph learning and scientific machine learning. We present a semi-implicit neural ODE approach that exploits the partitionable structure of the underlying dynamics. Our technique leads to an implicit neural network with significant computational advantages over existing approaches because of enhanced stability and efficient linear solves during time integration. We show that our approach outperforms existing approac

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32

Li, Xin, Chengli Zhao, Xue Zhang, and Xiaojun Duan. "Symbolic Neural Ordinary Differential Equations." Proceedings of the AAAI Conference on Artificial Intelligence 39, no. 17 (2025): 18511–19. https://doi.org/10.1609/aaai.v39i17.34037.

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Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great significance. In this study, we propose a novel learning framework of symbolic continuous-depth neural networks, termed Symbolic Neural Ordinary Differential Equations (SNODEs), to effectively and accurately learn the underlying dynamics of complex systems. Specifically, our learning framework comprises three stages: initially, pre-training a predefined symbolic neura

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33

Yousef, Ali Sulaiman Alsulaiman, and M. D. Al-Eybani Ahmad. "Solve the Second Order Ordinary Differential Equations by Adomian Decomposition Method." International Journal of Mathematics and Physical Sciences Research 11, no. 1 (2023): 6–9. https://doi.org/10.5281/zenodo.7907545.

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<strong>Abstract:</strong> The Adomian decomposition model (ADM) is described by Evans and Raslan (2005) as a semi-analytical model used to solve partial and ordinary non-linear differentials. ADM was designed by George Adomian between the 1970s and the 1990s (Hosseini & Nasabzadeh, 2007) whilst working at the University of Georgia’s Applied Mathematics department. <strong>Keywords:</strong> Ordinary Differential - Differential Equations - Adomian Decomposition Method. <strong>Title:</strong> Solve the Second Order Ordinary Differential Equations by Adomian Decomposition Method <stro

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34

W. Khairiyah Hulaini Wan Ramli, Nur Iffah Zulaikha Hamdan, Nur Ain Dayana Ruslan, et al. "The Solution of Third Order Ordinary Differential Equations using Adomian Decomposition Method and Variational Iteration Method." Journal of Mathematics and Computing Science 9, no. 2 (2023): 67–75. https://doi.org/10.24191/jmcs.v9i2.545.

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Solving ordinary differential equations is crucial in physics, engineering, and mathematics. Adomian decomposition method and variational iteration method are two of many methods in solving ordinary differential equations. Both techniques involve constructing the required iterative or recurrence formulas based on the equation under consideration and additional requirements, allowing the identification of succeeding iterations of a series or sequence approximately matching to solve the third order ordinary differential equations. In this study, adomian decomposition method and variational itera

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35

Jadhav, Changdev, Tanisha Dale, and Dr Vaijanath Chinchane. "A Method to solve ordinary fractional differential equations using Elzaki and Sumudu transform." Journal of Fractional Calculus and Nonlinear Systems 4, no. 1 (2023): 8–16. http://dx.doi.org/10.48185/jfcns.v4i1.757.

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The main objective of the paper is to solve ordinary fractional differential equations using Elzaki and Sumudu transform. Moreover some ordinary fractional differential equations are solved by using the presented method. Using different types of fractional differential operators existing methods have been extended and applied for ordinary fractional differential equations.

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36

Boykov, Ilya, Vladimir Roudnev, and Alla Boykova. "Approximate Methods for Solving Problems of Mathematical Physics on Neural Hopfield Networks." Mathematics 10, no. 13 (2022): 2207. http://dx.doi.org/10.3390/math10132207.

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A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficie

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37

Khalil, Omar, Hany El-Sharkawy, Maha Youssef, and Gerd Baumann. "Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations." Algorithms 15, no. 9 (2022): 320. http://dx.doi.org/10.3390/a15090320.

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We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximation, for which we derived an a priori error estimate for our adaptive method and showed its exponential convergence in the number of iterations. In this work, we show the exponential convergence in the number of iterations of the a priori error estima

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38

GEORGIEVA, A., and S. KOSTADINOV. "APPROXIMATION METHODS FOR THE SOLUTIONS OF IMPULSE DIFFERENTIAL EQUATIONS." Tamkang Journal of Mathematics 30, no. 4 (1999): 263–69. http://dx.doi.org/10.5556/j.tkjm.30.1999.4232.

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   In the present paper approximations for the solution of impulse differential equations by solutions of an appropriately constructed ordinary differential equations are found.   

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39

Luo, Dongsheng, Lianying Zhang, Peiyong Zhang, Hongyong Deng, and Xianghu Liu. "Several kinds of integral factors for first order nonlinear ODEs and some parts of their solutions." Journal of Physics: Conference Series 2905, no. 1 (2024): 012015. https://doi.org/10.1088/1742-6596/2905/1/012015.

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Abstract For the first-ordered nonlinear ordinary differential equations, we explore several kinds of integral factors in this paper, where the concrete formulas of the integral factors are acquired, and some of them are applied to solve corresponding nonlinear ordinary differential equations. At first, the sufficient and necessary condition of linearity for ordinary differential equations is proved that the ordinary differential equations are linear if and only if their integral factors only have a unique independent variable. Secondly, for several types of nonlinear ordinary differential equ

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40

Schlacher, Kurt, and Andreas Kugi. "SYMBOLIC METHODS FOR SYSTEMS OF IMPLICIT ORDINARY DIFFERENTIAL EQUATIONS*." Mechanics of Structures and Machines 30, no. 1 (2002): 103–21. http://dx.doi.org/10.1081/sme-120001479.

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41

Schlacher, Kurt, and Andreas Kugi. "SYMBOLIC METHODS FOR SYSTEMS OF IMPLICIT ORDINARY DIFFERENTIAL EQUATIONS*,†." Mechanics of Structures and Machines 30, no. 3 (2002): 411–29. http://dx.doi.org/10.1081/sme-120004424.

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42

Beccar-Varela, Maria P., Md Al Masum Bhuiyan, Maria C. Mariani, and Osei K. Tweneboah. "Analytic Methods for Solving Higher Order Ordinary Differential Equations." Mathematics 7, no. 9 (2019): 826. http://dx.doi.org/10.3390/math7090826.

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In this work, an analytic approach for solving higher order ordinary differential equations (ODEs) is developed. The techniques offer analytic flexibility in many research areas such as physics, engineering, and applied sciences and are effective for solving complex ODEs.

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43

Butcher, J. C., and Z. Jackiewicz. "Unconditionally Stable General Linear Methods for Ordinary Differential Equations." BIT Numerical Mathematics 44, no. 3 (2004): 557–70. http://dx.doi.org/10.1023/b:bitn.0000046804.67936.06.

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44

GOVORUKHIN, V. N., V. G. TSYBULIN, and B. KARASÖZEN. "DYNAMICS OF NUMERICAL METHODS FOR COSYMMETRIC ORDINARY DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 11, no. 09 (2001): 2339–57. http://dx.doi.org/10.1142/s0218127401003504.

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The dynamics of numerical approximation of cosymmetric ordinary differential equations with a continuous family of equilibria is investigated. Nonconservative and Hamiltonian model systems in two dimensions are considered and these systems are integrated with several first-order Runge–Kutta methods. The preservation of symmetry and cosymmetry, the stability of equilibrium points, spurious solutions and transition to chaos are investigated by presenting analytical and numerical results. The overall performance of the methods for different parameters is discussed.

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45

Garey, L. E., and C. J. Gladwin. "Unconditional stable methods for second order ordinary differential equations." International Journal of Computer Mathematics 41, no. 3-4 (1992): 181–88. http://dx.doi.org/10.1080/00207169208804038.

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46

Li, Changpin, and Fanhai Zeng. "The Finite Difference Methods for Fractional Ordinary Differential Equations." Numerical Functional Analysis and Optimization 34, no. 2 (2013): 149–79. http://dx.doi.org/10.1080/01630563.2012.706673.

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47

Rota, Gian-Carlo. "Geometrical methods in the theory of ordinary differential equations." Advances in Mathematics 80, no. 2 (1990): 269. http://dx.doi.org/10.1016/0001-8708(90)90030-q.

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48

D’Ambrosio, R., M. Ferro, Z. Jackiewicz, and B. Paternoster. "Two-step almost collocation methods for ordinary differential equations." Numerical Algorithms 53, no. 2-3 (2009): 195–217. http://dx.doi.org/10.1007/s11075-009-9280-5.

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49

Lopez, L. "Numerical methods for ordinary differential equations on matrix manifolds." Journal of Computational and Applied Mathematics 210, no. 1-2 (2007): 232–43. http://dx.doi.org/10.1016/j.cam.2006.10.066.

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50

Deng, Weihua, and Jan S. Hesthaven. "Local discontinuous Galerkin methods for fractional ordinary differential equations." BIT Numerical Mathematics 55, no. 4 (2014): 967–85. http://dx.doi.org/10.1007/s10543-014-0531-z.

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