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Статті в журналах з теми "Linear ODE"

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Автор: Grafiati

Опубліковано: 1 квітня 2023

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1

Deutscher, Joachim, Nicole Gehring, and Richard Kern. "Output feedback control of general linear heterodirectional hyperbolic ODE–PDE–ODE systems." Automatica 95 (September 2018): 472–80. http://dx.doi.org/10.1016/j.automatica.2018.06.021.

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2

Radnef, Sorin. "Analytic Solution of Non-Autonomous Linear ODE." PAMM 6, no. 1 (December 2006): 651–52. http://dx.doi.org/10.1002/pamm.200610306.

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3

Hu, Jie, Huihui Qin, and Xiaodan Fan. "Can ODE gene regulatory models neglect time lag or measurement scaling?" Bioinformatics 36, no. 13 (April 23, 2020): 4058–64. http://dx.doi.org/10.1093/bioinformatics/btaa268.

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Анотація:

Abstract Motivation Many ordinary differential equation (ODE) models have been introduced to replace linear regression models for inferring gene regulatory relationships from time-course gene expression data. But, since the observed data are usually not direct measurements of the gene products or there is an unknown time lag in gene regulation, it is problematic to directly apply traditional ODE models or linear regression models. Results We introduce a lagged ODE model to infer lagged gene regulatory relationships from time-course measurements, which are modeled as linear transformation of the gene products. A time-course microarray dataset from a yeast cell-cycle study is used for simulation assessment of the methods and real data analysis. The results show that our method, by considering both time lag and measurement scaling, performs much better than other linear and ODE models. It indicates the necessity of explicitly modeling the time lag and measurement scaling in ODE gene regulatory models. Availability and implementation R code is available at https://www.sta.cuhk.edu.hk/xfan/share/lagODE.zip.

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4

Lorber, Alfred A., Graham F. Carey, and Wayne D. Joubert. "ODE Recursions and Iterative Solvers for Linear Equations." SIAM Journal on Scientific Computing 17, no. 1 (January 1996): 65–77. http://dx.doi.org/10.1137/0917006.

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5

Shi-Da, Liu, Fu Zun-Tao, Liu Shi-Kuo, Xin Guo-Jun, Liang Fu-Ming, and Feng Bei-Ye. "Solitary Wave in Linear ODE with Variable Coefficients." Communications in Theoretical Physics 39, no. 6 (June 15, 2003): 643–46. http://dx.doi.org/10.1088/0253-6102/39/6/643.

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6

Ayadi, Habib. "Exponential stabilization of an ODE–linear KdV cascaded system with boundary input delay." IMA Journal of Mathematical Control and Information 37, no. 4 (September 23, 2020): 1506–23. http://dx.doi.org/10.1093/imamci/dnaa022.

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Abstract This paper considers the well posedness and the exponential stabilization problems of a cascaded ordinary differential equation (ODE)–partial differential equation (PDE) system. The considered system is governed by a linear ODE and the one-dimensional linear Korteweg–de Vries (KdV) equation posed on a bounded interval. For the whole system, a control input delay acts on the left boundary of the KdV domain by Dirichlet condition. Whereas, the KdV acts back on the ODE by Dirichlet interconnection on the right boundary. Firstly, we reformulate the system in question as an undelayed ODE-coupled KdV-transport system. Secondly, we use the so-called infinite dimensional backstepping method to derive an explicit feedback control law that transforms system under consideration to a well-posed and exponentially stable target system. Finally, by invertibility of such design, we use semigroup theory and Lyapunov analysis to prove the well posedness and the exponential stabilization in a suitable functional space of the original plant, respectively.

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7

Imoni, Sunday Obomeviekome, D. I. Lanlege, E. M. Atteh, and J. O. Ogbondeminu. "FORMULATION OF BLOCK SCHEMES WITH LINEAR MULTISTEP METHOD FOR THE APPROXIMATION OF FIRST-ORDER IVPS." FUDMA JOURNAL OF SCIENCES 4, no. 3 (September 24, 2020): 313–22. http://dx.doi.org/10.33003/fjs-2020-0403-260.

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ABSTRACT In this paper, formulation of an efficient numerical schemes for the approximation first-order initial value problems (IVPs) of ordinary differential equations (ODE) is presented. The method is a block scheme for some k-step linear multi-step methods (and) using the Hermite Polynomials a basis function. The continuous and discrete linear multi-step methods (LMM) are formulated through the technique of collocation and interpolation. Numerical examples of ODE have been examined and results obtained show that the proposed scheme can be efficient in solving initial value problems of first order ODE.

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8

POSPÍŠIL, JIŘÍ, ZDENĚK KOLKA, JANA HORSKÁ, and JAROMÍR BRZOBOHATÝ. "SIMPLEST ODE EQUIVALENTS OF CHUA'S EQUATIONS." International Journal of Bifurcation and Chaos 10, no. 01 (January 2000): 1–23. http://dx.doi.org/10.1142/s0218127400000025.

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The so-called elementary canonical state models of the third-order piecewise-linear (PWL) dynamical systems, as the simplest ODE equivalents of Chua's equations, are presented. Their mutual relations using the linear topological conjugacy are demonstrated in order to show in detail that Chua's equations and their canonical ODE equivalents represent various forms of qualitatively equivalent models of third-order dynamical systems. New geometrical aspects of the corresponding transformations together with examples of typical chaotic attractors in the stereoscopic view, give the possibility of a deeper insight into the third-order system dynamics.

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9

Mukhopadhyay, S., R. Picard, S. Trostorff, and M. Waurick. "A note on a two-temperature model in linear thermoelasticity." Mathematics and Mechanics of Solids 22, no. 5 (December 8, 2015): 905–18. http://dx.doi.org/10.1177/1081286515611947.

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We discuss the so-called two-temperature model in linear thermoelasticity and provide a Hilbert space framework for proving well-posedness of the equations under consideration. With the abstract perspective of evolutionary equations, the two-temperature model turns out to be a coupled system of the elastic equations and an abstract ordinary differential equation (ODE). Following this line of reasoning, we propose another model which is entirely an abstract ODE. We also highlight an alternative method for a two-temperature model, which might be of independent interest.

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10

Aksan, Emine. "An application of cubic B-Spline finite element method for the Burgers` equation." Thermal Science 22, Suppl. 1 (2018): 195–202. http://dx.doi.org/10.2298/tsci170613286a.

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It is difficult to achieve exact solution of non-linear PDE, directly. Sometimes, it is possible to convert non-linear PDE into equivalent linear PDE by applying a convenient transformation. Hence, Burgers? equation replaces with heat equation by means of the Hope-Cole transformation. In this study, Burgers? equation was converted to a set of non-linear ODE by keeping non-linear structure of Burgers? equation. In this case, solutions for each of the non-linear ODE were obtained by the help of the cubic B-spline finite element method. Model problems were considered to verify the efficiency of this method. Agreement of the solutions was shown with graphics and tables.

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11

Leitner, Johannes. "A Note on Credit Insurance." ASTIN Bulletin 36, no. 02 (November 2006): 347–60. http://dx.doi.org/10.2143/ast.36.2.2017925.

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In a simple stationary setting with constant interest rate, we derive pricing formulas for defaultable bonds with stochastic recovery rate using a replication argument. Replication is done by using an insurance contract (i.e. a kind of credit default swap), the price of which is determined by a dynamic premium calculation principle. We consider two cases, a linear one, where pricing amounts to solving an inhomogeneous linear ODE, and a super-linear case where a Riccati ODE has to be solved.

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12

Leitner, Johannes. "A Note on Credit Insurance." ASTIN Bulletin 36, no. 2 (November 2006): 347–60. http://dx.doi.org/10.1017/s0515036100014549.

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Анотація:

In a simple stationary setting with constant interest rate, we derive pricing formulas for defaultable bonds with stochastic recovery rate using a replication argument. Replication is done by using an insurance contract (i.e. a kind of credit default swap), the price of which is determined by a dynamic premium calculation principle. We consider two cases, a linear one, where pricing amounts to solving an inhomogeneous linear ODE, and a super-linear case where a Riccati ODE has to be solved.

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13

Dias, Ana Paula S., and Ian Stewart. "Linear equivalence and ODE-equivalence for coupled cell networks." Nonlinearity 18, no. 3 (February 10, 2005): 1003–20. http://dx.doi.org/10.1088/0951-7715/18/3/004.

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14

Calogero, Francesco. "A linear second-order ODE with only polynomial solutions." Journal of Differential Equations 255, no. 8 (October 2013): 2130–35. http://dx.doi.org/10.1016/j.jde.2013.06.007.

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15

Di Meglio, Florent, Federico Bribiesca Argomedo, Long Hu, and Miroslav Krstic. "Stabilization of coupled linear heterodirectional hyperbolic PDE–ODE systems." Automatica 87 (January 2018): 281–89. http://dx.doi.org/10.1016/j.automatica.2017.09.027.

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16

Yumaguzhin, Valeriy A. "Classification of 3rd order linear ODE up to equivalence." Differential Geometry and its Applications 6, no. 4 (December 1996): 343–50. http://dx.doi.org/10.1016/s0926-2245(96)00030-7.

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17

Xu, Xiang, Lu Liu, Miroslav Krstic, and Gang Feng. "Stabilization of chains of linear parabolic PDE–ODE cascades." Automatica 148 (February 2023): 110763. http://dx.doi.org/10.1016/j.automatica.2022.110763.

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18

Wu, Huai-Ning, and Jun-Wei Wang. "Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems." Automatica 50, no. 11 (November 2014): 2787–98. http://dx.doi.org/10.1016/j.automatica.2014.09.006.

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19

Duff, G. F. D., R. B. Leipnik, and C. E. M. Pearce. "Guide expansions for the recursive parametric solution of polynomial dynamical systems." ANZIAM Journal 47, no. 3 (January 2006): 387–96. http://dx.doi.org/10.1017/s1446181100009901.

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AbstractRecursive parametric series solutions are developed for polynomial ODE systems, based on expanding the system components in series of a form studied by Weiss. Individual terms involve first-order driven linear ODE systems with variable coefficients. We consider Lotka-Volterra systems as an example.

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20

Butusov, Denis, Aleksandra Tutueva, Petr Fedoseev, Artem Terentev, and Artur Karimov. "Semi-Implicit Multistep Extrapolation ODE Solvers." Mathematics 8, no. 6 (June 8, 2020): 943. http://dx.doi.org/10.3390/math8060943.

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Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep extrapolation method for solving ODEs based on the semi-implicit basic method of order 2. Considering several chaotic systems and van der Pol nonlinear oscillator as examples, we implemented a performance analysis of the proposed technique in comparison with well-known multistep methods: Adams–Bashforth, Adams–Moulton and the backward differentiation formula. We explicitly show that the multistep semi-implicit methods can outperform the classical linear multistep methods, providing more precision in the solutions for nonlinear differential equations. The analysis of stability regions reveals that the proposed methods are more stable than explicit linear multistep methods. The possible applications of the developed ODE solver are the long-term simulations of chaotic systems and processes, solving moderately stiff differential equations and advanced modeling systems.

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21

Dlala, Mohsen, and Abdallah Benabdallah. "Global Stabilization of Nonlinear Finite Dimensional System with Dynamic Controller Governed by 1 − d Heat Equation with Neumann Interconnection." Mathematics 10, no. 2 (January 12, 2022): 227. http://dx.doi.org/10.3390/math10020227.

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This paper deals with the stabilization of a class of uncertain nonlinear ordinary differential equations (ODEs) with a dynamic controller governed by a linear 1−d heat partial differential equation (PDE). The control operates at one boundary of the domain of the heat controller, while at the other end of the boundary, a Neumann term is injected into the ODE plant. We achieve the desired global exponential stabilization goal by using a recent infinite-dimensional backstepping design for coupled PDE-ODE systems combined with a high-gain state feedback and domination approach. The stabilization result of the coupled system is established under two main restrictions: the first restriction concerns the particular classical form of our ODE, which contains, in addition to a controllable linear part, a second uncertain nonlinear part verifying a lower triangular linear growth condition. The second restriction concerns the length of the domain of the PDE which is restricted.

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22

Berger, Thomas, and Timo Reis. "ODE observers for DAE systems." IMA Journal of Mathematical Control and Information 36, no. 4 (August 29, 2018): 1375–93. http://dx.doi.org/10.1093/imamci/dny032.

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Abstract We consider linear time-invariant differential-algebraic systems which are not necessarily regular. The following question is addressed: when does an (asymptotic) observer which is realized by an ordinary differential equation (ODE) system exist? In our main result we characterize the existence of such observers by means of a simple criterion on the system matrices. To be specific, we show that an ODE observer exists if, and only if, the completely controllable part of the system is impulse observable. Extending the observer design from earlier works we provide a procedure for the construction of (asymptotic) ODE observers.

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23

Da Silva Pinto, Alisson, Patricia Nunes da Silva, and André Luiz Cordeiro dos Santos. "A note about a new method for solving Riccati differential equations." International Journal for Innovation Education and Research 10, no. 4 (April 1, 2022): 123–29. http://dx.doi.org/10.31686/ijier.vol10.iss4.3715.

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Al Bastami, Belić, and Petrović (2010) proposed a new method to find solutions to some Riccati differential equations. Initially, they obtain a second-order linear ordinary differential equation (ODE) through a standard variable change in the Riccati equation. They then propose a new variable change and discuss the resolution of the resulting ODE in two cases. In the first one, the resulting ODE has constant coefficients. In the second case, they claim that it is possible to arbitrarily choose one of the resulting ODE coefficients and solve particular Riccati ODEs. We show in this work that all Riccati equations that belong to the first case can also be solved by Chini’s method. Furthermore, we show that any Riccati equation fits the second case and that the choice of the resulting ODE coefficients is not free.

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24

Darling, R. W. R. "Classroom Note:Converting Matrix Riccati Equations to Second-Order Linear ODE." SIAM Review 39, no. 3 (January 1997): 508–10. http://dx.doi.org/10.1137/s0036144595298574.

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25

Rychlik, Marek. "Renormalization of cocycles and linear ODE with almost-periodic coefficients." Inventiones Mathematicae 110, no. 1 (December 1992): 173–206. http://dx.doi.org/10.1007/bf01231330.

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26

Zhang, MeiRong. "Continuity in weak topology: higher order linear systems of ODE." Science in China Series A: Mathematics 51, no. 6 (June 2008): 1036–58. http://dx.doi.org/10.1007/s11425-008-0011-5.

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27

Lara, L. P., and M. Gadella. "An approximation to solutions of linear ODE by cubic interpolation." Computers & Mathematics with Applications 56, no. 6 (September 2008): 1488–95. http://dx.doi.org/10.1016/j.camwa.2008.03.024.

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28

Meng, Gang, and Mei Rong Zhang. "Continuity in weak topology: First order linear systems of ODE." Acta Mathematica Sinica, English Series 26, no. 7 (June 15, 2010): 1287–98. http://dx.doi.org/10.1007/s10114-010-8103-x.

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29

Cengizci, Süleyman. "An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations." International Journal of Differential Equations 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/7269450.

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In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

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30

Yang, Bin, and Yuehui Chen. "Overview of Gene Regulatory Network Inference Based on Differential Equation Models." Current Protein & Peptide Science 21, no. 11 (December 31, 2020): 1054–59. http://dx.doi.org/10.2174/1389203721666200213103350.

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: Reconstruction of gene regulatory networks (GRN) plays an important role in understanding the complexity, functionality and pathways of biological systems, which could support the design of new drugs for diseases. Because differential equation models are flexible androbust, these models have been utilized to identify biochemical reactions and gene regulatory networks. This paper investigates the differential equation models for reverse engineering gene regulatory networks. We introduce three kinds of differential equation models, including ordinary differential equation (ODE), time-delayed differential equation (TDDE) and stochastic differential equation (SDE). ODE models include linear ODE, nonlinear ODE and S-system model. We also discuss the evolutionary algorithms, which are utilized to search the optimal structures and parameters of differential equation models. This investigation could provide a comprehensive understanding of differential equation models, and lead to the discovery of novel differential equation models.

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31

Al-Hasan, Yasin. "Evaluation of MATLAB Methods used to Solve Second Order Linear ODE." Research Journal of Applied Sciences, Engineering and Technology 7, no. 13 (April 5, 2014): 2634–38. http://dx.doi.org/10.19026/rjaset.7.579.

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32

Cecchi, Mariella, Zuzana Dosla, Ondrej Dosly, and Mauro Marini. "On the integral characterization of principal solutions for half-linear ODE." Electronic Journal of Qualitative Theory of Differential Equations, no. 12 (2013): 1–14. http://dx.doi.org/10.14232/ejqtde.2013.1.12.

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33

Goltser, Ya, and E. Litsyn. "Non-linear Volterra IDE, infinite systems and normal forms of ODE." Nonlinear Analysis: Theory, Methods & Applications 68, no. 6 (March 2008): 1553–69. http://dx.doi.org/10.1016/j.na.2006.12.036.

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34

Shen, Jianhua. "Qualitative properties of solutions of second-order linear ODE with impulses." Mathematical and Computer Modelling 40, no. 3-4 (August 2004): 337–44. http://dx.doi.org/10.1016/j.mcm.2003.12.009.

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35

ANCO, S. C., and G. BLUMAN. "Erratum: Integrating factors and first integrals for ordinary differential equations." European Journal of Applied Mathematics 10, no. 2 (April 1999): 223. http://dx.doi.org/10.1017/s095679259900371x.

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Volume 9 (1998), pp. 245–259The last sentence of §3.2 should read as follows:For n=4, the splitting yields six such linear PDEs from the coefficients of the terms involving Y(6), Y(4), Y(5), Y(5), Y(4)3, Y(4)2 and Y(4).In the first paragraph of §6, the penultimate sentence should read as follows:For an nth-order scalar ODE the determining equations are a linear system of PDEs consisting of the adjoint of the determining equation for symmetries of the nth-order ODE and additional equations when n[ges ]2.We apologise for the errors in the above paper and hope that no inconvenience has been caused to readers.

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36

Hoai, Nguyen Thi. "Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable." Numerical Algebra, Control & Optimization 11, no. 4 (2021): 495. http://dx.doi.org/10.3934/naco.2020040.

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<p style='text-indent:20px;'>The direct scheme method is applied to construct an asymptotic approximation of any order to a solution of a singularly perturbed optimal problem with scalar state, controlled via a second-order linear ODE and two fixed end points. The error estimates for state and control variables and for the functional are obtained. An illustrative example is given.</p>

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37

Rehman, Mutti-Ur, and Jehad Alzabut. "On Instability Analysis of Linear Feedback Systems." Computation 8, no. 1 (March 6, 2020): 16. http://dx.doi.org/10.3390/computation8010016.

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The numerical approximation of the μ -value is key towards the measurement of instability, stability analysis, robustness, and the performance of linear feedback systems in system theory. The MATLAB function mussv available in MATLAB Control Toolbox efficiently computes both lower and upper bounds of the μ -value. This article deals with the numerical approximations of the lower bounds of μ -values by means of low-rank ordinary differential equation (ODE)-based techniques. The numerical simulation shows that approximated lower bounds of μ -values are much tighter when compared to those obtained by the MATLAB function mussv.

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38

Krovi, Hari. "Improved quantum algorithms for linear and nonlinear differential equations." Quantum 7 (February 2, 2023): 913. http://dx.doi.org/10.22331/q-2023-02-02-913.

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We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes the run time of quantum algorithms for linear ODEs opening the door to an application to a wider class of linear and nonlinear ODEs. In Berry et al., (2017), a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices. The algorithm here is also exponentially faster than the bounds derived in Berry et al., (2017) for certain classes of diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in Liu et al., (2021)). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by Xue et al., (2021), but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse, invertible matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas Liu et al., (2021) and Xue et al., (2021) additionally require normality.

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39

Diekmann, Odo, Mats Gyllenberg, and Johan A. J. Metz. "Finite dimensional state representation of physiologically structured populations." Journal of Mathematical Biology 80, no. 1-2 (December 21, 2019): 205–73. http://dx.doi.org/10.1007/s00285-019-01454-0.

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AbstractIn a physiologically structured population model (PSPM) individuals are characterised by continuous variables, like age and size, collectively called their i-state. The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition. The model consists of submodels for (i) the dynamics of the i-state, e.g. growth and maturation, (ii) survival, (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) functions of (i-state, environmental condition), like biomass or feeding rate, that integrated over the i-state distribution together produce the output of the population model. When the environmental condition is treated as a given function of time (input), the population model becomes linear in the state. Density dependence and interaction with other populations is captured by feedback via a shared environment, i.e., by letting the environmental condition be influenced by the populations’ outputs. This yields a systematic methodology for formulating community models by coupling nonlinear input–output relations defined by state-linear population models. For some combinations of submodels an (infinite dimensional) PSPM can without loss of relevant information be replaced by a finite dimensional ODE. We then call the model ODE-reducible. The present paper provides (a) a test for checking whether a PSPM is ODE reducible, and (b) a catalogue of all possible ODE-reducible models given certain restrictions, to wit: (i) the i-state dynamics is deterministic, (ii) the i-state space is one-dimensional, (iii) the birth rate can be written as a finite sum of environment-dependent distributions over the birth states weighted by environment independent ‘population outputs’. So under these restrictions our conditions for ODE-reducibility are not only sufficient but in fact necessary. Restriction (iii) has the desirable effect that it guarantees that the population trajectories are after a while fully determined by the solution of the ODE so that the latter gives a complete picture of the dynamics of the population and not just of its outputs.

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40

Li, Manna, and Weijie Mao. "Finite-Time Bounded Control for Coupled Parabolic PDE-ODE Systems Subject to Boundary Disturbances." Mathematical Problems in Engineering 2020 (December 7, 2020): 1–13. http://dx.doi.org/10.1155/2020/8882382.

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In this paper, the finite-time bounded control problem for coupled parabolic PDE-ODE systems subject to time-varying boundary disturbances and to time-invariant boundary disturbances is considered. First, the concept of finite-time boundedness is extended to coupled parabolic PDE-ODE systems. A Neumann boundary feedback controller is then designed in terms of the state variables. By applying the Lyapunov-like functional method, sufficient conditions which ensure the finite-time boundedness of closed-loop systems in the presence of time-varying boundary disturbances and time-invariant boundary disturbances are provided, respectively. Finally, the issues regarding the finite-time boundedness of coupled parabolic PDE-ODE systems are converted into the feasibility of linear matrix inequalities (LMIs), and the effectiveness of the proposed results is validated with two numerical simulations.

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41

Shahzad, Azeem, Bushra Habib, Muhammad Nadeem, Muhammad Kamran, Hijaz Ahma, Muhammad Atif, and Shafiq Ahmad. "Numerical analysis of flow and heat transfer in a thin film along an unsteady stretching cylinder." Thermal Science 25, Spec. issue 2 (2021): 441–48. http://dx.doi.org/10.2298/tsci21s2441s.

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In this framework, the boundary-layer mass and heat flow in a liquid film over an unsteady stretching cylinder are discussed under the influence of a magnetic field. By means of the similarity transformations the highly non-linear governing system of PDE is converted to ODE. We use the built-in function bvp4c in MATLAB to solve this system of ODE. The impact of distinctive parameters on velocity and temperature profile in the existence of an external magnetic field is depicted via graphs and deep analysis is also presented.

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42

Miletics, Edit. "Energy-Conservative Algorithm for the Numerical Solution of Initial-Value Hamiltonian System Problems." Journal of Advanced Computational Intelligence and Intelligent Informatics 8, no. 5 (September 20, 2004): 495–98. http://dx.doi.org/10.20965/jaciii.2004.p0495.

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The numerical treatment of ODE initial-value problems has been intensively researched. Energy-conservative algorithms are very important to dynamic systems. For the Hamiltonian system the symplectic algorithms are very effective. Powerful computers and algebraic software enable the creation of efficient numerical algorithms for solving ODE initial-value problems. In this paper, we propose an adaptive energy-conservative numerical-analytical algorithm for Hamiltonian systems. This algorithm is adaptable to initial-value problems where some quantities are preserved. The algorithm and its efficiency are presented for solving two-body and linear oscillator problems.

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43

Roman, Svetlana, and Artūras Štikonas. "Linear ODE with nonlocal boundary conditions and Green’s functions for such problems." Lietuvos matematikos rinkinys 51 (October 22, 2019): 379–84. http://dx.doi.org/10.15388/lmr.2010.14664.

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In this article we investigate a formula for the Green’s function for the n-orderlinear differential equation with n additional conditions. We use this formula for calculatingthe Green’s function for problems with nonlocal boundary conditions.

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44

Lie, Ivar. "Using Implicit ODE Methods with Iterative Linear Equation Solvers in Spectral Methods." SIAM Journal on Scientific Computing 14, no. 5 (September 1993): 1194–213. http://dx.doi.org/10.1137/0914071.

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45

AUZINGER, W., and A. EDER. "A NOTE ON LYAPUNOV TRANSFORMATION AND EXPONENTIAL DECAY IN LINEAR ODE SYSTEMS." Mathematical Models and Methods in Applied Sciences 11, no. 01 (February 2001): 23–31. http://dx.doi.org/10.1142/s0218202501000714.

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In this paper we consider a class of matrices A where all eigenvalues have negative real parts and are of a common magnitude O(1). Concerning the behavior of etA we provide a necessary and sufficient condition, via Lyapunov transformation, for an estimate of the form [Formula: see text] to be valid uniformly for t>0 with moderate-sized constants [Formula: see text] and [Formula: see text]. All relevant relations are quantitatively specified.

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46

Liu, Zhijie, Jinkun Liu, and Wei He. "Robust adaptive fault tolerant control for a linear cascaded ODE-beam system." Automatica 98 (December 2018): 42–50. http://dx.doi.org/10.1016/j.automatica.2018.09.021.

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47

Deutscher, Joachim, and Jakob Gabriel. "A backstepping approach to output regulation for coupled linear wave–ODE systems." Automatica 123 (January 2021): 109338. http://dx.doi.org/10.1016/j.automatica.2020.109338.

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48

Šátek, V., F. Kocina, J. Kunovský, and A. Schirrer. "Taylor Series Based Solution of Linear ODE Systems and MATLAB Solvers Comparison." IFAC-PapersOnLine 48, no. 1 (2015): 693–94. http://dx.doi.org/10.1016/j.ifacol.2015.05.210.

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49

Gadella, M., and L. P. Lara. "On the determination of approximate periodic solutions of some non-linear ODE." Applied Mathematics and Computation 218, no. 10 (January 2012): 6038–44. http://dx.doi.org/10.1016/j.amc.2011.11.085.

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50

Leake, Carl, Hunter Johnston, Lidia Smith, and Daniele Mortari. "Analytically Embedding Differential Equation Constraints into Least Squares Support Vector Machines Using the Theory of Functional Connections." Machine Learning and Knowledge Extraction 1, no. 4 (October 9, 2019): 1058–83. http://dx.doi.org/10.3390/make1040060.

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Differential equations (DEs) are used as numerical models to describe physical phenomena throughout the field of engineering and science, including heat and fluid flow, structural bending, and systems dynamics. While there are many other techniques for finding approximate solutions to these equations, this paper looks to compare the application of the Theory of Functional Connections (TFC) with one based on least-squares support vector machines (LS-SVM). The TFC method uses a constrained expression, an expression that always satisfies the DE constraints, which transforms the process of solving a DE into solving an unconstrained optimization problem that is ultimately solved via least-squares (LS). In addition to individual analysis, the two methods are merged into a new methodology, called constrained SVMs (CSVM), by incorporating the LS-SVM method into the TFC framework to solve unconstrained problems. Numerical tests are conducted on four sample problems: One first order linear ordinary differential equation (ODE), one first order nonlinear ODE, one second order linear ODE, and one two-dimensional linear partial differential equation (PDE). Using the LS-SVM method as a benchmark, a speed comparison is made for all the problems by timing the training period, and an accuracy comparison is made using the maximum error and mean squared error on the training and test sets. In general, TFC is shown to be slightly faster (by an order of magnitude or less) and more accurate (by multiple orders of magnitude) than the LS-SVM and CSVM approaches.

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