Published June 30, 2021 | Version v1
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Integrating linear ordinary fourth-order differential equations in the MAPLE programming environment

Creators

  • 1. Belgorod State University
  • 2. Kharkiv National Automobile and Highway University

Description

This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.

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References

  • Trikomi, F. (1962). Differentsial'nye uravneniya. Moscow: Izdatel'stvo inostrannoy literatury, 352.
  • Berezin, I. S., Zhidkov, N. P. (1962). Metody vychisleniy differentsial'nyh uravneniy. Moscow: Gos. izdatel'stvo fiz.-mat. literatury, 620.
  • Bahvalov, N. S. (1973). Chislennye metody (analiz, algebra, obyknovennye differentsial'nye uravneniya). Moscow: Nauka. Glavnaya redaktsiya fiz.-mat. literatury, 632.
  • Kollatts, L. (1968). Zadachi na sobstvennye znacheniya. Moscow: Nauka, 504.
  • Demihovskiy, V. Ya. (2000). Fizika kvantovyh nizkorazmernyh struktur. Moscow: Logos, 248.
  • Dong, L., Alotaibi, A., Mohiuddine, S. A., Atluri, S. N. (2014). Computational methods in engineering: A variety of primal & mixed methods, with global & local interpolations, for well-posed or ill-posed BCs. CMES - Computer Modeling in Engineering and Sciences, 99 (1), 1–85. Available at: https://www.scopus.com/record/display.uri?eid=2-s2.0-84904022089&origin=inward&txGid=31ec3491db056863e37edd98aa82519c
  • Polyanin, A., Zaitsev, V. (2018). Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, 1496. doi: https://doi.org/10.1201/9781315117638
  • Dzhakal'ya, G. E. O. (1979). Metody teorii vozmuscheniy dlya nelineynyh sistem. Moscow: Nauka. Glavnaya redaktsiya fiz.-mat. literatury, 320.
  • Nayfe, A. (1976). Metody vozmuscheniy. Moscow: Izd-vo "Mir", 456.
  • Grebenikov, E. A. (1986). Metod usredneniya v prikladnyh zadachah. Moscow: Nauka. Glavnaya redaktsiya fiz.-mat. literatury, 256.
  • Marchuk, G. I. (1977). Metody vychislitel'noy matematiki. Moscow: Nauka. Glavnaya redaktsiya fiz.-mat. literatury, 456.
  • Abramov, A., Berkovich, L. M., Hantzschmann, K. (1990). Extended possibilities of some computer algebra algorithms for solving linear differential and difference equations. IV International Conference on Computer Algebra in Physical Research. Dubna.
  • Jator, S. N. (2008). Numerical integrators for fourth order initial and boundary value problems. International Journal of Pure and Applied Mathematics, 47 (4), 563–576.
  • Alomari, A. K., Anakira, N. R., Bataineh, A. S., Hashim, I. (2013). Approximate Solution of Nonlinear System of BVP Arising in Fluid Flow Problem. Mathematical Problems in Engineering, 2013, 1–7. doi: https://doi.org/10.1155/2013/136043
  • Poslavsky, S. (2019). Rings: An efficient Java/Scala library for polynomial rings. Computer Physics Communications, 235, 400–413. doi: https://doi.org/10.1016/j.cpc.2018.09.005
  • Kayal, N., Nair, V., Saha, C. (2019). Average-case linear matrix factorization and reconstruction of low width algebraic branching programs. Computational Complexity, 28 (4), 749–828. doi: https://doi.org/10.1007/s00037-019-00189-0
  • England, M., Florescu, D. (2019). Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition. Intelligent Computer Mathematics, 93–108. doi: https://doi.org/10.1007/978-3-030-23250-4_7
  • Grudo, Y. O., Kalinin, A. I. (2006). Asymptotic optimization method for a quasilinear system with multidimensional controls. Differential Equations, 42 (12), 1674–1681. doi: https://doi.org/10.1134/s0012266106120020
  • Galanin, M. P., Sorokin, D. L. (2020). Solving Exterior Boundary Value Problems for the Laplace Equation. Differential Equations, 56 (7), 890–899. doi: https://doi.org/10.1134/s0012266120070083
  • Mozzhorina, T. Yu. (2017). Numerical solution to problems of optimal control with switching by means of the shooting method. Matematicheskoe Modelirovanie i Chislennye Metody, 14, 94–106. Available at: http://www.mathnet.ru/links/6d5f8ecc7e0c0b1543da6fc6984b9ec4/mmcm101.pdf
  • Hussain, K., Ismail, F., Senu, N. (2016). Solving directly special fourth-order ordinary differential equations using Runge–Kutta type method. Journal of Computational and Applied Mathematics, 306, 179–199. doi: https://doi.org/10.1016/j.cam.2016.04.002
  • You, X., Chen, Z. (2013). Direct integrators of Runge–Kutta type for special third-order ordinary differential equations. Applied Numerical Mathematics, 74, 128–150. doi: https://doi.org/10.1016/j.apnum.2013.07.005
  • Islam, M. A. (2015). Accurate Solutions of Initial Value Problems for Ordinary Differential Equations with the Fourth Order Runge Kutta Method. Journal of Mathematics Research, 7 (3). doi: https://doi.org/10.5539/jmr.v7n3p41
  • Waeleh, N., Majid, Z. A., Ismail, F., Suleiman, M. (2012). Numerical Solution of Higher Order Ordinary Differential Equations by Direct Block Code. Journal of Mathematics and Statistics, 8 (1), 77–81. doi: https://doi.org/10.3844/jmssp.2012.77.81
  • Waeleh, N., Majid, Z. A., Ismail, F. (2011). A new algorithm for solving higher order IVPs of ODEs. Applied Mathematical Sciences, 5, 2795–2805. Available at: http://www.m-hikari.com/ams/ams-2011/ams-53-56-2011/majidAMS53-56-2011.pdf
  • Bulavina, I. V., Kirichenko, I. K., Chekanov, N. N., Chekanova, N. A. (2011). Calculations the eigenvalues and functions for Mathieu equation by means of the maple mathematical package. Vestnik Hersonskogo natsional'nogo tekhnicheskogo universiteta, 3 (42), 115–118.
  • Chekanova, N. N., Chekanov, N. A. (2013). Invarianty odnomernogo garmonicheskogo ostsillyatora s zavisyaschey ot vremeni chastotoy. Vestnik Hersonskogo natsional'nogo tekhnicheskogo universiteta, 2 (47), 372–374.
  • Bogachev, V. E., Kirichenko, I. K., Chekanova, N. N., Chekanov, N. A. (2015). Issledovanie nelineynoy gamil'tonovoy sistemy metodom normal'noy formy Birkgofa-Gustavsona. Visnyk Kharkivskoho natsionalnoho universytetu imeni V. N. Karazina. Seriya: Matematychne modeliuvannia. Informatsiyni tekhnolohiyi. Avtomatyzovani systemy upravlinnia, 1156, 17–28.