Symbolic Analysis of Second-order Ordinary Differential Equations with Polynomial Coefficients

Year 2022, , 281 - 291, 30.12.2022

Tolga Birkandan

https://doi.org/10.47000/tjmcs.1025121

Cited By: 2

Abstract

The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. It is shown that the $\theta$-operator method can be used as a symbolic computational approach to obtain the indicial equation and the recurrence relation. Consequently, the singularity structure leads to the transformations that yield a solution in terms of a special function, if the equation is suitable. Hypergeometric and Heun-type equations are mostly employed in physical applications. Thus, only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.

Keywords

Ordinary differential equations, symbolic analysis, special functions

References

• Allen, G., Some efficient methods for obtaining infinite series solutions of nth-order linear ordinary differential equations, NASA Technical Report (NASA TR-R-390), 1972.
• Birkandan, T., Cvetiˇc, M., Conformal invariance and near-extreme rotating AdS black holes, Phys. Rev. D, 84(2011), 044018. [arXiv:1106.4329 [hep-th]].
• Birkandan, T., Cvetiˇc, M., Addentum to: Conformal invariance and near-extreme rotating AdS black holes, Phys. Rev. D, 90(6)(2014), 067504. [arXiv:1406.5208 [hep-th]].
• Birkandan, T.,Cvetiˇc, M., An analysis of the wave equation for the U(1)2 gauged supergravity black hole, Class. Quant. Grav., 32(8)(2015), 085007. [arXiv:1501.03144 [hep-th]].
• Birkandan, T., Hortacsu, M., Dirac equation in the background of the Nutku helicoid metric, J. Math. Phys., 48(2007), 092301. [arXiv:0706.2543 [gr-qc]].
• Birkandan, T., Hortacsu, M., Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces, J. Phys. A, 40(2007), 1105–1116. [arXiv:gr-qc/0607108 [gr-qc]].
• Birkandan, T., G¨uzelg¨un, C., S¸ irin, E., Uslu, M.C., Symbolic and numerical analysis in general relativity with open source computer algebrasystems, Gen. Rel. Grav., 51(1)(2019), 4. [arXiv:1703.09738 [gr-qc]].
• Birkandan, T., Giscard, P.L., Tamar, A., Computations of general Heun functions from their integral series representations, IEEE XPlore, Proceedings of the International Conference Days on Diffraction 2021(2021), 12—18. [arXiv: 2106.13729 [math.NA]].
• Birkandan, T., The symODE2 package, Accessed 12 June 2022, https://github.com/tbirkandan/symODE2
• Bronstein, M., Lafaille, S., Solutions of linear ordinary differential equations in terms of special functions, Proceedings of the 2002 InternationalSymposium on Symbolic and algebraic computation(2002), 23.
• Chan, L., Cheb-Terrab, E.S., Non-liouvillian solutions for second order Linear ODEs, Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation(2004), 80. [arXiv:math-ph/0402063].
• Cunha, M. S., Christiansen, H.R., Confluent Heun functions in gauge theories on thick braneworlds, Phys. Rev. D 84(2011), 085002. [arXiv:1109.3486 [hep-th]].
• Debeerst, R., van Hoeij, M., Koepf,W., Solving differential equations in terms of Bessel functions, Proceedings of the Twenty-first International Symposium on Symbolic and Algebraic Computation, (2008), 39.
• Derezinski, J., Hypergeometric type functions and their symmetries, Ann. Henri Poincare, 15(2014), 1569. [arXiv:1305.3113 [math.CA]].
• Dong, Q., Sun, G.H., Aoki, M.A., Chen, C.Y., Dong, S.H., Exact solutions of a quartic potential, Mod. Phys. Lett. A 34(26)(2019), 1950208.
• Duval, A., Loday-Richaud, M., Kovacic’s algorithm and its application to some families of special functions, AAECC, 3(1992), 211.
• Fiziev, P.P., Exact solutions of Regge-Wheeler equation and quasi-normal modes of compact objects, Class. Quant. Grav., 23(2006), 2447-2468. [arXiv:gr-qc/0509123 [gr-qc]].
• Giscard, P.L.,Tamar, A., Elementary integral series for Heun functions: Application to black-hole perturbation theory, J. Math. Phys., 63(2022), 063501. [arXiv:2010.03919 [math-ph]].
• GNU Octave Version 5.2.0, Scientific Programming Language, https://www.gnu.org/software/octave/, 2020.
• Grabmeier, J., Kaltofen, E., Weispfennig, U. (eds.), Computer Algebra Handbook, Springer, Berlin, 2003.
• Gourgoulhon, E., Bejger, M., Mancini, M., Tensor calculus with open-source software: the SageManifolds project, J. Phys. Conf. Ser., 600(2015), 012002. [arXiv:1412.4765 [gr-qc]].
• Gourgoulhon, E., Mancini, M., Symbolic tensor calculus on manifolds: a SageMath implementation, Les cours du CIRM, 6(2018), I. [arXiv:1804.07346 [gr-qc]].
• “HeunG”, Accessed 12 June 2022, https://reference.wolfram.com/language/ref/HeunG.html.
• “HeunB”, Accessed 12 June 2022, https://reference.wolfram.com/language/ref/HeunB.html.
• “HeunT”, Accessed 12 June 2022, https://reference.wolfram.com/language/ref/HeunT.html.
• “Heun and Related Functions”, Accessed 12 June 2022, https://reference.wolfram.com/language/guide/HeunAndRelatedFunctions.html
• Hortacsu, M., Explicit examples on conformal invariance, Int. J. Theor. Phys. 42(2003), 49. [arXiv:hep-th/0106080 [hep-th]].
• Hortacsu, M., Heun functions and some of their applications in physics, Advances in High Energy Physics, 2018(2018), 8621573. [arXiv:1101.0471 [math-ph]].
• “Hypergeometric2F1”, Accessed 12 June 2022, https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/13/01/01/01/.
• “HypergeometricU”, Accessed 12 June 2022, https://reference.wolfram.com/language/ref/HypergeometricU.html.
• Imamoglu, E.,van Hoeij, M., Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases, J. Symbolic Computation, 83(2017), 254. [arXiv:1606.01576 [cs.SC]].
• Ishkhanyan, T., From Sine to Heun: 5 New Functions for Mathematics and Physics in the Wolfram Language, Accessed 12 June 2022, https://blog.wolfram.com/2020/05/06/from-sine-to-heun-5-new-functions-for-mathematics-and-physics-in-the-wolfram-language/
• Karayer, H., Demirhan, D., Büyükkılıç, F., Extension of Nikiforov-Uvarov method for the solution of Heun equation, J. Math. Phys., 56(2015), 063504. [arXiv:1504.03518 [math-ph]].
• Karayer, H., Demirhan, D., Exact analytical solution of Schrodinger equation for a generalized noncentral potential, Eur. Phys. J. Plus 137(2022), 527.
• Karayer, H., Demirhan, D., Atman K.G., Analytical exact solutions for the Razavy type potential, Mathematical Methods in the Applied Sciences, 43(15)(2020), 9185–9194.
• Kovacic, J.J., An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Computation, 2(1986), 3.
• Kristensson, G., Second Order Differential Equations: Special Functions and Their Classification, Springer, New York, 2010.
• Kunwar, V., van Hoeij, M., Second order differential equations with hypergeometric solutions of degree three, Proceedings of the Twenty-first International Symposium on Symbolic and Algebraic Computation, (2008), 235.
• MacCallum, M.A.H., Computer algebra in gravity research, Living Rev. Rel., 21(1)(2018), 6.
• Maier, R.S., On reducing the Heun equation to the hypergeometric equation, J. Differential Equations, 213(2005), 171. [arXiv:math/0203264 [math.CA]].
• Maier, R.S., The 192 solutions of the Heun equation, Math. Comp., 76(2007), 811. [arXiv:math/0408317 [math.CA]].
• Maple 2020, Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario, 2020.
• Mathematica Version 12.1, Wolfram Research, Inc., Champaign, IL, 2020.
• Maxima, a Computer Algebra System, Version 5.44.0, http://maxima.sourceforge.net/, 2020.
• Motygin, O.V., On numerical evaluation of the Heun functions, IEEE XPlore, Proceedings of the International Conference Days on Diffraction 2015, (2015), 222. [arXiv:1506.03848 [math.NA]].
• Motygin, O.V., On evaluation of the confluent Heun functions, IEEE XPlore, Proceedings of the International Conference Days on Diffraction 2018, (2018), 223. [arXiv:1804.01007 [math.NA]].
• Nasheeha, R.N., Thirukkanesh, S., Ragel, F.C., Anisotropic generalization of isotropic models via hypergeometric equation, Eur. Phys. J. C, 80(1)(2020), 6.
• Olver, F.W.J., Asymptotics and Special Functions, Academic Press, New York, 1974.
• Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F. et all. (eds.), NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.27, 2020.
• Petroff, D., Slowly rotating homogeneous stars and the Heun equation, Class. Quant. Grav., 24(2007), 1055-1068. [arXiv:gr-qc/0701081 [grqc]].
• REDUCE, a portable general-purpose computer algebra system, https://reduce-algebra.sourceforge.io/, 2020.
• Ronveaux, A. (ed.), Heun’s Differential Equations, Oxford University Press, New York, 1995.
• SageMath, The Sage Mathematics Software System (Version 9.2), The Sage Developers, https://www.sagemath.org, 2020.
• Sakallı, İ., Jusufi, K., Övgün, A., Analytical solutions in a cosmic string born-infeld-dilaton black hole geometry: Quasinormal modes and quantization, Gen. Rel. Grav., 50(10)(2018), 125. [arXiv:1803.10583 [gr-qc]].
• Slavyanov, S.Yu., Lay, W., Special Functions, A Unified Theory Based on Singularities, Oxford University Press, New York, 2000.
• “Solving Some Second Order Linear ODEs that Admit Hypergeometric 2F1, 1F1, and 0F1 Function Solutions”, Accessed 12 June 2022, https://www.maplesoft.com/support/help/Maple/view.aspx?path=dsolve/hyper3.
• “The five Second Order Linear Heun equations and the corresponding Heun function solutions”, Accessed 12 June 2022, https://www.maplesoft.com/support/help/Maple/view.aspx?path=Heun.
• Vieira, H.S., Resonant frequencies of the hydrodynamic vortex, Int. J. Mod. Phys. D 26, no.04(2016), 1750035. [arXiv:1510.08298 [gr-qc]].
• Vit´oria, R.L.L., Furtado, C. and Bakke, K., On a relativistic particle and a relativistic position-dependent mass particle subject to the Klein–Gordon oscillator and the Coulomb potential, Annals Phys. 370(2016), 128-136. [arXiv:1511.05072 [quant-ph]].