Thus, only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution.
📊
Key Findings & Data:Consequently, the singularity structure leads to the transformations that yield a solution in terms of a special function, if the equation is suitable.
💡
Conclusion & Impact:Thus, only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Ordinary differential equations · symbolic analysis · special functions
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.
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., A