Removing false singular points as a method of solving ordinary differential equations

A. V. SHANIN; R. V. CRASTER

doi:10.1017/S0956792502004916

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

A general formalism is described whereby some regular singular points are effectively removed and substantial simplifications ensue for a class of Fuchsian ordinary differential equations, and related confluent equations. These simplifications follow provided the exponents at the singular points satisfy certain relations; explicit, illustrative examples are constructed to demonstrate the ideas.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Cheb-Terrab, E S 2004. Solutions for the general, confluent and biconfluent Heun equations and their connection with Abel equations. Journal of Physics A: Mathematical and General, Vol. 37, Issue. 42, p. 9923.

Kazakov, A. Ya. and Sirota, J. N. 2006. Mobius Transforms for Linear Ordinary Differential Equations. Journal of Mathematical Sciences, Vol. 132, Issue. 1, p. 37.

Takemura, Kouichi 2012. Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial. Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue. 8, p. 085211.

Novaes, Fábio and da Cunha, Bruno Carneiro 2014. Isomonodromy, Painlevé transcendents and scattering off of black holes. Journal of High Energy Physics, Vol. 2014, Issue. 7,

Kazakov, A. Ya. and Slavyanov, S. Yu. 2015. Representations and Use of Symbolic Computations in the Theory of Heun Equations. Journal of Mathematical Sciences, Vol. 209, Issue. 6, p. 910.

Slavyanov, S. Yu. Satco, D. A. Ishkhanyan, A. M. and Rotinyan, T. A. 2016. Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients. Theoretical and Mathematical Physics, Vol. 189, Issue. 3, p. 1726.

Славянов, Сергей Юрьевич Slavyanov, Sergei Yur'evich Шатько, Дарья Алексеевна Shat'ko, Dar'ya Alekseevna Ишханян, Артур Михайлович Ishkhanyan, Artur Mikhailovich Ротинян, Татьяна Александровна and Rotinyan, Tat'yana Aleksandrovna 2016. Генерация и удаление ложных особенностей в линейных обыкновенных дифференциальных уравнениях с полиномиальными коэффициентами. Теоретическая и математическая физика, Vol. 189, Issue. 3, p. 371.

Kraniotis, G V 2016. The Klein–Gordon–Fock equation in the curved spacetime of the Kerr–Newman (anti) de Sitter black hole. Classical and Quantum Gravity, Vol. 33, Issue. 22, p. 225011.

Kazakov, A. Ya. 2016. Integral Symmetry for the Confluent Heun Equation with an Added Apparent Singularity. Journal of Mathematical Sciences, Vol. 214, Issue. 3, p. 268.

Slavyanov, S. Yu. 2017. Symmetries and apparent singularities for the simplest Fuchsian equations. Theoretical and Mathematical Physics, Vol. 193, Issue. 3, p. 1754.

TAKEMURA, Kouichi 2017. Integral transformation of Heun's equation and some applications. Journal of the Mathematical Society of Japan, Vol. 69, Issue. 2, p. 849.

Ishkhanyan, T. A. Shahverdyan, T. A. and Ishkhanyan, A. M. 2018. Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients. Advances in High Energy Physics, Vol. 2018, Issue. , p. 1.

Melas, Evangelos 2018. On the Liouvillian solutions to the perturbation equations of the Schwarzschild black hole. Journal of Mathematical Physics, Vol. 59, Issue. 8,

Ishkhanyan, A. M. 2019. SERIES SOLUTIONS OF CONFLUENT HEUN EQUATIONS IN TERMS OF INCOMPLETE GAMMA-FUNCTIONS. Journal of Applied Analysis & Computation, Vol. 9, Issue. 1, p. 118.

Ishkhanyan, A. M. 2020. Generalized Hypergeometric Solutions of the Heun Equation. Theoretical and Mathematical Physics, Vol. 202, Issue. 1, p. 1.

Melikdzhanian, D.Yu. and Ishkhanyan, A.M. 2021. A note on the generalized-hypergeometric solutions of general and single-confluent Heun equations. Journal of Mathematical Analysis and Applications, Vol. 499, Issue. 2, p. 125037.

Assier, R C and Shanin, A V 2021. Vertex Green’s Functions of a Quarter-Plane: Links Between the Functional Equation, Additive Crossing and Lamé Functions. The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 74, Issue. 3, p. 251.

Chiang, Yik-Man Ching, Avery and Tsang, Chiu-Yin 2021. Resolving Singularities and Monodromy Reduction of Fuchsian Connections. Annales Henri Poincaré, Vol. 22, Issue. 9, p. 3051.

de Oliveira, M. D. and Schmidt, Alexandre G. M. 2024. Analysis of the three possible event horizons of anti-de Sitter space–time in the Klein–Gordon equation. International Journal of Modern Physics A, Vol. 39, Issue. 29,

Hopper, Scott J. Wood, Toby S. and Bushby, Paul J. 2024. Stratified Resistive Tearing Instability. Journal of Fluid Mechanics, Vol. 994, Issue. ,

Download full list

Article contents

Removing false singular points as a method of solving ordinary differential equations

Abstract

Information

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

Removing false singular points as a method of solving ordinary differential equations

Abstract

Information

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests