On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application
The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients...
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doaj-bd8da55c29c84338a91f1e24dd9789ae2020-11-25T04:10:53ZengMDPI AGAlgorithms1999-48932020-11-011328628610.3390/a13110286On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, ApplicationKyle R. Bryenton0Andrew R. Cameron1Keegan L. A. Kirk2Nasser Saad3Patrick Strongman4Nikita Volodin5School of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, PE C1A 4P3, CanadaDepartment of Physics, University of Prince Edward Island, Charlottetown, PE C1A 4P3, CanadaSchool of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, PE C1A 4P3, CanadaSchool of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, PE C1A 4P3, CanadaSchool of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, PE C1A 4P3, CanadaSchool of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, PE C1A 4P3, CanadaThe analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree <i>n</i>, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> respectively. We show that for <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula> the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted <i>n</i> generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary <i>n</i>, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula>. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations.https://www.mdpi.com/1999-4893/13/11/286symbolic computationalgorithmpolynomial solutionsScheffé criteriaHeun equationDirac equation |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Kyle R. Bryenton Andrew R. Cameron Keegan L. A. Kirk Nasser Saad Patrick Strongman Nikita Volodin |
spellingShingle |
Kyle R. Bryenton Andrew R. Cameron Keegan L. A. Kirk Nasser Saad Patrick Strongman Nikita Volodin On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application Algorithms symbolic computation algorithm polynomial solutions Scheffé criteria Heun equation Dirac equation |
author_facet |
Kyle R. Bryenton Andrew R. Cameron Keegan L. A. Kirk Nasser Saad Patrick Strongman Nikita Volodin |
author_sort |
Kyle R. Bryenton |
title |
On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application |
title_short |
On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application |
title_full |
On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application |
title_fullStr |
On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application |
title_full_unstemmed |
On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application |
title_sort |
on the solutions of second-order differential equations with polynomial coefficients: theory, algorithm, application |
publisher |
MDPI AG |
series |
Algorithms |
issn |
1999-4893 |
publishDate |
2020-11-01 |
description |
The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree <i>n</i>, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> respectively. We show that for <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula> the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted <i>n</i> generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary <i>n</i>, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula>. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations. |
topic |
symbolic computation algorithm polynomial solutions Scheffé criteria Heun equation Dirac equation |
url |
https://www.mdpi.com/1999-4893/13/11/286 |
work_keys_str_mv |
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