Wave polynomials, transmutations and Cauchy’s problem for the Klein–Gordon equation

We prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein–Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discov...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 399; no. 1; pp. 191 - 212
Main Authors Khmelnytskaya, Kira V., Kravchenko, Vladislav V., Torba, Sergii M., Tremblay, Sébastien
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.03.2013
Subjects
Cauchy problem
Completeness
Klein–Gordon equation
Numerical method
Transmutation operators
Transmutation operators
Numerical method
Klein–Gordon equation
Completeness
Cauchy problem
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Summary:We prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein–Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the Cauchy problem for the Klein–Gordon equation. Based on this result we develop a numerical method for solving the Cauchy problem and test its performance.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2012.10.013