Wave polynomials, transmutations and Cauchy's problem for the Klein-Gordon equation
arXiv:1208.5984 · doi:10.1016/j.jmaa.2012.10.013
Abstract
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.
31 pages, 8 figures (16 graphs)
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- Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems
- Modified spectral parameter power series representations for solutions of Sturm-Liouville equations and their applications
- Analytic approximation of solutions of parabolic partial differential equations with variable coefficients
- Analytic approximation of transmutation operators and related systems of functions
- On Sturm-Liouville Equations with Several Spectral Parameters
- On two-dimensional supersymmetric quantum mechanics, pseudoanalytic functions and transmutation operators
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