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We consider an approximate integration method of the Cauchy problem for the generalized Liouville equation using symbolic and numeric computer computations. This method is based on the probability density function orthonormal series expansion in the small and initial time space domains. We are investigating several expansions and determine their convergence conditions to ensure the convergence of the asymptotic expansion to the solution of the considered problem.To illustrate the applicability of the introduced asymptotic orthogonal decompositions 18 we took the describing bidimensional integrable dispersive shallow water equation developed by Roberto Camassa and Darryl D. Holm, Los Alamos National Laboratory. Since CH-equation solutionsare represented by a superposition of arbitrary number of peakons peaked solitons 9,16, one can compare the coincidence of the peakon- solutions character provided by numerical modeling along some trajectories for truncated asymptotic series expansions obtained by symbolic computations.

Tipo de documento: Artículo - Article

Palabras clave: Liouville equation, orthonormal system, eigenfunction, strong and weak convergence, mean convergence, Camassa- Holm equation, Hermite functions.





Source: http://www.bdigital.unal.edu.co


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Boletı́n de Matemáticas Nueva Serie, Volumen XIV No.
2 (2007), pp.
129–172 AN APPROXIMATE ORTHOGONAL DECOMPOSITION METHOD FOR THE SOLUTION OF THE GENERALIZED LIOUVILLE EQUATION EUGENE V.
DULOV (*) ALEXANDRE V.
SINITSYN (**) Abstract.
We consider an approximate integration method of the Cauchy problem for the generalized Liouville equation using symbolic and numeric computer computations.
This method is based on the probability density function orthonormal series expansion in the small and initial time space domains.
We are investigating several expansions and determine their convergence conditions to ensure the convergence of the asymptotic expansion to the solution of the considered problem. To illustrate the applicability of the introduced asymptotic orthogonal decompositions [18] we took the describing bidimensional integrable dispersive shallow water equation developed by Roberto Camassa and Darryl D.
Holm, Los Alamos National Laboratory.
Since CH-equation solutions are represented by a superposition of arbitrary number of peakons (peaked solitons) [9],[16], one can compare the coincidence of the “peakon” solutions character provided by numerical modeling along some trajectories for truncated asymptotic series expansions obtained by symbolic computations. Key Words and Phrases. Liouville equation, orthonormal system, eigenfunction, strong and weak convergence, mean convergence, CamassaHolm equation, Hermite functions. 2000 Mathematics Subject Classification: 82C10, 35C10, 35C20, 35F10, 42C05. (*) Eugene V.
Dulov, Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional de Colombia, Bogotá, Colombia.
E-mail: dulov@yahoo.com (**) Alexandre V.
Sinitsyn, Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional de Colombia, Bogotá, Colombia.
E-mail: avsinitsyn@yahoo.com. 129 130 EUGENE V.
DULOV AND ALEXANDRE V.
SINITSYN Resumen.
Consideramos un método de integración aproximada del problema de Cauchy para la ...






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