Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation
The generalized forced Boussinesq equation, utt−uxx+[f(u)]xx+uxxxx=h0, and its periodic traveling wave solutions are considered. Using the transform z=x−ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the o...
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| Format: | Article |
| Language: | English |
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Wiley
1999-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171299226439 |
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| _version_ | 1850176869859590144 |
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| author | Kenneth L. Jones Yunkai Chen |
| author_facet | Kenneth L. Jones Yunkai Chen |
| author_sort | Kenneth L. Jones |
| collection | DOAJ |
| description | The generalized forced Boussinesq equation, utt−uxx+[f(u)]xx+uxxxx=h0, and its periodic traveling wave solutions are considered. Using the transform
z=x−ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral equation is then established by using the Green's function method. This integral equation generates compact operators in a Banach space of real-valued continuous
periodic functions with a given period 2T. The Schauder's fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant
periodic traveling wave solutions to the generalized forced Boussinesq equation is established. |
| format | Article |
| id | doaj-art-b8127e4e37db42c0819e4692de8472f2 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1999-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-b8127e4e37db42c0819e4692de8472f22025-08-20T02:19:11ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251999-01-0122364364810.1155/S0161171299226439Existence of periodic traveling wave solutions to the generalized forced Boussinesq equationKenneth L. Jones0Yunkai Chen1Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville 28301-4298, North Carolin, USADepartment of Mathematics and Computer Science, Fayetteville State University, Fayetteville 28301-4298, North Carolin, USAThe generalized forced Boussinesq equation, utt−uxx+[f(u)]xx+uxxxx=h0, and its periodic traveling wave solutions are considered. Using the transform z=x−ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral equation is then established by using the Green's function method. This integral equation generates compact operators in a Banach space of real-valued continuous periodic functions with a given period 2T. The Schauder's fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant periodic traveling wave solutions to the generalized forced Boussinesq equation is established.http://dx.doi.org/10.1155/S0161171299226439Existence theoremperiodic traveling wave solutionsgeneralized forced Boussinesq equation. |
| spellingShingle | Kenneth L. Jones Yunkai Chen Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation International Journal of Mathematics and Mathematical Sciences Existence theorem periodic traveling wave solutions generalized forced Boussinesq equation. |
| title | Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation |
| title_full | Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation |
| title_fullStr | Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation |
| title_full_unstemmed | Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation |
| title_short | Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation |
| title_sort | existence of periodic traveling wave solutions to the generalized forced boussinesq equation |
| topic | Existence theorem periodic traveling wave solutions generalized forced Boussinesq equation. |
| url | http://dx.doi.org/10.1155/S0161171299226439 |
| work_keys_str_mv | AT kennethljones existenceofperiodictravelingwavesolutionstothegeneralizedforcedboussinesqequation AT yunkaichen existenceofperiodictravelingwavesolutionstothegeneralizedforcedboussinesqequation |