{"id":{"repo_id":"siu-theses","oai_identifier":"oai:opensiuc.lib.siu.edu:dissertations-1977"},"canonical_url":"https://search.dev.ndltd.org/etd/siu-theses/oai:opensiuc.lib.siu.edu:dissertations-1977","repository":{"repo_id":"siu-theses","name":"Southern Illinois University","base_url":"https://opensiuc.lib.siu.edu/do/oai/"},"display":{"title":"Modified Stochastic Sine-Gordon Equation","abstract":"The main focus of my dissertation is the Modified Stochastic Sine-Gordon Equation: utt = 2uxx − ut − sin(|u|^(&gamma)) + b(u, du/dt)dW/dt where &gamma > 0 is the parameter of the power of non-linearity, &delta &ge 0 is the magnitude of non-linearity, &alpha> 0 be the damping parameter, and &sigma the diffusion intensity, on one dimensional domain. We analyze the properties of the solution of the SPDE by the eigenfunctions approach allowing us to truncate the infinite-dimensional stochastic system (i.e the SDEs of Fourier coefficients related to the SPDE), to control its energy, existence, uniqueness, continuity and stability. The analysis relies on the investigation of expected Lyapunov functional of the energy in terms of all system-parameters. We simulate the model with respect to all system-parameters to visualize our conclusions.","abstract_html":"The main focus of my dissertation is the Modified Stochastic Sine-Gordon Equation: utt = 2uxx − ut − sin(|u|^(&gamma)) + b(u, du/dt)dW/dt where &gamma > 0 is the parameter of the power of non-linearity, &delta &ge 0 is the magnitude of non-linearity, &alpha> 0 be the damping parameter, and &sigma the diffusion intensity, on one dimensional domain. We analyze the properties of the solution of the SPDE by the eigenfunctions approach allowing us to truncate the infinite-dimensional stochastic system (i.e the SDEs of Fourier coefficients related to the SPDE), to control its energy, existence, uniqueness, continuity and stability. The analysis relies on the investigation of expected Lyapunov functional of the energy in terms of all system-parameters. We simulate the model with respect to all system-parameters to visualize our conclusions.","abstract_has_math":false,"creators":["Talafha, Abdallah M."],"institution":null,"degree_name":"Doctor of Philosophy","degree_level":"Campus Only Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Schurz, Henri"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-01T08:00:00Z","date_published":"2014-12-01T08:00:00Z","updated_at":"2026-07-24T04:34:36Z","subjects":["Fourier solutions","Lyapunov functions","MODIFIED STOCHASTIC SINE-GORDON EQUATION","Q-regular space-time noise","total energy functional","trace formula"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://opensiuc.lib.siu.edu/dissertations/973","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Schurz, Henri"]},{"key":"dc:creator","label":"Author","values":["Talafha, Abdallah M."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Only Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Fourier solutions","Lyapunov functions","MODIFIED STOCHASTIC SINE-GORDON EQUATION","Q-regular space-time noise","total energy functional","trace formula"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://opensiuc.lib.siu.edu/dissertations/973"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The main focus of my dissertation is the Modified Stochastic Sine-Gordon Equation: utt = 2uxx − ut − sin(|u|^(&gamma)) + b(u, du/dt)dW/dt where &gamma > 0 is the parameter of the power of non-linearity, &delta &ge 0 is the magnitude of non-linearity, &alpha> 0 be the damping parameter, and &sigma the diffusion intensity, on one dimensional domain. We analyze the properties of the solution of the SPDE by the eigenfunctions approach allowing us to truncate the infinite-dimensional stochastic system (i.e the SDEs of Fourier coefficients related to the SPDE), to control its energy, existence, uniqueness, continuity and stability. The analysis relies on the investigation of expected Lyapunov functional of the energy in terms of all system-parameters. We simulate the model with respect to all system-parameters to visualize our conclusions."]},{"key":"dc:title","label":"Title","values":["Modified Stochastic Sine-Gordon Equation"]}]}],"canonical_facts":{"dc:contributor":["Schurz, Henri"],"dc:creator":["Talafha, Abdallah M."],"dc:description.abstract":["The main focus of my dissertation is the Modified Stochastic Sine-Gordon Equation: utt = 2uxx − ut − sin(|u|^(&gamma)) + b(u, du/dt)dW/dt where &gamma > 0 is the parameter of the power of non-linearity, &delta &ge 0 is the magnitude of non-linearity, &alpha> 0 be the damping parameter, and &sigma the diffusion intensity, on one dimensional domain. We analyze the properties of the solution of the SPDE by the eigenfunctions approach allowing us to truncate the infinite-dimensional stochastic system (i.e the SDEs of Fourier coefficients related to the SPDE), to control its energy, existence, uniqueness, continuity and stability. The analysis relies on the investigation of expected Lyapunov functional of the energy in terms of all system-parameters. We simulate the model with respect to all system-parameters to visualize our conclusions."],"dc:identifier":["https://opensiuc.lib.siu.edu/dissertations/973"],"dc:subject":["Fourier solutions","Lyapunov functions","MODIFIED STOCHASTIC SINE-GORDON EQUATION","Q-regular space-time noise","total energy functional","trace formula"],"dc:title":["Modified Stochastic Sine-Gordon Equation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Only Dissertation"],"thesis:degree_name":["Doctor of Philosophy"]},"updated_at":"2026-07-24T04:34:36Z"}