Partial Differential Equation is one of the major influential and useful subjects in Mathematical Sciences. In this thesis, I mainly use Fourier analysis and PDE methods to study the solution to an inhomogeneous wave equation on Euclidean spaces. I obtained the existence, uniqueness, and regularity for the solution. In the classical case, the datum involved is required to have up to C2 smoothness; my results treat the low regularity case and only require Hs smoothness with fractional order s. The main tools include Fourier transform, Duhamel principle, and the method of energy estimates. The results have potential applications to the solutions of nonlinear wave equations with low regularity datum, which have background and implications in Physics and Engineering.
","abstract_html":"<p>Partial Differential Equation is one of the major influential and useful subjects in Mathematical Sciences. In this thesis, I mainly use Fourier analysis and PDE methods to study the solution to an inhomogeneous wave equation on Euclidean spaces. I obtained the existence, uniqueness, and regularity for the solution. In the classical case, the datum involved is required to have up to C2 smoothness; my results treat the low regularity case and only require Hs smoothness with fractional order s. The main tools include Fourier transform, Duhamel principle, and the method of energy estimates. The results have potential applications to the solutions of nonlinear wave equations with low regularity datum, which have background and implications in Physics and Engineering.</p>","abstract_has_math":false,"creators":["Uchida, Masahiko"],"institution":null,"degree_name":"Master of Science in Mathematics (M.S.)","degree_level":"Thesis (restricted to Georgia Southern)","degree_discipline":"Department of Mathematical Sciences","degree_department":null,"school":null,"contributors":["Sze-Man Ngai","Alex Stokolos","Yan Wu"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-01-01T08:00:00Z","date_published":"2010-01-01T08:00:00Z","updated_at":"2026-07-24T02:27:19Z","subjects":["ETD","Fourier analysis","wave equation","Sobolov spaces","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.georgiasouthern.edu/etd/684","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Sze-Man Ngai","Alex Stokolos","Yan Wu"]},{"key":"dc:creator","label":"Author","values":["Uchida, Masahiko"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2013-08-07T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis (restricted to Georgia Southern)"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science in Mathematics (M.S.)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["ETD","Fourier analysis","wave equation","Sobolov spaces","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.georgiasouthern.edu/etd/684"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Partial Differential Equation is one of the major influential and useful subjects in Mathematical Sciences. In this thesis, I mainly use Fourier analysis and PDE methods to study the solution to an inhomogeneous wave equation on Euclidean spaces. I obtained the existence, uniqueness, and regularity for the solution. In the classical case, the datum involved is required to have up to C2 smoothness; my results treat the low regularity case and only require Hs smoothness with fractional order s. The main tools include Fourier transform, Duhamel principle, and the method of energy estimates. The results have potential applications to the solutions of nonlinear wave equations with low regularity datum, which have background and implications in Physics and Engineering.
"]},{"key":"dc:title","label":"Title","values":["Fourier Analysis of Wave Equations"]}]}],"canonical_facts":{"dc:contributor":["Sze-Man Ngai","Alex Stokolos","Yan Wu"],"dc:creator":["Uchida, Masahiko"],"dc:date.available":["2013-08-07T07:00:00Z"],"dc:description.abstract":["Partial Differential Equation is one of the major influential and useful subjects in Mathematical Sciences. In this thesis, I mainly use Fourier analysis and PDE methods to study the solution to an inhomogeneous wave equation on Euclidean spaces. I obtained the existence, uniqueness, and regularity for the solution. In the classical case, the datum involved is required to have up to C2 smoothness; my results treat the low regularity case and only require Hs smoothness with fractional order s. The main tools include Fourier transform, Duhamel principle, and the method of energy estimates. The results have potential applications to the solutions of nonlinear wave equations with low regularity datum, which have background and implications in Physics and Engineering.
"],"dc:identifier":["https://digitalcommons.georgiasouthern.edu/etd/684"],"dc:subject":["ETD","Fourier analysis","wave equation","Sobolov spaces","Mathematics"],"dc:title":["Fourier Analysis of Wave Equations"],"thesis:degree_discipline":["Department of Mathematical Sciences"],"thesis:degree_level":["Thesis (restricted to Georgia Southern)"],"thesis:degree_name":["Master of Science in Mathematics (M.S.)"]},"updated_at":"2026-07-24T02:27:19Z"}