Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, or lack thereof, of each technique. These techniques are integral in applied mathematics and it is shown that correct employment allows us to see the behavior of a differential equation when the exact solution may not be attainable.
","abstract_html":"<p>Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, or lack thereof, of each technique. These techniques are integral in applied mathematics and it is shown that correct employment allows us to see the behavior of a differential equation when the exact solution may not be attainable.</p>","abstract_has_math":false,"creators":["Weymier, Emily Jean"],"institution":null,"degree_name":"Master of Science - Mathematical Sciences","degree_level":"Thesis","degree_discipline":"Mathematics and Statistics","degree_department":null,"school":null,"contributors":["Matthew A. Beauregard","William Clark","Thomas Judson"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-02-01T08:00:00Z","date_published":"2018-02-01T08:00:00Z","updated_at":"2026-07-24T04:30:15Z","subjects":["differential equations","analysis","solution","eigenvalue","nonlinear equation","Non-linear Dynamics","Numerical Analysis and Computation","Ordinary Differential Equations and Applied Dynamics","Other Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.sfasu.edu/etds/145","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Matthew A. Beauregard","William Clark","Thomas Judson"]},{"key":"dc:creator","label":"Author","values":["Weymier, Emily Jean"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2018-04-06T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics and Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science - Mathematical Sciences"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["differential equations","analysis","solution","eigenvalue","nonlinear equation","Non-linear Dynamics","Numerical Analysis and Computation","Ordinary Differential Equations and Applied Dynamics","Other Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.sfasu.edu/etds/145"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, or lack thereof, of each technique. These techniques are integral in applied mathematics and it is shown that correct employment allows us to see the behavior of a differential equation when the exact solution may not be attainable.
"]},{"key":"dc:title","label":"Title","values":["Theoretical Analysis of Nonlinear Differential Equations"]}]}],"canonical_facts":{"dc:contributor":["Matthew A. Beauregard","William Clark","Thomas Judson"],"dc:creator":["Weymier, Emily Jean"],"dc:date.available":["2018-04-06T07:00:00Z"],"dc:description.abstract":["Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, or lack thereof, of each technique. These techniques are integral in applied mathematics and it is shown that correct employment allows us to see the behavior of a differential equation when the exact solution may not be attainable.
"],"dc:identifier":["https://scholarworks.sfasu.edu/etds/145"],"dc:subject":["differential equations","analysis","solution","eigenvalue","nonlinear equation","Non-linear Dynamics","Numerical Analysis and Computation","Ordinary Differential Equations and Applied Dynamics","Other Mathematics"],"dc:title":["Theoretical Analysis of Nonlinear Differential Equations"],"thesis:degree_discipline":["Mathematics and Statistics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science - Mathematical Sciences"]},"updated_at":"2026-07-24T04:30:15Z"}