For this thesis, Krylov Subspace Spectral (KSS) methods, developed by Dr. James Lambers, will be used to solve a one-dimensional, heat equation with non-homogenous boundary conditions. While current methods such as Finite Difference are able to carry out these computations efficiently, their accuracy and scalability can be improved. We will solve the heat equation in one-dimension with two cases to observe the behaviors of the errors using KSS methods. The first case will implement KSS methods with trigonometric initial conditions, then another case where the initial conditions are polynomial functions. We will also look at both the time-independent and time-dependent cases for both sets of initial conditions for discrepancies in accuracy and efficiency. Our numerical results will be compared to the results given by Finite Difference methods to show that accuracy can be improved without sacrificing efficiency.
","abstract_html":"<p>For this thesis, Krylov Subspace Spectral (KSS) methods, developed by Dr. James Lambers, will be used to solve a one-dimensional, heat equation with non-homogenous boundary conditions. While current methods such as Finite Difference are able to carry out these computations efficiently, their accuracy and scalability can be improved. We will solve the heat equation in one-dimension with two cases to observe the behaviors of the errors using KSS methods. The first case will implement KSS methods with trigonometric initial conditions, then another case where the initial conditions are polynomial functions. We will also look at both the time-independent and time-dependent cases for both sets of initial conditions for discrepancies in accuracy and efficiency. Our numerical results will be compared to the results given by Finite Difference methods to show that accuracy can be improved without sacrificing efficiency.</p>","abstract_has_math":false,"creators":["Hendley, Abbie"],"institution":null,"degree_name":"Master of Science (MS)","degree_level":"Masters Thesis","degree_discipline":null,"degree_department":null,"school":null,"contributors":["James Lambers","Haiyan Tian","Huiqing Zhu"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-08-01T07:00:00Z","date_published":"2019-08-01T07:00:00Z","updated_at":"2026-07-24T05:45:12Z","subjects":["partial differential equation","numerical","Fast Fourier Transform","Dirichlet","analysis","Numerical Analysis and Computation","Numerical Analysis and Scientific Computing","Partial Differential Equations"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://aquila.usm.edu/masters_theses/674","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["James Lambers","Haiyan Tian","Huiqing Zhu"]},{"key":"dc:creator","label":"Author","values":["Hendley, Abbie"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2019-06-21T07:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MS)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["partial differential equation","numerical","Fast Fourier Transform","Dirichlet","analysis","Numerical Analysis and Computation","Numerical Analysis and Scientific Computing","Partial Differential Equations"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://aquila.usm.edu/masters_theses/674"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["For this thesis, Krylov Subspace Spectral (KSS) methods, developed by Dr. James Lambers, will be used to solve a one-dimensional, heat equation with non-homogenous boundary conditions. While current methods such as Finite Difference are able to carry out these computations efficiently, their accuracy and scalability can be improved. We will solve the heat equation in one-dimension with two cases to observe the behaviors of the errors using KSS methods. The first case will implement KSS methods with trigonometric initial conditions, then another case where the initial conditions are polynomial functions. We will also look at both the time-independent and time-dependent cases for both sets of initial conditions for discrepancies in accuracy and efficiency. Our numerical results will be compared to the results given by Finite Difference methods to show that accuracy can be improved without sacrificing efficiency.
"]},{"key":"dc:title","label":"Title","values":["Krylov Subspace Spectral Methods with Non-homogenous Boundary Conditions"]}]}],"canonical_facts":{"dc:contributor":["James Lambers","Haiyan Tian","Huiqing Zhu"],"dc:creator":["Hendley, Abbie"],"dc:date.available":["2019-06-21T07:00:00Z"],"dc:description.abstract":["For this thesis, Krylov Subspace Spectral (KSS) methods, developed by Dr. James Lambers, will be used to solve a one-dimensional, heat equation with non-homogenous boundary conditions. While current methods such as Finite Difference are able to carry out these computations efficiently, their accuracy and scalability can be improved. We will solve the heat equation in one-dimension with two cases to observe the behaviors of the errors using KSS methods. The first case will implement KSS methods with trigonometric initial conditions, then another case where the initial conditions are polynomial functions. We will also look at both the time-independent and time-dependent cases for both sets of initial conditions for discrepancies in accuracy and efficiency. Our numerical results will be compared to the results given by Finite Difference methods to show that accuracy can be improved without sacrificing efficiency.
"],"dc:identifier":["https://aquila.usm.edu/masters_theses/674"],"dc:subject":["partial differential equation","numerical","Fast Fourier Transform","Dirichlet","analysis","Numerical Analysis and Computation","Numerical Analysis and Scientific Computing","Partial Differential Equations"],"dc:title":["Krylov Subspace Spectral Methods with Non-homogenous Boundary Conditions"],"thesis:degree_level":["Masters Thesis"],"thesis:degree_name":["Master of Science (MS)"]},"updated_at":"2026-07-24T05:45:12Z"}