We propose to create a new numerical method for a class of time-dependent PDEs (second-order, one space dimension, Dirichlet boundary conditions) that can be used to obtain more accurate and reliable solutions than traditional methods. Previously, it was shown that conventional time-stepping methods could be avoided for time-dependent mathematical models featuring a finite number of homogeneous materials, thus assuming general piecewise constant coefficients. This proposed method will avoid the modeling shortcuts that are traditionally taken, and it will generalize the piecewise constant case of energy diffusion and wave propagation to work for an infinite number of smaller pieces, or a smoothly varying coefficient. We hypothesize that by treating a smoothly varying function as a piecewise constant function with infinitely many pieces, this potential method can be realized. Through the Uncertainty Principle, we will formulate highly accurate estimates of eigenvalues, and then through the QR algorithm, we will use these estimates to formulate highly accurate eigenvalues and eigenfunctions. Ultimately, we will produce a more efficient solution method that avoids traditional time-stepping.
","abstract_html":"<p>We propose to create a new numerical method for a class of time-dependent PDEs (second-order, one space dimension, Dirichlet boundary conditions) that can be used to obtain more accurate and reliable solutions than traditional methods. Previously, it was shown that conventional time-stepping methods could be avoided for time-dependent mathematical models featuring a finite number of homogeneous materials, thus assuming general piecewise constant coefficients. This proposed method will avoid the modeling shortcuts that are traditionally taken, and it will generalize the piecewise constant case of energy diffusion and wave propagation to work for an infinite number of smaller pieces, or a smoothly varying coefficient. We hypothesize that by treating a smoothly varying function as a piecewise constant function with infinitely many pieces, this potential method can be realized. Through the Uncertainty Principle, we will formulate highly accurate estimates of eigenvalues, and then through the QR algorithm, we will use these estimates to formulate highly accurate eigenvalues and eigenfunctions. Ultimately, we will produce a more efficient solution method that avoids traditional time-stepping.</p>","abstract_has_math":false,"creators":["Walker, Carley"],"institution":null,"degree_name":"Master of Science (MS)","degree_level":"Masters Thesis","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Dr. James Lambers","Dr. Haiyan Tian","Dr. Huiqing Zhu"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-12-01T08:00:00Z","date_published":"2020-12-01T08:00:00Z","updated_at":"2026-07-24T05:45:19Z","subjects":["partial","differential","equation","smoothly-varying","second-order","one-dimensional","Partial Differential Equations"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://aquila.usm.edu/masters_theses/778","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dr. James Lambers","Dr. Haiyan Tian","Dr. Huiqing Zhu"]},{"key":"dc:creator","label":"Author","values":["Walker, Carley"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2020-10-09T07:00:00Z"]},{"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","smoothly-varying","second-order","one-dimensional","Partial Differential Equations"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://aquila.usm.edu/masters_theses/778"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We propose to create a new numerical method for a class of time-dependent PDEs (second-order, one space dimension, Dirichlet boundary conditions) that can be used to obtain more accurate and reliable solutions than traditional methods. Previously, it was shown that conventional time-stepping methods could be avoided for time-dependent mathematical models featuring a finite number of homogeneous materials, thus assuming general piecewise constant coefficients. This proposed method will avoid the modeling shortcuts that are traditionally taken, and it will generalize the piecewise constant case of energy diffusion and wave propagation to work for an infinite number of smaller pieces, or a smoothly varying coefficient. We hypothesize that by treating a smoothly varying function as a piecewise constant function with infinitely many pieces, this potential method can be realized. Through the Uncertainty Principle, we will formulate highly accurate estimates of eigenvalues, and then through the QR algorithm, we will use these estimates to formulate highly accurate eigenvalues and eigenfunctions. Ultimately, we will produce a more efficient solution method that avoids traditional time-stepping.
"]},{"key":"dc:title","label":"Title","values":["Rapid Implicit Diagonalization of Variable-Coefficient Differential Operators Using the Uncertainty Principle"]}]}],"canonical_facts":{"dc:contributor":["Dr. James Lambers","Dr. Haiyan Tian","Dr. Huiqing Zhu"],"dc:creator":["Walker, Carley"],"dc:date.available":["2020-10-09T07:00:00Z"],"dc:description.abstract":["We propose to create a new numerical method for a class of time-dependent PDEs (second-order, one space dimension, Dirichlet boundary conditions) that can be used to obtain more accurate and reliable solutions than traditional methods. Previously, it was shown that conventional time-stepping methods could be avoided for time-dependent mathematical models featuring a finite number of homogeneous materials, thus assuming general piecewise constant coefficients. This proposed method will avoid the modeling shortcuts that are traditionally taken, and it will generalize the piecewise constant case of energy diffusion and wave propagation to work for an infinite number of smaller pieces, or a smoothly varying coefficient. We hypothesize that by treating a smoothly varying function as a piecewise constant function with infinitely many pieces, this potential method can be realized. Through the Uncertainty Principle, we will formulate highly accurate estimates of eigenvalues, and then through the QR algorithm, we will use these estimates to formulate highly accurate eigenvalues and eigenfunctions. Ultimately, we will produce a more efficient solution method that avoids traditional time-stepping.
"],"dc:identifier":["https://aquila.usm.edu/masters_theses/778"],"dc:subject":["partial","differential","equation","smoothly-varying","second-order","one-dimensional","Partial Differential Equations"],"dc:title":["Rapid Implicit Diagonalization of Variable-Coefficient Differential Operators Using the Uncertainty Principle"],"thesis:degree_level":["Masters Thesis"],"thesis:degree_name":["Master of Science (MS)"]},"updated_at":"2026-07-24T05:45:19Z"}