In this thesis, we develop a highly accurate and efficient algorithm for computing the solution of a partial differential equation defined on a two-dimensional domain with discontinuous coefficients. An example of such a problem is for modeling the diffusion of heat energy in two space dimensions, in the case where the spatial domain represents a medium consisting of two different but homogeneous materials, with periodic boundary conditions.
Since diffusivity changes based on the material, it will be represented using a piecewise constant function, and this results in the formation of a complicated mathematical model. Such a model is impossible to solve analytically, and is very difficult to solve using existing numerical methods, thus the implementation of an alternative approach.
In this thesis, we take an approach that represents the solution as a linear combination of functions which change frequencies at the interfaces between different materials. Unlike previous work in the one-dimensional case, these functions are not all wave functions, like sine and cosine since we also have sinh and cosh functions. It will be demonstrated that by computing the eigenvalues and eigenfunctions to construct a basis of such functions- both independently and simultaneously, in conjunction with the secant method- a mathematical model for heat diffusion through different materials in two space dimensions can be solved much more efficiently and accurately than using conventional time-stepping methods.
","abstract_html":"<p>In this thesis, we develop a highly accurate and efficient algorithm for computing the solution of a partial differential equation defined on a two-dimensional domain with discontinuous coefficients. An example of such a problem is for modeling the diffusion of heat energy in two space dimensions, in the case where the spatial domain represents a medium consisting of two different but homogeneous materials, with periodic boundary conditions.</p> <p>Since diffusivity changes based on the material, it will be represented using a piecewise constant function, and this results in the formation of a complicated mathematical model. Such a model is impossible to solve analytically, and is very difficult to solve using existing numerical methods, thus the implementation of an alternative approach.</p> <p>In this thesis, we take an approach that represents the solution as a linear combination of functions which change frequencies at the interfaces between different materials. Unlike previous work in the one-dimensional case, these functions are not all wave functions, like sine and cosine since we also have sinh and cosh functions. It will be demonstrated that by computing the eigenvalues and eigenfunctions to construct a basis of such functions- both independently and simultaneously, in conjunction with the secant method- a mathematical model for heat diffusion through different materials in two space dimensions can be solved much more efficiently and accurately than using conventional time-stepping methods.</p>","abstract_has_math":false,"creators":["Aurko, Abdullah Muheel Momit"],"institution":null,"degree_name":"Master of Science (MS)","degree_level":"Masters Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["James V. Lambers","Haiyan Tian","Huiqing Zhu"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-08-01T07:00:00Z","date_published":"2017-08-01T07:00:00Z","updated_at":"2026-07-24T05:44:50Z","subjects":["heat equation","interface problem","eigenfunction expansion","Numerical Analysis and Computation","Partial Differential Equations"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://aquila.usm.edu/masters_theses/316","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["James V. Lambers","Haiyan Tian","Huiqing Zhu"]},{"key":"dc:creator","label":"Author","values":["Aurko, Abdullah Muheel Momit"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2017-06-28T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"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":["heat equation","interface problem","eigenfunction expansion","Numerical Analysis and Computation","Partial Differential Equations"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://aquila.usm.edu/masters_theses/316"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, we develop a highly accurate and efficient algorithm for computing the solution of a partial differential equation defined on a two-dimensional domain with discontinuous coefficients. An example of such a problem is for modeling the diffusion of heat energy in two space dimensions, in the case where the spatial domain represents a medium consisting of two different but homogeneous materials, with periodic boundary conditions.
Since diffusivity changes based on the material, it will be represented using a piecewise constant function, and this results in the formation of a complicated mathematical model. Such a model is impossible to solve analytically, and is very difficult to solve using existing numerical methods, thus the implementation of an alternative approach.
In this thesis, we take an approach that represents the solution as a linear combination of functions which change frequencies at the interfaces between different materials. Unlike previous work in the one-dimensional case, these functions are not all wave functions, like sine and cosine since we also have sinh and cosh functions. It will be demonstrated that by computing the eigenvalues and eigenfunctions to construct a basis of such functions- both independently and simultaneously, in conjunction with the secant method- a mathematical model for heat diffusion through different materials in two space dimensions can be solved much more efficiently and accurately than using conventional time-stepping methods.
"]},{"key":"dc:title","label":"Title","values":["Eignefunctions for Partial Differential Equations on Two-Dimensional Domains With Piecewise Constant Coefficients"]}]}],"canonical_facts":{"dc:contributor":["James V. Lambers","Haiyan Tian","Huiqing Zhu"],"dc:creator":["Aurko, Abdullah Muheel Momit"],"dc:date.available":["2017-06-28T07:00:00Z"],"dc:description.abstract":["In this thesis, we develop a highly accurate and efficient algorithm for computing the solution of a partial differential equation defined on a two-dimensional domain with discontinuous coefficients. An example of such a problem is for modeling the diffusion of heat energy in two space dimensions, in the case where the spatial domain represents a medium consisting of two different but homogeneous materials, with periodic boundary conditions.
Since diffusivity changes based on the material, it will be represented using a piecewise constant function, and this results in the formation of a complicated mathematical model. Such a model is impossible to solve analytically, and is very difficult to solve using existing numerical methods, thus the implementation of an alternative approach.
In this thesis, we take an approach that represents the solution as a linear combination of functions which change frequencies at the interfaces between different materials. Unlike previous work in the one-dimensional case, these functions are not all wave functions, like sine and cosine since we also have sinh and cosh functions. It will be demonstrated that by computing the eigenvalues and eigenfunctions to construct a basis of such functions- both independently and simultaneously, in conjunction with the secant method- a mathematical model for heat diffusion through different materials in two space dimensions can be solved much more efficiently and accurately than using conventional time-stepping methods.
"],"dc:identifier":["https://aquila.usm.edu/masters_theses/316"],"dc:subject":["heat equation","interface problem","eigenfunction expansion","Numerical Analysis and Computation","Partial Differential Equations"],"dc:title":["Eignefunctions for Partial Differential Equations on Two-Dimensional Domains With Piecewise Constant Coefficients"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Masters Thesis"],"thesis:degree_name":["Master of Science (MS)"]},"updated_at":"2026-07-24T05:44:50Z"}