Reactive-diffusive systems modeling physical phenomena in certain situations develop a singularity at a finite value of the independent variable referred to as \"blow-up.\" The attempt to find the blow-up time analytically is most often impossible, thus requiring a numerical determination of the value. The numerical methods often use a priori knowledge of the blow-up solution such as monotonicity or self-similarity. For equations where such a priori knowledge is unavailable, ad hoc methods were constructed. The object of this research is to develop a simple and consistent approach to find numerically the blow-up solution without having a priori knowledge or resorting to other ad hoc methods. The proposed method allows the investigator the ability to distinguish whether a singular solution or a non-singular solution exists on a given interval. Step size in the vicinity of a singular solution is automatically adjusted. The programming of the proposed method is simple and uses well-developed software for most of the auxiliary routines. The proposed numerical method is mainly concerned with the integration of nonlinear integral equations with Abel-type kernels developed from combustion problems, but may be used on similar equations from other fields. To demonstrate the flexibility of the proposed method, it is applied to ordinary differential equations with blow-up solutions or to ordinary differential equations which exhibit extremely stiff structure.
","abstract_html":"<p>Reactive-diffusive systems modeling physical phenomena in certain situations develop a singularity at a finite value of the independent variable referred to as "blow-up." The attempt to find the blow-up time analytically is most often impossible, thus requiring a numerical determination of the value. The numerical methods often use <em>a priori</em> knowledge of the blow-up solution such as monotonicity or self-similarity. For equations where such a priori knowledge is unavailable, <em>ad hoc</em> methods were constructed. The object of this research is to develop a simple and consistent approach to find numerically the blow-up solution without having <em>a priori</em> knowledge or resorting to other <em>ad hoc</em> methods. The proposed method allows the investigator the ability to distinguish whether a singular solution or a non-singular solution exists on a given interval. Step size in the vicinity of a singular solution is automatically adjusted. The programming of the proposed method is simple and uses well-developed software for most of the auxiliary routines. The proposed numerical method is mainly concerned with the integration of nonlinear integral equations with Abel-type kernels developed from combustion problems, but may be used on similar equations from other fields. To demonstrate the flexibility of the proposed method, it is applied to ordinary differential equations with blow-up solutions or to ordinary differential equations which exhibit extremely stiff structure.</p>","abstract_has_math":false,"creators":["Touron, Charles F."],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["David G. Lasseigne","John Adam","Richard Noren","Robert Ash"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-07-01T07:00:00Z","date_published":"2009-07-01T07:00:00Z","updated_at":"2026-07-24T03:35:00Z","subjects":["Blow-up solutions","Combustion","Reaction-diffusion systems","Volterra equations","Algebraic Geometry","Ordinary Differential Equations and Applied Dynamics"],"languages":[],"rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["9781109335811"],"render_values":[{"text":"9781109335811","href":null,"code":true}]}]},"links":{"outbound_url":"https://digitalcommons.odu.edu/mathstat_etds/62","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["David G. Lasseigne","John Adam","Richard Noren","Robert Ash"]},{"key":"dc:creator","label":"Author","values":["Touron, Charles F."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2019-06-13T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics & Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Blow-up solutions","Combustion","Reaction-diffusion systems","Volterra equations","Algebraic Geometry","Ordinary Differential Equations and Applied Dynamics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
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"]},{"key":"dc:title","label":"Title","values":["An Adaptive Method for Calculating Blow-Up Solutions"]}]}],"canonical_facts":{"dc:contributor":["David G. Lasseigne","John Adam","Richard Noren","Robert Ash"],"dc:creator":["Touron, Charles F."],"dc:date.available":["2019-06-13T07:00:00Z"],"dc:description.abstract":["Reactive-diffusive systems modeling physical phenomena in certain situations develop a singularity at a finite value of the independent variable referred to as \"blow-up.\" The attempt to find the blow-up time analytically is most often impossible, thus requiring a numerical determination of the value. The numerical methods often use a priori knowledge of the blow-up solution such as monotonicity or self-similarity. For equations where such a priori knowledge is unavailable, ad hoc methods were constructed. The object of this research is to develop a simple and consistent approach to find numerically the blow-up solution without having a priori knowledge or resorting to other ad hoc methods. The proposed method allows the investigator the ability to distinguish whether a singular solution or a non-singular solution exists on a given interval. Step size in the vicinity of a singular solution is automatically adjusted. The programming of the proposed method is simple and uses well-developed software for most of the auxiliary routines. The proposed numerical method is mainly concerned with the integration of nonlinear integral equations with Abel-type kernels developed from combustion problems, but may be used on similar equations from other fields. To demonstrate the flexibility of the proposed method, it is applied to ordinary differential equations with blow-up solutions or to ordinary differential equations which exhibit extremely stiff structure.
"],"dc:identifier":["9781109335811","https://digitalcommons.odu.edu/mathstat_etds/62"],"dc:rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"],"dc:subject":["Blow-up solutions","Combustion","Reaction-diffusion systems","Volterra equations","Algebraic Geometry","Ordinary Differential Equations and Applied Dynamics"],"dc:title":["An Adaptive Method for Calculating Blow-Up Solutions"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:35:00Z"}