{"id":{"repo_id":"wku-diss","oai_identifier":"oai:digitalcommons.wku.edu:theses-1163"},"canonical_url":"https://search.dev.ndltd.org/etd/wku-diss/oai:digitalcommons.wku.edu:theses-1163","repository":{"repo_id":"wku-diss","name":"Western Kentucky University","base_url":"https://digitalcommons.wku.edu/do/oai/"},"display":{"title":"Discrete Fractional Calculus and Its Applications to Tumor Growth","abstract":"Almost every theory of mathematics has its discrete counterpart that makes it conceptually easier to understand and practically easier to use in the modeling process of real world problems. For instance, one can take the \"difference\" of any function, from 1st order up to the <i>n</i>-th order with discrete calculus. However, it is also possible to extend this theory by means of discrete fractional calculus and make <i>n</i>- any real number such that the ½-th order difference is well defined. This thesis is comprised of five chapters that demonstrate some basic definitions and properties of discrete fractional calculus while developing the simplest discrete fractional variational theory. Some applications of the theory to tumor growth are also studied. <br> The first chapter is a brief introduction to discrete fractional calculus that presents some important mathematical functions widely used in the theory. The second chapter shows the main fractional difference and sum operators as well as their important properties. In the third chapter, a new proof for Leibniz formula is given and summation by parts for discrete fractional calculus is stated and proved. The simplest variational problem in discrete calculus and the related Euler-Lagrange equation are developed in the fourth chapter. In the fifth chapter, the fractional Gompertz difference equation is introduced. <br> First, the existence and uniqueness of the solution is shown and then the equation is solved by the method of successive approximation. Finally, applications of the theory to tumor and bacterial growth are presented.","abstract_html":"Almost every theory of mathematics has its discrete counterpart that makes it conceptually easier to understand and practically easier to use in the modeling process of real world problems. For instance, one can take the &quot;difference&quot; of any function, from 1st order up to the &lt;i&gt;n&lt;/i&gt;-th order with discrete calculus. However, it is also possible to extend this theory by means of discrete fractional calculus and make &lt;i&gt;n&lt;/i&gt;- any real number such that the ½-th order difference is well defined. This thesis is comprised of five chapters that demonstrate some basic definitions and properties of discrete fractional calculus while developing the simplest discrete fractional variational theory. Some applications of the theory to tumor growth are also studied. &lt;br&gt; The first chapter is a brief introduction to discrete fractional calculus that presents some important mathematical functions widely used in the theory. The second chapter shows the main fractional difference and sum operators as well as their important properties. In the third chapter, a new proof for Leibniz formula is given and summation by parts for discrete fractional calculus is stated and proved. The simplest variational problem in discrete calculus and the related Euler-Lagrange equation are developed in the fourth chapter. In the fifth chapter, the fractional Gompertz difference equation is introduced. &lt;br&gt; First, the existence and uniqueness of the solution is shown and then the equation is solved by the method of successive approximation. Finally, applications of the theory to tumor and bacterial growth are presented.","abstract_has_math":false,"creators":["Sengul, Sevgi"],"institution":null,"degree_name":"Master of Science","degree_level":null,"degree_discipline":"Department of Mathematics and Computer Science","degree_department":null,"school":null,"contributors":["Dr. Ferhan Atici (Director), Dr. John Spraker, Dr. Mark Robinson, Dr. Nancy Rice"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-05-01T07:00:00Z","date_published":"2010-05-01T07:00:00Z","updated_at":"2026-07-24T06:07:10Z","subjects":["Leibniz formula","Euler-Lagrange equation","Gompertz difference equation","tumor and bacterial growth","Cell Biology","Discrete Mathematics and Combinatorics","Other Applied Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/161","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dr. Ferhan Atici (Director), Dr. John Spraker, Dr. Mark Robinson, Dr. Nancy Rice"]},{"key":"dc:creator","label":"Author","values":["Sengul, Sevgi"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics and Computer Science"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Leibniz formula","Euler-Lagrange equation","Gompertz difference equation","tumor and bacterial growth","Cell Biology","Discrete Mathematics and Combinatorics","Other Applied Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wku.edu/theses/161"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Almost every theory of mathematics has its discrete counterpart that makes it conceptually easier to understand and practically easier to use in the modeling process of real world problems. For instance, one can take the \"difference\" of any function, from 1st order up to the <i>n</i>-th order with discrete calculus. However, it is also possible to extend this theory by means of discrete fractional calculus and make <i>n</i>- any real number such that the ½-th order difference is well defined. This thesis is comprised of five chapters that demonstrate some basic definitions and properties of discrete fractional calculus while developing the simplest discrete fractional variational theory. Some applications of the theory to tumor growth are also studied. <br> The first chapter is a brief introduction to discrete fractional calculus that presents some important mathematical functions widely used in the theory. The second chapter shows the main fractional difference and sum operators as well as their important properties. In the third chapter, a new proof for Leibniz formula is given and summation by parts for discrete fractional calculus is stated and proved. The simplest variational problem in discrete calculus and the related Euler-Lagrange equation are developed in the fourth chapter. In the fifth chapter, the fractional Gompertz difference equation is introduced. <br> First, the existence and uniqueness of the solution is shown and then the equation is solved by the method of successive approximation. Finally, applications of the theory to tumor and bacterial growth are presented."]},{"key":"dc:title","label":"Title","values":["Discrete Fractional Calculus and Its Applications to Tumor Growth"]}]}],"canonical_facts":{"dc:contributor":["Dr. Ferhan Atici (Director), Dr. John Spraker, Dr. Mark Robinson, Dr. Nancy Rice"],"dc:creator":["Sengul, Sevgi"],"dc:description.abstract":["Almost every theory of mathematics has its discrete counterpart that makes it conceptually easier to understand and practically easier to use in the modeling process of real world problems. For instance, one can take the \"difference\" of any function, from 1st order up to the <i>n</i>-th order with discrete calculus. However, it is also possible to extend this theory by means of discrete fractional calculus and make <i>n</i>- any real number such that the ½-th order difference is well defined. This thesis is comprised of five chapters that demonstrate some basic definitions and properties of discrete fractional calculus while developing the simplest discrete fractional variational theory. Some applications of the theory to tumor growth are also studied. <br> The first chapter is a brief introduction to discrete fractional calculus that presents some important mathematical functions widely used in the theory. The second chapter shows the main fractional difference and sum operators as well as their important properties. In the third chapter, a new proof for Leibniz formula is given and summation by parts for discrete fractional calculus is stated and proved. The simplest variational problem in discrete calculus and the related Euler-Lagrange equation are developed in the fourth chapter. In the fifth chapter, the fractional Gompertz difference equation is introduced. <br> First, the existence and uniqueness of the solution is shown and then the equation is solved by the method of successive approximation. Finally, applications of the theory to tumor and bacterial growth are presented."],"dc:identifier":["https://digitalcommons.wku.edu/theses/161"],"dc:subject":["Leibniz formula","Euler-Lagrange equation","Gompertz difference equation","tumor and bacterial growth","Cell Biology","Discrete Mathematics and Combinatorics","Other Applied Mathematics"],"dc:title":["Discrete Fractional Calculus and Its Applications to Tumor Growth"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics and Computer Science"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:07:10Z"}