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Algorithms for Solving Linear Differential Equations with Rational Function Coefficients

Title: Algorithms for Solving Linear Differential Equations with Rational Function Coefficients.
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Name(s): Imamoglu, Erdal, author
van Hoeij, Mark, professor directing dissertation
van Engelen, Robert, university representative
Agashe, Amod S. (Amod Sadanand), committee member
Aldrovandi, Ettore, committee member
Aluffi, Paolo, 1960-, committee member
Florida State University, degree granting institution
College of Arts and Sciences, degree granting college
Department of Mathematics, degree granting department
Type of Resource: text
Genre: Text
Doctoral Thesis
Issuance: monographic
Date Issued: 2017
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource (70 pages)
Language(s): English
Abstract/Description: This thesis introduces two new algorithms to find hypergeometric solutions of second order regular singular differential operators with rational function or polynomial coefficients. Algorithm 3.2.1 searches for solutions of type: exp(∫ r dx) ⋅ ₂F₁ (a₁,a₂;b₁;f) and Algorithm 5.2.1 searches for solutions of type exp(∫ r dx) (r₀ ⋅ ₂F₁(a₁,a₂;b₁;f) + r₁ ⋅ ₂F´₁ (a₁,a₂;b₁;f)) where f, r, r₀, r₁ ∈ ℚ̅(̅x̅)̅ and a₁,a₂,b₁ ∈ ℚ and denotes the Gauss hypergeometric function. The algorithms use modular reduction, Hensel lifting, rational function reconstruction, and rational number reconstruction to do so. Numerous examples from different branches of science (mostly from combinatorics and physics) showed that the algorithms presented in this thesis are very effective. Presently, Algorithm 5.2.1 is the most general algorithm in the literature to find hypergeometric solutions of such operators. This thesis also introduces a fast algorithm (Algorithm 4.2.3) to find integral bases for arbitrary order regular singular differential operators with rational function or polynomial coefficients. A normalized (Algorithm 4.3.1) integral basis for a differential operator provides us transformations that convert the differential operator to its standard forms (Algorithm 5.1.1) which are easier to solve.
Identifier: FSU_SUMMER2017_Imamoglu_fsu_0071E_13942 (IID)
Submitted Note: A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Summer Semester 2017.
Date of Defense: May 25, 2017.
Keywords: Differential Operators, Hypergeometric Solutions, Integral Bases
Bibliography Note: Includes bibliographical references.
Advisory Committee: Mark van Hoeij, Professor Directing Dissertation; Robert A. van Engelen, University Representative; Amod S. Agashe, Committee Member; Ettore Aldrovandi, Committee Member; Paolo B. Aluffi, Committee Member.
Subject(s): Mathematics
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_SUMMER2017_Imamoglu_fsu_0071E_13942
Owner Institution: FSU

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Imamoglu, E. (2017). Algorithms for Solving Linear Differential Equations with Rational Function Coefficients. Retrieved from http://purl.flvc.org/fsu/fd/FSU_SUMMER2017_Imamoglu_fsu_0071E_13942