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Theory and Numerics for Some Types o...

Slepoi, Jeffrey.

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Theory and Numerics for Some Types of Fractional Differential Equations.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Theory and Numerics for Some Types of Fractional Differential Equations./
Author:	Slepoi, Jeffrey.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2020,
Description:	128 p.
Notes:	Source: Dissertations Abstracts International, Volume: 82-08, Section: B.
Contained By:	Dissertations Abstracts International82-08B.
Subject:	Applied mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28148277
ISBN:	9798569980574

Theory and Numerics for Some Types of Fractional Differential Equations.
Slepoi, Jeffrey.

Theory and Numerics for Some Types of Fractional Differential Equations. - Ann Arbor : ProQuest Dissertations & Theses, 2020 - 128 p.

Source: Dissertations Abstracts International, Volume: 82-08, Section: B.

Thesis (Ph.D.)--Stevens Institute of Technology, 2020.

This item must not be sold to any third party vendors.

Fractional differential equations (FrDEs) are better suited for the description of properties of various real materials. It has been demonstrated that models with fractional derivatives are often more adequate in description of these properties and processes than the models with integer derivatives. This is the main advantage of the fractional derivative models over the integer derivative models. Physical considerations for the use of models with fractional derivatives are given in many works including [20], [24], [34].A number of methods were developed to numerically solve FrDEs (e.g. [2], [7], [33]). This work contains two methods for numerically solving fractional differential equations, necessity of multiple approaches is demonstrated to assure their validity. The substitution method for Riemann-Liouville and Caputo derivatives is presented and used in this work as a base for all calculations.Solutions for Bessel equation in fractional derivatives were attempted before and the topic interests many scientists (e.g. [14], [19], [26], [29]). A theory is developed in this work on how to solve the generalized fractional Bessel equation, conditions for existence of a solution an its uniqueness for equations with Caputo derivatives are identified and proven. A step further is taken and resolved by expanding the Bessel equation into a more general version, quasi-linear fractional Bessel equation, where the matching of powers and the order of the derivatives is not required for all terms but one.The natural step for the use of the discovered methodology for quasi-linear Bessel equation is its application to the homogeneous equations with constant coefficients and fractional equations with power function factors at each derivative. The simple equations in this domain were considered before [17], the analytical solutions were identified. We expand the class of equations and check how the the methodology achieves the same results for simpler equations for which analytical solutions were previously identified.Some cases of fractional Cauchy Euler equations were addressed in the past ([4], [35], [18]). Generalized fractional Cauchy Euler equations for both Riemann-Liouville and Caputo fractional derivatives are analyzed next in this work. The solutions similar to the classical Cauchy Euler equation are identified. Independence of solutions is proved.

ISBN: 9798569980574Subjects--Topical Terms:

2122814
Applied mathematics.
Subjects--Index Terms:

Fractional Bessel equation

Theory and Numerics for Some Types of Fractional Differential Equations.
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245 1 0 $a Theory and Numerics for Some Types of Fractional Differential Equations.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2020
300 $a 128 p.
500 $a Source: Dissertations Abstracts International, Volume: 82-08, Section: B.
500 $a Advisor: Dubovski, Pavel B.;Boyadjiev, Lyubomir.
502 $a Thesis (Ph.D.)--Stevens Institute of Technology, 2020.
506 $a This item must not be sold to any third party vendors.
520 $a Fractional differential equations (FrDEs) are better suited for the description of properties of various real materials. It has been demonstrated that models with fractional derivatives are often more adequate in description of these properties and processes than the models with integer derivatives. This is the main advantage of the fractional derivative models over the integer derivative models. Physical considerations for the use of models with fractional derivatives are given in many works including [20], [24], [34].A number of methods were developed to numerically solve FrDEs (e.g. [2], [7], [33]). This work contains two methods for numerically solving fractional differential equations, necessity of multiple approaches is demonstrated to assure their validity. The substitution method for Riemann-Liouville and Caputo derivatives is presented and used in this work as a base for all calculations.Solutions for Bessel equation in fractional derivatives were attempted before and the topic interests many scientists (e.g. [14], [19], [26], [29]). A theory is developed in this work on how to solve the generalized fractional Bessel equation, conditions for existence of a solution an its uniqueness for equations with Caputo derivatives are identified and proven. A step further is taken and resolved by expanding the Bessel equation into a more general version, quasi-linear fractional Bessel equation, where the matching of powers and the order of the derivatives is not required for all terms but one.The natural step for the use of the discovered methodology for quasi-linear Bessel equation is its application to the homogeneous equations with constant coefficients and fractional equations with power function factors at each derivative. The simple equations in this domain were considered before [17], the analytical solutions were identified. We expand the class of equations and check how the the methodology achieves the same results for simpler equations for which analytical solutions were previously identified.Some cases of fractional Cauchy Euler equations were addressed in the past ([4], [35], [18]). Generalized fractional Cauchy Euler equations for both Riemann-Liouville and Caputo fractional derivatives are analyzed next in this work. The solutions similar to the classical Cauchy Euler equation are identified. Independence of solutions is proved.
590 $a School code: 0733.
650 4 $a Applied mathematics. $3 2122814
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
653 $a Fractional Bessel equation
653 $a Fractional Cauchy Euler equation
653 $a Fractional differential equations
653 $a Fractional quasi Bessel equation
653 $a Solution existence
653 $a Solution uniqueness
690 $a 0364
690 $a 0405
690 $a 0642
710 2 $a Stevens Institute of Technology. $b Mathematics. $3 2096766
773 0 $t Dissertations Abstracts International $g 82-08B.
790 $a 0733
791 $a Ph.D.
792 $a 2020
793 $a English
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28148277