The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation
An elastic beam equation (EBEq) described by a fourth-order fractional difference equation is proposed in this work with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. New sufficient conditions ensuring the solutions’ existence and uniqueness of the p...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/5/789 |
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| author | Jehad Alzabut A. George Maria Selvam R. Dhineshbabu Mohammed K. A. Kaabar |
| author_facet | Jehad Alzabut A. George Maria Selvam R. Dhineshbabu Mohammed K. A. Kaabar |
| author_sort | Jehad Alzabut |
| collection | DOAJ |
| description | An elastic beam equation (EBEq) described by a fourth-order fractional difference equation is proposed in this work with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. New sufficient conditions ensuring the solutions’ existence and uniqueness of the proposed problem are established. The findings are obtained by employing properties of discrete fractional equations, Banach contraction, and Brouwer fixed-point theorems. Further, we discuss our problem’s results concerning <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>yers–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula>lam (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">HU</mi></semantics></math></inline-formula>), generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>yers–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula>lam (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">GHU</mi></semantics></math></inline-formula>), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>yers–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula>lam–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>assias (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">HUR</mi></semantics></math></inline-formula>), and generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>yers–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula>lam–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>assias (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">GHUR</mi></semantics></math></inline-formula>) stability. Specific examples with graphs and numerical experiment are presented to demonstrate the effectiveness of our results. |
| first_indexed | 2024-03-10T11:44:23Z |
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| id | doaj.art-80f2ab81569b41bbae4f5a78e0a656b7 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T11:44:23Z |
| publishDate | 2021-05-01 |
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| series | Symmetry |
| spelling | doaj.art-80f2ab81569b41bbae4f5a78e0a656b72023-11-21T18:13:52ZengMDPI AGSymmetry2073-89942021-05-0113578910.3390/sym13050789The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam EquationJehad Alzabut0A. George Maria Selvam1R. Dhineshbabu2Mohammed K. A. Kaabar3Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaDepartment of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635 601, Tamil Nadu, IndiaDepartment of Mathematics, Sri Venkateswara College of Engineering and Technology (Autonomous), Chittoor 517 127, Andhra Pradesh, IndiaDepartment of Mathematics and Statistics, Washington State University, Pullman, WA 99163, USAAn elastic beam equation (EBEq) described by a fourth-order fractional difference equation is proposed in this work with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. New sufficient conditions ensuring the solutions’ existence and uniqueness of the proposed problem are established. The findings are obtained by employing properties of discrete fractional equations, Banach contraction, and Brouwer fixed-point theorems. Further, we discuss our problem’s results concerning <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>yers–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula>lam (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">HU</mi></semantics></math></inline-formula>), generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>yers–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula>lam (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">GHU</mi></semantics></math></inline-formula>), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>yers–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula>lam–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>assias (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">HUR</mi></semantics></math></inline-formula>), and generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>yers–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">U</mi></semantics></math></inline-formula>lam–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>assias (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">GHUR</mi></semantics></math></inline-formula>) stability. Specific examples with graphs and numerical experiment are presented to demonstrate the effectiveness of our results.https://www.mdpi.com/2073-8994/13/5/789Riemann–Liouville fractional difference operatorboundary value problemdiscrete fractional calculusexistence and uniquenessUlam stabilityelastic beam problem |
| spellingShingle | Jehad Alzabut A. George Maria Selvam R. Dhineshbabu Mohammed K. A. Kaabar The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation Symmetry Riemann–Liouville fractional difference operator boundary value problem discrete fractional calculus existence and uniqueness Ulam stability elastic beam problem |
| title | The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation |
| title_full | The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation |
| title_fullStr | The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation |
| title_full_unstemmed | The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation |
| title_short | The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation |
| title_sort | existence uniqueness and stability analysis of the discrete fractional three point boundary value problem for the elastic beam equation |
| topic | Riemann–Liouville fractional difference operator boundary value problem discrete fractional calculus existence and uniqueness Ulam stability elastic beam problem |
| url | https://www.mdpi.com/2073-8994/13/5/789 |
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