Fractional Sturm–Liouville Eigenvalue Problems, II
We continue the study of a non-self-adjoint fractional three-term Sturm–Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left Riemann–Liouville fractional integral under <i>Dirichlet type</i> boundary conditions. We study the existen...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-08-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/9/487 |
| _version_ | 1827660840602435584 |
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| author | Mohammad Dehghan Angelo B. Mingarelli |
| author_facet | Mohammad Dehghan Angelo B. Mingarelli |
| author_sort | Mohammad Dehghan |
| collection | DOAJ |
| description | We continue the study of a non-self-adjoint fractional three-term Sturm–Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left Riemann–Liouville fractional integral under <i>Dirichlet type</i> boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, there is a finite set of real eigenvalues and that, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> near <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>, there may be none at all. As <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>→</mo><msup><mn>1</mn><mo>−</mo></msup></mrow></semantics></math></inline-formula> we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm–Liouville problem with the composition of the operators becoming the operator of second order differentiation. |
| first_indexed | 2024-03-09T23:59:09Z |
| format | Article |
| id | doaj.art-3a7eedf2b7f24c7db370fa9ec724f1b6 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T23:59:09Z |
| publishDate | 2022-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-3a7eedf2b7f24c7db370fa9ec724f1b62023-11-23T16:19:28ZengMDPI AGFractal and Fractional2504-31102022-08-016948710.3390/fractalfract6090487Fractional Sturm–Liouville Eigenvalue Problems, IIMohammad Dehghan0Angelo B. Mingarelli1School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, CanadaSchool of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, CanadaWe continue the study of a non-self-adjoint fractional three-term Sturm–Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left Riemann–Liouville fractional integral under <i>Dirichlet type</i> boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, there is a finite set of real eigenvalues and that, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> near <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>, there may be none at all. As <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>→</mo><msup><mn>1</mn><mo>−</mo></msup></mrow></semantics></math></inline-formula> we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm–Liouville problem with the composition of the operators becoming the operator of second order differentiation.https://www.mdpi.com/2504-3110/6/9/487Fractional Sturm–Liouvillefractional calculusLaplace transformMittag-Leffler functioneigenvaluesasymptotics |
| spellingShingle | Mohammad Dehghan Angelo B. Mingarelli Fractional Sturm–Liouville Eigenvalue Problems, II Fractal and Fractional Fractional Sturm–Liouville fractional calculus Laplace transform Mittag-Leffler function eigenvalues asymptotics |
| title | Fractional Sturm–Liouville Eigenvalue Problems, II |
| title_full | Fractional Sturm–Liouville Eigenvalue Problems, II |
| title_fullStr | Fractional Sturm–Liouville Eigenvalue Problems, II |
| title_full_unstemmed | Fractional Sturm–Liouville Eigenvalue Problems, II |
| title_short | Fractional Sturm–Liouville Eigenvalue Problems, II |
| title_sort | fractional sturm liouville eigenvalue problems ii |
| topic | Fractional Sturm–Liouville fractional calculus Laplace transform Mittag-Leffler function eigenvalues asymptotics |
| url | https://www.mdpi.com/2504-3110/6/9/487 |
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