Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus
We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-05-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/5/2/43 |
| _version_ | 1797534820635508736 |
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| author | Gerd Baumann |
| author_facet | Gerd Baumann |
| author_sort | Gerd Baumann |
| collection | DOAJ |
| description | We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order. |
| first_indexed | 2024-03-10T11:34:53Z |
| format | Article |
| id | doaj.art-aea91f0222464e938ca63bb66ba9ba9c |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T11:34:53Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-aea91f0222464e938ca63bb66ba9ba9c2023-11-21T18:56:58ZengMDPI AGFractal and Fractional2504-31102021-05-01524310.3390/fractalfract5020043Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional CalculusGerd Baumann0Mathematics Department, German University in Cairo, New Cairo City, EgyptWe shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.https://www.mdpi.com/2504-3110/5/2/43Sinc methodsinverse Laplace transformindefinite integralsfractional calculusMittag-Leffler functionPrabhakar function |
| spellingShingle | Gerd Baumann Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus Fractal and Fractional Sinc methods inverse Laplace transform indefinite integrals fractional calculus Mittag-Leffler function Prabhakar function |
| title | Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus |
| title_full | Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus |
| title_fullStr | Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus |
| title_full_unstemmed | Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus |
| title_short | Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus |
| title_sort | sinc based inverse laplace transforms mittag leffler functions and their approximation for fractional calculus |
| topic | Sinc methods inverse Laplace transform indefinite integrals fractional calculus Mittag-Leffler function Prabhakar function |
| url | https://www.mdpi.com/2504-3110/5/2/43 |
| work_keys_str_mv | AT gerdbaumann sincbasedinverselaplacetransformsmittaglefflerfunctionsandtheirapproximationforfractionalcalculus |