Mittag–Leffler Memory Kernel in Lévy Flights
In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag−Leffler kernel in the context of Atangana−Baleanu operator. The Lévy process has several application...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/9/766 |
| _version_ | 1818147269123244032 |
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| author | Maike A. F. dos Santos |
| author_facet | Maike A. F. dos Santos |
| author_sort | Maike A. F. dos Santos |
| collection | DOAJ |
| description | In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag−Leffler kernel in the context of Atangana−Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana−Baleanu in time derivative, i.e., <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mrow></mrow> <mrow> <mspace width="0.277778em"></mspace> <mspace width="0.277778em"></mspace> <mi>a</mi> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="script">D</mi> <mi>t</mi> <mi>α</mi> </msubsup> <mi>ρ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="script">K</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mspace width="4pt"></mspace> <msubsup> <mi>∂</mi> <mi>x</mi> <mi>μ</mi> </msubsup> <mi>ρ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. We present the exact solution to model and discuss how the Mittag−Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems. |
| first_indexed | 2024-12-11T12:32:33Z |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-11T12:32:33Z |
| publishDate | 2019-08-01 |
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| spelling | doaj.art-453ad5b5d2a94ad8b8bca545a49678442022-12-22T01:07:12ZengMDPI AGMathematics2227-73902019-08-017976610.3390/math7090766math7090766Mittag–Leffler Memory Kernel in Lévy FlightsMaike A. F. dos Santos0Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems–Rua Dr. Xavier Sigaud 150, Rio de Janeiro 22290-180, BrazilIn this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag−Leffler kernel in the context of Atangana−Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana−Baleanu in time derivative, i.e., <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mrow></mrow> <mrow> <mspace width="0.277778em"></mspace> <mspace width="0.277778em"></mspace> <mi>a</mi> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="script">D</mi> <mi>t</mi> <mi>α</mi> </msubsup> <mi>ρ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="script">K</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mspace width="4pt"></mspace> <msubsup> <mi>∂</mi> <mi>x</mi> <mi>μ</mi> </msubsup> <mi>ρ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. We present the exact solution to model and discuss how the Mittag−Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems.https://www.mdpi.com/2227-7390/7/9/766fractional calculuscontinuous time random walksLévy processexact solutions |
| spellingShingle | Maike A. F. dos Santos Mittag–Leffler Memory Kernel in Lévy Flights Mathematics fractional calculus continuous time random walks Lévy process exact solutions |
| title | Mittag–Leffler Memory Kernel in Lévy Flights |
| title_full | Mittag–Leffler Memory Kernel in Lévy Flights |
| title_fullStr | Mittag–Leffler Memory Kernel in Lévy Flights |
| title_full_unstemmed | Mittag–Leffler Memory Kernel in Lévy Flights |
| title_short | Mittag–Leffler Memory Kernel in Lévy Flights |
| title_sort | mittag leffler memory kernel in levy flights |
| topic | fractional calculus continuous time random walks Lévy process exact solutions |
| url | https://www.mdpi.com/2227-7390/7/9/766 |
| work_keys_str_mv | AT maikeafdossantos mittaglefflermemorykernelinlevyflights |