A Cotangent Fractional Derivative with the Application
In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/6/444 |
| _version_ | 1827737229405978624 |
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| author | Lakhlifa Sadek |
| author_facet | Lakhlifa Sadek |
| author_sort | Lakhlifa Sadek |
| collection | DOAJ |
| description | In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>D</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula> and Caputo cotangent fractional derivatives <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mi>C</mi></msup><msup><mi>D</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula>, respectively, and their corresponding integral <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>I</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula>. The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> we obtain the Riemann–Liouville FD (RL-FD), Caputo FD (C-FD), and Riemann–Liouville fractional integral (RL-FI). We give some theorems and lemmas, and we give solutions to linear cotangent fractional differential equations using the Laplace transform of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>D</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mi>C</mi></msup><msup><mi>D</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>I</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula>. Finally, we give the application of this new type on the SIR model. This new type of fractional calculus can help other researchers who still work on the actual subject. |
| first_indexed | 2024-03-11T02:27:16Z |
| format | Article |
| id | doaj.art-2363b68e27bd4b748571fc76f087e9cb |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T02:27:16Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-2363b68e27bd4b748571fc76f087e9cb2023-11-18T10:29:22ZengMDPI AGFractal and Fractional2504-31102023-05-017644410.3390/fractalfract7060444A Cotangent Fractional Derivative with the ApplicationLakhlifa Sadek0Department of Mathematics, Faculty of Sciences, Chouaib Doukkali University, El Jadida 24000, MoroccoIn this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>D</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula> and Caputo cotangent fractional derivatives <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mi>C</mi></msup><msup><mi>D</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula>, respectively, and their corresponding integral <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>I</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula>. The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> we obtain the Riemann–Liouville FD (RL-FD), Caputo FD (C-FD), and Riemann–Liouville fractional integral (RL-FI). We give some theorems and lemmas, and we give solutions to linear cotangent fractional differential equations using the Laplace transform of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>D</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mi>C</mi></msup><msup><mi>D</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>I</mi><mrow><mi>σ</mi><mo>,</mo><mi>γ</mi></mrow></msup></semantics></math></inline-formula>. Finally, we give the application of this new type on the SIR model. This new type of fractional calculus can help other researchers who still work on the actual subject.https://www.mdpi.com/2504-3110/7/6/444cotangent fractional integralcotangent fractional derivativeRiemann–Liouville cotangent fractional derivativeCaputo cotangent fractional derivative |
| spellingShingle | Lakhlifa Sadek A Cotangent Fractional Derivative with the Application Fractal and Fractional cotangent fractional integral cotangent fractional derivative Riemann–Liouville cotangent fractional derivative Caputo cotangent fractional derivative |
| title | A Cotangent Fractional Derivative with the Application |
| title_full | A Cotangent Fractional Derivative with the Application |
| title_fullStr | A Cotangent Fractional Derivative with the Application |
| title_full_unstemmed | A Cotangent Fractional Derivative with the Application |
| title_short | A Cotangent Fractional Derivative with the Application |
| title_sort | cotangent fractional derivative with the application |
| topic | cotangent fractional integral cotangent fractional derivative Riemann–Liouville cotangent fractional derivative Caputo cotangent fractional derivative |
| url | https://www.mdpi.com/2504-3110/7/6/444 |
| work_keys_str_mv | AT lakhlifasadek acotangentfractionalderivativewiththeapplication AT lakhlifasadek cotangentfractionalderivativewiththeapplication |