Fractal Differential Equations of 2<i>α</i>-Order
In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>α...
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MDPI AG
2024-11-01
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| Online Access: | https://www.mdpi.com/2075-1680/13/11/786 |
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| author | Alireza Khalili Golmankhaneh Donatella Bongiorno |
| author_facet | Alireza Khalili Golmankhaneh Donatella Bongiorno |
| author_sort | Alireza Khalili Golmankhaneh |
| collection | DOAJ |
| description | In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>α</mi></mrow></semantics></math></inline-formula>-order fractal differential equation with constant coefficients across different scenarios. We propose a uniqueness theorem for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>α</mi></mrow></semantics></math></inline-formula>-order fractal linear differential equations. We define the solution space as a vector space with non-integer orders. We establish precise conditions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>α</mi></mrow></semantics></math></inline-formula>-order fractal linear differential equations and derive the corresponding fractal adjoint differential equation. |
| format | Article |
| id | doaj-art-6c27ab1cf8d346ff934fe2daf58191b2 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-6c27ab1cf8d346ff934fe2daf58191b22025-08-20T01:53:49ZengMDPI AGAxioms2075-16802024-11-01131178610.3390/axioms13110786Fractal Differential Equations of 2<i>α</i>-OrderAlireza Khalili Golmankhaneh0Donatella Bongiorno1Department of Physics, Islamic Azad University, Urmia Branch, Urmia 63896, West Azerbaijan, IranDepartment of Engineering, University of Palermo, 90100 Palermo, ItalyIn this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>α</mi></mrow></semantics></math></inline-formula>-order fractal differential equation with constant coefficients across different scenarios. We propose a uniqueness theorem for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>α</mi></mrow></semantics></math></inline-formula>-order fractal linear differential equations. We define the solution space as a vector space with non-integer orders. We establish precise conditions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>α</mi></mrow></semantics></math></inline-formula>-order fractal linear differential equations and derive the corresponding fractal adjoint differential equation.https://www.mdpi.com/2075-1680/13/11/786fractal calculus2α-order fractal linear differential equationsfractal adjoint differential equation |
| spellingShingle | Alireza Khalili Golmankhaneh Donatella Bongiorno Fractal Differential Equations of 2<i>α</i>-Order Axioms fractal calculus 2α-order fractal linear differential equations fractal adjoint differential equation |
| title | Fractal Differential Equations of 2<i>α</i>-Order |
| title_full | Fractal Differential Equations of 2<i>α</i>-Order |
| title_fullStr | Fractal Differential Equations of 2<i>α</i>-Order |
| title_full_unstemmed | Fractal Differential Equations of 2<i>α</i>-Order |
| title_short | Fractal Differential Equations of 2<i>α</i>-Order |
| title_sort | fractal differential equations of 2 i α i order |
| topic | fractal calculus 2α-order fractal linear differential equations fractal adjoint differential equation |
| url | https://www.mdpi.com/2075-1680/13/11/786 |
| work_keys_str_mv | AT alirezakhaliligolmankhaneh fractaldifferentialequationsof2iaiorder AT donatellabongiorno fractaldifferentialequationsof2iaiorder |