Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains
This study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>F</...
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MDPI AG
2025-07-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/7/434 |
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| author | Djelloul Ziane Mountassir Hamdi Cherif Carlo Cattani Abdelhamid Mohammed Djaouti |
| author_facet | Djelloul Ziane Mountassir Hamdi Cherif Carlo Cattani Abdelhamid Mohammed Djaouti |
| author_sort | Djelloul Ziane |
| collection | DOAJ |
| description | This study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>F</mi><msubsup><mi>L</mi><mrow><mi>ζ</mi></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>) and rigorously investigates its foundational properties, including linearity, differentiation, and convolution. The proposed method is formulated via double local fractional integrals, enabling a robust mechanism for addressing local fractional partial differential equations defined on fractal domains, particularly Cantor sets. Through a series of illustrative examples, we demonstrate the applicability and efficacy of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>F</mi><msubsup><mi>L</mi><mrow><mi>ζ</mi></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> transform in solving complex local fractional partial differential equation models. Special emphasis is placed on the local fractional Laplace equation, the linear local fractional Klein–Gordon equation, and other models, wherein the method reveals significant computational and analytical advantages. The results substantiate the method’s potential as a powerful tool for broader classes of problems governed by local fractional dynamics on fractal geometries. |
| format | Article |
| id | doaj-art-259db62191d741778537febda0fb088a |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-259db62191d741778537febda0fb088a2025-08-20T02:45:42ZengMDPI AGFractal and Fractional2504-31102025-07-019743410.3390/fractalfract9070434Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal DomainsDjelloul Ziane0Mountassir Hamdi Cherif1Carlo Cattani2Abdelhamid Mohammed Djaouti3Department of Mathematics, Faculty of Mathematics and Material Sciences, Kasdi Merbah University of Ouargla, Ouargla 30000, AlgeriaLaboratory of Complex Systems, Higher School of Electrical and Energetic Engineering of Oran (ESGEE-Oran), Oran 31000, AlgeriaDepartment of Economics, Engineering, Society and Business Organization (DEIM), University of Tuscia, 01100 Viterbo, ItalyDepartement of Mathematics and Statistics, Faculty of Sciences, King Faisal University, Hofuf 31982, Saudi ArabiaThis study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>F</mi><msubsup><mi>L</mi><mrow><mi>ζ</mi></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>) and rigorously investigates its foundational properties, including linearity, differentiation, and convolution. The proposed method is formulated via double local fractional integrals, enabling a robust mechanism for addressing local fractional partial differential equations defined on fractal domains, particularly Cantor sets. Through a series of illustrative examples, we demonstrate the applicability and efficacy of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>F</mi><msubsup><mi>L</mi><mrow><mi>ζ</mi></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> transform in solving complex local fractional partial differential equation models. Special emphasis is placed on the local fractional Laplace equation, the linear local fractional Klein–Gordon equation, and other models, wherein the method reveals significant computational and analytical advantages. The results substantiate the method’s potential as a powerful tool for broader classes of problems governed by local fractional dynamics on fractal geometries.https://www.mdpi.com/2504-3110/9/7/434local fractional calculusYang–Laplace transform methodlocal fractional partial differential equationsfractal domains |
| spellingShingle | Djelloul Ziane Mountassir Hamdi Cherif Carlo Cattani Abdelhamid Mohammed Djaouti Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains Fractal and Fractional local fractional calculus Yang–Laplace transform method local fractional partial differential equations fractal domains |
| title | Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains |
| title_full | Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains |
| title_fullStr | Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains |
| title_full_unstemmed | Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains |
| title_short | Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains |
| title_sort | double local fractional yang laplace transform for local fractional pdes on fractal domains |
| topic | local fractional calculus Yang–Laplace transform method local fractional partial differential equations fractal domains |
| url | https://www.mdpi.com/2504-3110/9/7/434 |
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