Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains

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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Bibliographic Details
Main Authors: Djelloul Ziane, Mountassir Hamdi Cherif, Carlo Cattani, Abdelhamid Mohammed Djaouti
Format: Article
Language:English
Published: MDPI AG 2025-07-01
Series:Fractal and Fractional
Subjects:
local fractional calculus
Yang–Laplace transform method
local fractional partial differential equations
fractal domains
Online Access:https://www.mdpi.com/2504-3110/9/7/434
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Summary: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.
ISSN:2504-3110

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