Khan Hira, Rahman Gauhar, Samraiz Muhammad, Shah Kamal, Abdeljawad Thabet
Department of Mathematics and Statistics, Hazara University, Mansehra, 21300, Pakistan.
Department of Mathematics, University of Sargodha, P.O. Box 40100, Sargodha, Pakistan.
Heliyon. 2025 Jan 23;11(4):e42144. doi: 10.1016/j.heliyon.2025.e42144. eCollection 2025 Feb 28.
In recent years, the Atangana-Baleanu (AB) fractal-fractional derivatives are widely used in many fields. In 2017, Atangana defined such operators by utilizing one parameter Mittag-Leffler function (M-L) function. Such operators have not yet been studied for three parameters M-L function. In this paper, we discuss further modifications of Caputo Fabrizio (CF), AB and generalized Hattaf fractal-fractional (GHF) operators. We used the modified three parameters M-L function to define the generalized fractal-fractional (GFF) differential and integral operators. We study an innovative class of new generalized weighted differential and integral operators. We define the generalized fractal-fractional (GFF) differential and integral operators with generalized Mittag-Leffler (M-L) kernels, which are used to simulate the complex dynamics of several natural and physical phenomena in a variety of scientific and engineering domains. There are a few established features of the newly defined operators. An example of an application for this new class of GFF integral is presented. Also, we discussed the graphical comparison of this new GFF operator with the existing GHF, AB and CF derivatives. Our case is the more general case compared with the existing fractal-fractional operators. We have presented some novel results for the new operators both analytically and graphically. Also, we discussed some special cases by giving specific value to the parameter . All the classical operators are restored by applying certain conditions on parameters.
近年来，阿坦加纳 - 巴莱亚努（AB）分形 - 分数阶导数在许多领域得到广泛应用。2017年，阿坦加纳通过利用单参数米塔格 - 莱夫勒（M - L）函数定义了此类算子。此类算子尚未针对三参数M - L函数进行研究。在本文中，我们讨论了卡普托·法布里齐奥（CF）、AB和广义哈塔夫分形 - 分数阶（GHF）算子的进一步修正。我们使用修正后的三参数M - L函数来定义广义分形 - 分数阶（GFF）微分和积分算子。我们研究了一类创新的新型广义加权微分和积分算子。我们定义了具有广义米塔格 - 莱夫勒（M - L）核的广义分形 - 分数阶（GFF）微分和积分算子，其用于模拟各种科学和工程领域中几种自然和物理现象的复杂动力学。新定义的算子有一些既定特征。给出了这类新型GFF积分的一个应用示例。此外，我们讨论了这种新型GFF算子与现有GHF、AB和CF导数的图形比较。与现有的分形 - 分数阶算子相比，我们的情况更具一般性。我们从解析和图形两方面给出了新算子的一些新颖结果。此外，我们通过给参数赋予特定值讨论了一些特殊情况。通过对参数施加某些条件可恢复所有经典算子。
