Alazman Ibtehal, Ibrahim Rabha W
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia.
Department of Mathematics, Near East Boulevard, Mathematics Research Center, Near East University, PC: 99138, Mersin, Nicosia 10, Turkey.
MethodsX. 2024 Mar 28;12:102684. doi: 10.1016/j.mex.2024.102684. eCollection 2024 Jun.
Numerous special functions, including the beta function, hypergeometric functions, and other orthogonal polynomials, are closely connected to the gamma function. Recently, gamma function has been enhanced by adding a new parameter. As a consequence, this gamma is called the parametric gamma function or b-gamma function. By utilizing the enhanced gamma function (or the parametric gamma function), we have present a generalization for the special function Rabotnov function. Consequently, new fractal-fractional operators (derivative and integral) involving the generalized Rabotnov function are defined. Analysis is introduced to discover the main properties of the suggested operators. Since the main challenge in the calculus of fractal-fractional operators is to design examples, we illustrate a set of examples including power series. We explore the boundedness of the recommended operators. There are many gains for checking the boundedness of fractal operators in general and fractal-fractional operators in particular. Moreover, as an application, we establish the existence and uniqueness solution of abstract fractal-fractional equation. Examples are presented at the end of the effort. Graphics and computations are observed using MATHEMATICA 13.3 software.•By using the parametric gamma function, the Rabotnov special function is generalized. New fractal-fractional operators are presented using the generalized Rabotnov function with examples;•Boundedness of these operators is investigated, where bounded operators give a strong foundation for understanding linear transformations across normed spaces, with many applications in science and mathematics.•Conditions of the existence and uniqueness of solutions of fractal-fractional differential abstract equation are established.
许多特殊函数，包括贝塔函数、超几何函数和其他正交多项式，都与伽马函数密切相关。最近，伽马函数通过添加一个新参数得到了扩展。因此，这种伽马函数被称为参数伽马函数或b - 伽马函数。通过利用扩展后的伽马函数（或参数伽马函数），我们对特殊函数拉博特诺夫函数进行了推广。因此，定义了涉及广义拉博特诺夫函数的新的分形 - 分数阶算子（导数和积分）。引入分析以发现所建议算子的主要性质。由于分形 - 分数阶算子微积分中的主要挑战是设计示例，我们给出了一组包括幂级数的示例。我们探讨了所建议算子的有界性。一般来说，检查分形算子特别是分形 - 分数阶算子的有界性有很多益处。此外，作为一个应用，我们建立了抽象分形 - 分数阶方程解的存在性和唯一性。在工作结束时给出了示例。使用MATHEMATICA 13.3软件进行了图形绘制和计算。
•通过使用参数伽马函数，推广了拉博特诺夫特殊函数。使用广义拉博特诺夫函数给出了新的分形 - 分数阶算子及示例；
•研究了这些算子的有界性，有界算子为理解赋范空间中的线性变换提供了坚实基础，在科学和数学中有许多应用。
•建立了分形 - 分数阶微分抽象方程解的存在性和唯一性条件。
