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<p>This paper combines the concepts of the 1st-level fractional derivative, the general fractional calculus with Sonine kernels, and the weighted fractional calculus with respect to functions. We introduce a generalized formulation that includes all of these types of fractional calculus by incorporating function-based transmutation with Sonine kernels and 1st-level fractional derivatives. This extends the concept of the 1st-level general fractional derivative, the concept of weighted general fractional derivatives with respect to functions, and the concept of weighted 1st-level fractional derivatives with respect to functions, all of which have been introduced in the recent literature. We explore the new generalized fractional operators and their inversion relations, establishing a unified framework that encompasses various existing definitions in fractional calculus.</p>

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Weighted 1st-level fractional derivatives with Sonine Kernels with respect to functions

  • 期刊论文

  • 发表日期：

    2025年4月10日

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  • Mahendrakumar N. Dabhi,
  • Krunal B. Kachhia

摘要

This paper combines the concepts of the 1st-level fractional derivative, the general fractional calculus with Sonine kernels, and the weighted fractional calculus with respect to functions. We introduce a generalized formulation that includes all of these types of fractional calculus by incorporating function-based transmutation with Sonine kernels and 1st-level fractional derivatives. This extends the concept of the 1st-level general fractional derivative, the concept of weighted general fractional derivatives with respect to functions, and the concept of weighted 1st-level fractional derivatives with respect to functions, all of which have been introduced in the recent literature. We explore the new generalized fractional operators and their inversion relations, establishing a unified framework that encompasses various existing definitions in fractional calculus.