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<p>Although the theory of fractional operators has numerous definitions in the literature, it is not easy to know which operator is best used for it. One way to try to get around this problem is to propose more general operators where, based on the choice of parameters involved in this new operator, it is possible to obtain the maximum number of definitions of fractional operators in particular cases. This paper is concerned with the calculus of generalized <i>k</i>-fractional derivatives with respective to monotonic functions namely <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_678_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\((k, \Phi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Riemann-Liouville fractional derivative, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_678_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\((k, \Phi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Caputo fractional derivative and most generalized one <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_678_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\((k, \Phi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Hilfer fractional derivative. In this sense, we discuss a wide class of important results in the area and deal with particular cases and important comments for the work.</p>

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Calculus of k-fractional derivative with respect to monotonic functions

  • 期刊论文

  • 发表日期：

    2025年1月30日

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  • Ashwini D. Mali,
  • Kishor D. Kucche,
  • J. Vanterler C. Sousa

摘要

Although the theory of fractional operators has numerous definitions in the literature, it is not easy to know which operator is best used for it. One way to try to get around this problem is to propose more general operators where, based on the choice of parameters involved in this new operator, it is possible to obtain the maximum number of definitions of fractional operators in particular cases. This paper is concerned with the calculus of generalized k-fractional derivatives with respective to monotonic functions namely \((k, \Phi )\) ( k , Φ ) -Riemann-Liouville fractional derivative, \((k, \Phi )\) ( k , Φ ) -Caputo fractional derivative and most generalized one \((k, \Phi )\) ( k , Φ ) -Hilfer fractional derivative. In this sense, we discuss a wide class of important results in the area and deal with particular cases and important comments for the work.