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<p>Fractional differential equations (FDEs) constitute an essential field of mathematics which has been increasingly studied over recent decades, hence the importance of finding solutions for such systems. In this paper, we prove the existence and the uniqueness of the mild solution for a class of time-fractional evolution systems with an order of differentiation <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1897_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(1&lt;\alpha &lt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>α</mi> <mo>&lt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We make use of different properties of the cosine and sine families as well as the Laplace transform (LT) to obtain a simplified expression of the mild solution, which will be given as an integral formula involving the Mainardi’s wright-type function. Moreover, many useful properties related to the operator appearing in the mild solution are given with proofs. Finally, we provide an illustrative example along with the expression for its mild solution.</p>

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Existence and Uniqueness Results for Mild Solutions of Riemann–Liouville Time-Fractional Systems With Order \(1<\alpha <2\)

  • 期刊论文

  • 发表日期：

    2025年4月18日

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  • Hamza Ben Brahim,
  • Khalid Zguaid,
  • Fatima-Zahrae El Alaoui

摘要

Fractional differential equations (FDEs) constitute an essential field of mathematics which has been increasingly studied over recent decades, hence the importance of finding solutions for such systems. In this paper, we prove the existence and the uniqueness of the mild solution for a class of time-fractional evolution systems with an order of differentiation \(1<\alpha <2\) 1 < α < 2 . We make use of different properties of the cosine and sine families as well as the Laplace transform (LT) to obtain a simplified expression of the mild solution, which will be given as an integral formula involving the Mainardi’s wright-type function. Moreover, many useful properties related to the operator appearing in the mild solution are given with proofs. Finally, we provide an illustrative example along with the expression for its mild solution.