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<p>In this work, we present the quantum Adams-Bashforth-Moulton fractional method for solving the Caputo quantum fractional differential equation on the time scale set <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2509_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{T}_q\)</EquationSource> </InlineEquation><Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2509_Article_Equa.gif" Format="GIF" Height="54" Rendition="HTML" Resolution="72" Type="Linedraw" Width="309" /> </MediaObject> <EquationSource Format="TEX">\(\left\{ \begin{array}{ll}^cD_{q}^{\varpi}\Xi(\Theta)&amp;=f(\Theta, \Xi(\Theta)), \\ \Xi^{(k)}(a)&amp;=\Xi_a^{(k)}, \ k=0,1, \ldots, \lceil\varpi\rceil-1, \end{array}\right.\)</EquationSource> </Equation>&#xa0;&#xa0;</p><p>with <i>ϖ</i> &gt; 0, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2509_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(0 &lt; q &lt; 1\)</EquationSource> </InlineEquation> and the differential operator is the Caputo type quantum derivative. We study the stability of the solution, also give a detailed error analysis. Finally, numerical examples including linear and nonlinear are provided to illustrate the robustness of our method.</p>

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The quantum Adams-Bashforth-Moulton fractional method for solving quantum fractional differential equations

  • 期刊论文

  • 发表日期：

    2025年5月14日

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  • Said Chablaoui,
  • Lakhlifa Sadek,
  • El Mostafa Sadek

摘要

In this work, we present the quantum Adams-Bashforth-Moulton fractional method for solving the Caputo quantum fractional differential equation on the time scale set \(\mathbb{T}_q\) \(\left\{ \begin{array}{ll}^cD_{q}^{\varpi}\Xi(\Theta)&=f(\Theta, \Xi(\Theta)), \\ \Xi^{(k)}(a)&=\Xi_a^{(k)}, \ k=0,1, \ldots, \lceil\varpi\rceil-1, \end{array}\right.\)   

with ϖ > 0, \(0 < q < 1\) and the differential operator is the Caputo type quantum derivative. We study the stability of the solution, also give a detailed error analysis. Finally, numerical examples including linear and nonlinear are provided to illustrate the robustness of our method.