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<p>The solution and source term of nonlinear fractional differential equations (NFDEs) with initial values generally have the initial singularity. As is known that numerical methods for NFDEs usually occur the phenomenon of order reduction due to the existence of initial singularity. In this paper, an improved fractional predictor-corrector (PC) method is developed for NFDEs based on the technique of variable transformation. This improved fractional PC method can achieve the optimal convergence order, i.e., the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_371_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(1+\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>+</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> order convergence rate for fractional order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_371_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, of the classical fractional PC method under the high smoothness requirement on the solution and source term. Furthermore, the detailed error analysis also exhibits the relationship between the convergence rate of the improved fractional PC method and the regularities of the solution and source term. Finally, the theoretical error estimate is verified through numerical experiments.</p>

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An improved fractional predictor-corrector method for nonlinear fractional differential equations with initial singularity

  • 期刊论文

  • 发表日期：

    2025年1月17日

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  • Jianfei Huang,
  • Junlan Lv,
  • Sadia Arshad

摘要

The solution and source term of nonlinear fractional differential equations (NFDEs) with initial values generally have the initial singularity. As is known that numerical methods for NFDEs usually occur the phenomenon of order reduction due to the existence of initial singularity. In this paper, an improved fractional predictor-corrector (PC) method is developed for NFDEs based on the technique of variable transformation. This improved fractional PC method can achieve the optimal convergence order, i.e., the \(1+\alpha \) 1 + α order convergence rate for fractional order \(\alpha \in (0,1)\) α ∈ ( 0 , 1 ) , of the classical fractional PC method under the high smoothness requirement on the solution and source term. Furthermore, the detailed error analysis also exhibits the relationship between the convergence rate of the improved fractional PC method and the regularities of the solution and source term. Finally, the theoretical error estimate is verified through numerical experiments.