<p>The inverse source problem for a time-space fractional diffusion equation with Caputo–Hadamard derivative presents significant computational challenges due to its inherent ill-posedness. This research introduces a novel methodology integrating a modified quasi-boundary value regularization approach and a logarithmic reconstruction technique to address these complexities under the fundamental assumption of a priori norm-bounded solutions. Our theoretical framework critically relies on the knowledge of an upper bound <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2533_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2533_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">$\Vert f \Vert _{\mathbb {H}^{\tau }(\Omega )} \le \mathcal {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>τ</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≤</mo> <mi mathvariant="script">M</mi> </mrow> </math></EquationSource> </InlineEquation> for the sought source function, representing an essential constraint for the validity of our error estimation methodology.</p><p>By systematically examining the regularization solution within this norm-constrained framework, we establish rigorous convergence rates through both a priori and a posteriori parameter selection strategies. Leveraging sophisticated mathematical tools including the Fourier method and Sobolev embedding theorem, we derive comprehensive error estimations within <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2533_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L^b$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>b</mi> </msup> </math></EquationSource> </InlineEquation> function spaces. The theoretical framework demonstrates particular robustness when applied to observational data modeled as Gaussian white noise, showcasing both theoretical depth and practical applicability within the specified boundedness constraint. The proposed norm-constrained regularization approach advances mathematical understanding of inverse source problems by providing a sophisticated computational framework for reconstructing source functions in complex spatiotemporal systems, specifically applicable when a priori solution boundedness information is available. This work bridges advanced mathematical theory with practical numerical implementation while explicitly delineating the methodological assumptions essential for theoretical validity and error control.</p>

Inverse source problem for time-fractional diffusion equation: norm-constrained regularization and error estimation under a priori boundedness assumptions

期刊论文
发表日期：

2025年6月27日

查看全文

Le Dinh Long,
Mahmoud A. Zaky,
B. Parsa Moghaddam,
Yusuf Gürefe

摘要

The inverse source problem for a time-space fractional diffusion equation with Caputo–Hadamard derivative presents significant computational challenges due to its inherent ill-posedness. This research introduces a novel methodology integrating a modified quasi-boundary value regularization approach and a logarithmic reconstruction technique to address these complexities under the fundamental assumption of a priori norm-bounded solutions. Our theoretical framework critically relies on the knowledge of an upper bound $\mathcal {M}$ $M$ such that $\Vert f \Vert _{\mathbb {H}^{\tau }(\Omega )} \le \mathcal {M}$ ${‖ f ‖}_{H^{τ} (Ω)} \leq M$ for the sought source function, representing an essential constraint for the validity of our error estimation methodology.

By systematically examining the regularization solution within this norm-constrained framework, we establish rigorous convergence rates through both a priori and a posteriori parameter selection strategies. Leveraging sophisticated mathematical tools including the Fourier method and Sobolev embedding theorem, we derive comprehensive error estimations within $L^b$ $L^{b}$ function spaces. The theoretical framework demonstrates particular robustness when applied to observational data modeled as Gaussian white noise, showcasing both theoretical depth and practical applicability within the specified boundedness constraint. The proposed norm-constrained regularization approach advances mathematical understanding of inverse source problems by providing a sophisticated computational framework for reconstructing source functions in complex spatiotemporal systems, specifically applicable when a priori solution boundedness information is available. This work bridges advanced mathematical theory with practical numerical implementation while explicitly delineating the methodological assumptions essential for theoretical validity and error control.