Pith. sign in

REVIEW 3 major objections 7 minor 30 references

Inverse source problem of sub-diffusion of variable exponent

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Local time-series data uniquely fix a diffusion source

desk verdict The inverse-source uniqueness idea is new and probably repairable, but the submitted proof of the central theorem rests on a false Duhamel lemma and should go back for major revision. read the letter →

arxiv 2501.18228 v1 pith:TVKJSW7R submitted 2025-01-30 math.NA cs.NAmath-phmath.MP

Zhiyuan Li , Chunlong Sun , Xiangcheng Zheng This is my paper · ORCID
classification math.NAcs.NAmath-phmath.MP MSC 35R1135R3065M32
keywords inversesourceproblemvariable-exponentsub-diffusionweakuniquecontinuationperturbationmethodfractionaldiffusionLipschitzstabilityiterativeregularization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the variable-exponent sub-diffusion equation, in which the fractional order $\alpha(t)$ changes with time, and asks whether the space-dependent source $f(x)$ can be recovered from observing the solution on a small subdomain over time. The main claim is uniqueness: if the observed solution vanishes on $\Omega_0\times(0,T)$, then $f$ must vanish everywhere in $\Omega$, provided the known time factor $\beta$ is nonzero and integrably differentiable. The paper also proves a conditional Lipschitz stability estimate in a specially designed weak norm, meaning small data errors produce small source errors in that norm. The route is a perturbation method that rewrites the variable-exponent model as a constant-order fractional equation with a convolution correction, followed by analytic continuation, a weak unique-continuation argument, and a variational identity with the adjoint problem. Numerical experiments with iterative regularization show smooth and piecewise-constant sources reconstructed from local interior data.

What carries the argument

The central device is the perturbation method, which rewrites the variable-exponent kernel $t^{-\alpha(t)}/\Gamma(1-\alpha(t))$ as the constant-order kernel $t^{-\alpha_0}/\Gamma(1-\alpha_0)$ plus a correction $\tilde g(t)$, turning (1.1) into the constant-order equation $c\partial^{\alpha_0}_t u-\Delta u+\tilde g'*u=\tilde g u_0+F$. This makes the solution analytically extendable beyond $T$, so Laplace transforms and eigenfunction expansions apply. The uniqueness argument then uses Duhamel's representation $u=\theta*v$ with $J^{1-\alpha_0}\theta=\beta$ and $v$ solving an auxiliary problem with initial value $f$; the weak unique-continuation principle forces $v\equiv 0$, hence $f=0$. Stability is carried by the variational identity $\langle \beta f,\varphi[\omega]\rangle=\langle u[f],\omega\rangle$ connecting the inversion data to a weak norm $\|\cdot\|_B$ built from adjoint solutions.

What would settle it

Substitute $u=\theta*v$ into the first equation of (3.1) and use the auxiliary equation defining $v$; the left side should reduce to $\beta f$, but a direct computation gives $\beta f+\theta*(\tilde g f)$. Whether this extra term is nonzero for a non-constant $\alpha(t)$ and a nonzero $f$, checked symbolically or numerically, settles Lemma 3.1 as stated.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that for the zero-initial-data problem $c\partial^{\alpha_0}_t u-\Delta u+\tilde g'*u=f(x)\beta(t)$ with zero boundary data, measuring $u$ on any nonempty subdomain $\Omega_0$ over the whole time interval determines $f$ uniquely: Theorem 3 states that if $\beta\ne 0$ and $\beta\in W^{1,1}(0,T)$, then $u=0$ on $\Omega_0\times(0,T)$ implies $f=0$ in $\Omega$. Theorem 4 gives the same conclusion from vanishing Neumann data on a subboundary. The stability result (Theorem 5) bounds the difference of two sources in the weak norm $\|\cdot\|_B$ by the $L^1$ difference of their interior data, and Theorem 6 does the same for flux data. The proof chain relies on extending solutions analytically in time to an infinite interval, applying the Laplace transform to get a weak unique-continuation principle, and using Duhamel's principle to carry the observation from $u$ to an auxiliary solution $v$ whose initial value is $f$.

Load-bearing premise

The load-bearing premise is Lemma 3.1's Duhamel representation $u=\theta*v$, with $v$ solving the auxiliary problem whose forcing is $\tilde g f$; if the representation actually produces an extra convolution term $\theta*(\tilde g f)$, the uniqueness and stability results do not follow from the proof as written.

Editorial extensions

If this is right

  • A single interior observation over time is, in principle, enough to determine the whole spatial source; no boundary or full-domain measurement is required.
  • The same uniqueness holds from flux measurements on an open part of the boundary.
  • Noise in the data propagates linearly: the source error in the weak $B$-norm is at most a constant times the $L^1$ data error.
  • Analytic extensibility of solutions brings Laplace-transform and eigenfunction tools to variable-exponent models, matching the toolkit used for constant-order sub-diffusion.
  • The iterative thresholding and Nesterov-type schemes reconstruct smooth and non-smooth sources from local data in the numerical tests.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the uniqueness proof depends on Lemma 3.1's Duhamel representation, the first check for a reader should be whether substituting $u=\theta*v$ into (3.1) leaves an extra term $\theta*(\tilde g f)$; if it does, the theorem needs a revised auxiliary problem rather than the one stated.
  • The analytic-extensibility result suggests a stronger inverse statement than the paper proves: observations on a short time interval, or at finitely many times, may already determine $f$, since analyticity in $t$ propagates local time data.
  • The weak norm $\|\cdot\|_B$ is defined through adjoint solutions and is not what the numerical algorithms minimize; connecting the stability guarantee to the $L^2$-Tikhonov and TV functionals actually used is an open step.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The paper studies the variable-exponent sub-diffusion model (1.1). It first converts the equation to an equivalent constant-exponent form with a convolution perturbation (Lemma 2.1), proves time-analyticity of the solution (Lemma 2.4), and derives a weak unique continuation principle (Theorems 1–2). These tools are then applied to the inverse space-dependent source problem: Theorem 3 claims uniqueness of the source f from interior observations u|_{\Omega_0\times(0,T)} under a nonzero temporal factor \beta(t), Theorem 4 claims the analogous result from boundary flux data, and Theorems 5–6 establish conditional Lipschitz stability in specially defined weak norms. A numerical section tests iterative reconstruction algorithms for smooth and non-smooth sources.

Significance. If the results hold, the paper would make a useful contribution: it extends inverse-source uniqueness and stability theory to sub-diffusion models with time-dependent fractional order, using a tractable perturbation transformation and analytic-continuation techniques. The weak-norm stability estimates are of interest, and the numerical experiments, while heuristic, illustrate the feasibility of the proposed reconstruction approach. However, the central proof currently rests on a false lemma and a formally invalid application of Titchmarsh's theorem, so the theoretical claims are not established in the submitted version.

major comments (3)
  1. [Section 3.1, Lemma 3.1] The auxiliary problem in Lemma 3.1 is stated with right-hand side g̃f, but substituting u = θ*v into (3.1) with v solving that problem yields c∂^{α0}_t u − ∆u + g̃′*u = β(t)f(x) + (θ*(g̃f))(t), because J^{1−α0}θ = β, v(0)=f, and the differential expression applied to v equals g̃f, not zero. Thus the claimed Duhamel representation does not solve (3.1) unless g̃f ≡ 0. The correct statement should take v to solve the homogeneous equation c∂^{α0}_t v − ∆v + g̃′*v = 0 with the same initial and boundary conditions. Since Theorem 3 and the subsequent stability results rely directly on this lemma, the error is load-bearing and must be fixed.
  2. [Section 3.1, proof of Theorem 3] After deriving 0 = β*v on Ω0×(0,T), the paper invokes Titchmarsh's convolution theorem to conclude that v vanishes on all of Ω0×(0,T). The finite-interval form of Titchmarsh's theorem only ensures that there exist a,b ≥ 0 with a+b ≥ T such that β = 0 a.e. on (0,a) and v = 0 a.e. on (0,b); it does not imply v = 0 on the full interval. To make the argument valid, the authors must use the t-analyticity of v supplied by Lemma 2.4 to extend from the subinterval to (0,T). As written, this step is not justified.
  3. [Section 2.3, proof of Theorem 1] The proof introduces η = s^{α0} + s g̃(s) and asserts that the identity Σ (u0,φn)φn/(λn+η) = 0 holds for η in an open set O, but it does not show that the image of the relevant s-domain under η contains an open set. Since η is analytic and nonconstant, this follows from the open mapping theorem after a short argument (e.g., g̃(s)→0 as |s|→∞ while s^{α0} is nonconstant), but the paper omits this justification. Additionally, the final reduction to u0 = 0 is only cited to [19]; it should be stated in enough detail for the reader to verify the argument.
minor comments (7)
  1. [Section 2.1 (title)] The section heading contains a typo: 'well-posdeness' should be 'well-posedness'.
  2. [Lemma 2.1 proof] The sentence 'Notice the zero initial condition' is misleading; the boundary term in the integration by parts vanishes because g̃(0)=0, not because the initial value u0 is zero.
  3. [Section 2.3, Laplace-transformed equation] In the displayed equation following the Laplace transform, a closing bracket is missing: the right-hand side should read (s^{α0−1} + g̃(s))u0.
  4. [Corollary 2.3] The symbol β is used for a power bound in the statement of the corollary, while Section 3 uses β(t) for the temporal factor of the source; this notation clash should be resolved (e.g., by using ρ or γ for the power).
  5. [Theorem 3 proof] The final sentence of the proof reads 'We finish the proof of the lemma' but should be 'the theorem'.
  6. [Section 4] There are small language errors: 'date-match functional' should be 'data-match functional', and 'characterization function' should be 'characteristic function'.
  7. [Lemma 3.1] The existence of θ satisfying J^{1−α0}θ = β for arbitrary β ∈ W^{1,1}(0,T) is asserted without proof or sufficient regularity conditions; a brief justification (e.g., taking θ as an appropriate fractional derivative of β) should be added.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inverse-source uniqueness and stability results are derived from the forward model, adjoint identities, and standard theorems, while the self-citations provide independent prior support.

full rationale

The derivation chain is not circular. Lemma 2.1 establishes an exact equivalence between the original variable-exponent model (1.1) and the transformed form (2.1) by a direct kernel splitting, and this calculation is self-contained. The inverse-source uniqueness in Theorem 3 is obtained through a Duhamel representation, Titchmarsh's convolution theorem, and the weak unique continuation principle; none of these steps assumes the target result. The weak norm in (3.5) is defined through the adjoint solution and the variational identity (3.3), and the Lipschitz estimate in Theorem 5 follows directly from that identity; the norm's nondegeneracy is shown using the already-proved uniqueness rather than being assumed. The self-citations [17], [24], [25], and [29] are used for forward well-posedness, operator estimates, and a norm-construction method, and none of them contains the inverse-source uniqueness or stability statement proved here, so their use is independent support rather than circular import. No fitted parameters are renamed as predictions, and no ansatz is smuggled in by citation. The apparent algebraic inconsistency in the proof of Lemma 3.1, where the stated auxiliary right-hand side g̃f would leave an extra θ*(g̃f) term when substituted into the representation, is a mathematical correctness issue in the submitted proof, not a circular reduction of the conclusion to the hypothesis; the same applies to the finite-interval form of the Titchmarsh step. Because no central claim reduces by construction to its own inputs, the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The theoretical claims introduce no fitted constants. The paper depends on the cited perturbation framework and operator estimates from the same research group, plus an unstated assumption about the existence of θ in the Duhamel representation. No new physical entities are postulated.

free parameters (3)
  • A (iterative thresholding tuning parameter) = 30
    Hand-chosen parameter in the iterative algorithm (Section 4). Not part of the theoretical claim.
  • rho (stopping tolerance) = 2e-4
    Hand-chosen tolerance for terminating the iterative thresholding algorithm (Section 4).
  • tau (discrepancy principle constant) = 1.05
    Hand-chosen parameter in the discrepancy principle for the nonsmooth reconstruction (Section 4).
assumptions (5)
  • domain assumption Assumptions H1 and H2: α ∈ C^3 with bounded derivatives and α analytically extendable to sector |arg z| < θ, |α(z)| ≤ α(0) < 1.
    Used throughout for the perturbation kernel estimates and the analytic extension of the solution (Lemmas 2.2, 2.4).
  • domain assumption The perturbation equivalence (Lemma 2.1) and well-posedness of the transformed problem are assumed from the authors' previous work [29] without reproducing the proof.
    The paper states the equivalence follows and cites [29]; it is the foundation of all subsequent Laplace and UCP arguments.
  • domain assumption Estimates (7) from Li, Imanuvilov and Yamamoto [17] for the operators S1 and S2 are used without proof.
    Used in the induction proof of analytic extensibility in Lemma 2.4.
  • ad hoc to paper For β ∈ W^{1,1}(0,T), there exists θ with J^{1-α0}θ = β.
    The Duhamel representation in Lemma 3.1 requires this, but no proof or condition is given; β ∈ W^{1,1} alone does not guarantee membership in the range of the fractional integral.
  • standard math Titchmarsh convolution theorem and its finite-interval form.
    Invoked in Theorem 3 to conclude v vanishes from β*v=0; the paper applies it more strongly than the finite-interval version supports.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Inverse source problem of sub-diffusion of variable exponent." pith.science (2026). https://pith.science/paper/TVKJSW7R

@misc{pith2026250118228,
  author       = {Pith},
  title        = {Pith review of: Inverse source problem of sub-diffusion of variable exponent},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TVKJSW7R}},
  note         = {Machine review of arXiv:2501.18228}
}
read the original abstract

This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which transfers the original model to an equivalent but more tractable form, the analytical extensibility of the solutions and the weak unique continuation principle are proved, which results in the uniqueness of the inverse space-dependent source problem from local internal observation. Then, based on the variational identity connecting the inversion input data with the unknown source function, we propose a weak norm and prove the conditional stability for the inverse problem in this norm. The iterative thresholding algorithm and Nesterov iteration scheme are employed to numerically reconstruct the smooth and non-smooth sources, respectively. Numerical experiments are performed to investigate their effectiveness.

Figures

Figures reproduced from arXiv: 2501.18228 by the authors.

Figure 1
Figure 1. The exact and reconstructed sources from interior [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. The exact and reconstructed sources from interior [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 28 canonical work pages

  1. [19]

    Sakamoto and M

    K. Sakamoto and M. Yamamoto, Initial value/boundary va lue problems for fractional diffusion-wave equations and applications to some inverse p roblems, J. Math. Anal. Appl., 382 (2011), 426–447

    work page 2011

  2. [1]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , Dover, New York, 1972

    work page 1972

  3. [2]

    Adams and J

    R. Adams and J. Fournier, Sobolev Spaces, Elsevier, San Diego, 2003

    2003

  4. [3]

    Chepizhko and F

    O. Chepizhko and F. Peruani, Diffusion, subdiffusion, and trapping of active particles in heterogeneous media. Phys. Rev. Lett. , 111 (2013), 160604

    work page 2013

  5. [4]

    W. Deng, R. Hou, W. Wang, P. Xu, Modeling anomalous diffusion—from statistics to mathematics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2020

    work page 2020

  6. [5]

    Evans, Partial Differential Equations , Graduate Studies in Mathematics, V 19, Amer- ican Mathematical Society, Rhode Island, 1998

    L. Evans, Partial Differential Equations , Graduate Studies in Mathematics, V 19, Amer- ican Mathematical Society, Rhode Island, 1998

    work page 1998

  7. [6]

    Fujishiro, M

    K. Fujishiro, M. Yamamoto, Approximate controllabilit y for fractional diffusion equa- tions by interior control. Appl. Anal. , 93 (2014), 1793–1810 19

    work page 2014

  8. [7]

    J. Hong, B. Jin, Y. Kian, Identification of a spatially-dep endent variable order in one- dimensional subdiffusion. arXiv:2407.02193

    work page Pith review arXiv

Show all 30 references
  1. [8]

    Jin, Fractional differential equations–an approach via fractio nal derivatives , Appl

    B. Jin, Fractional differential equations–an approach via fractio nal derivatives , Appl. Math. Sci. 206, Springer, Cham, 2021

    2021

  2. [9]

    B. Jin, Y. Kian, Z. Zhou, Inverse problems for subdiffusion from observation at an unknown terminal time. SIAM J. Appl. Math. , 83 (2023), 1496–1517

    2023

  3. [10]

    Jin and W

    Q. Jin and W. Wang. Landweber iteration of kaczmarz type with general non-smooth convex penalty functionals. Inverse Probl., 29 (2013), 085011

    2013

  4. [11]

    Jiang, Z

    D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique conti nuation property and a related inverse source problem for time-fractional diffusi on-advection equations, Inverse Probl., 33 (2017), 055013

    2017

  5. [12]

    Kaltenbacher and W

    B. Kaltenbacher and W. Rundell, Inverse problems for fractional partial differential equations, Grad. Stud. Math., 230 American Mathematical Society, Pro vidence, RI, 2023

    2023

  6. [13]

    Y. Kian, M. Slodička, É. Soccorsi and K. Bockstal, On time -fractional partial differential equations of time-dependent piecewise constant order. Math. Meth. Appl. Sci. 48 (2025), 2354–2369

    2025

  7. [14]

    Y. Kian, E. Soccorsi and M. Yamamoto, On time-fractiona l diffusion equations with space-dependent variable order, Ann. Henri Poincaré , 19 (2018), 3855–3881

    2018

  8. [15]

    Y. Kian, Z. Li, Y. Liu and M. Yamamoto, The uniqueness of i nverse problems for a fractional equation with a single measurement. Math. Ann. , 380 (2021), 1465–1495

    2021

  9. [16]

    Le and M

    K. Le and M. Stynes, An α-robust semidiscrete finite element method for a Fokker- Planck initial-boundary value problem with variable-orde r fractional time derivative. J. Sci. Comput. , 86 (2021), 22

    2021

  10. [17]

    Z. Li, O. Y. Imanuvilov and M. Yamamoto, Uniqueness in in verse boundary value problems for fractional diffusion equations. Inverse Problems, 32 (2016), 015004

    2016

  11. [18]

    Lorenzo and T

    C. Lorenzo and T. Hartley, Variable order and distribut ed order fractional operators, Nonlinear Dyn. , 29 (2002), 57–98

    2002

  12. [20]

    Slodička, Some direct and inverse source problems in nonlinear evolutionary PDEs with Volterra operators

    M. Slodička, Some direct and inverse source problems in nonlinear evolutionary PDEs with Volterra operators. Inverse Probl. , 38 (2022), 124001

    2022

  13. [21]

    H. Sun, W. Chen, H. Sheng, Y. Chen, On mean square displac ement behaviors of anomalous diffusions with variable and random orders, Phys. Lett. A , 374 (2010), 906– 910

    2010

  14. [22]

    H. Sun, W. Chen and Y. Chen, Variable-order fractional d ifferential operators in anoma- lous diffusion modeling. Physica A , 388 (2009), 4586–4592. 20

    2009

  15. [23]

    H. Sun, A. Chang, Y. Zhang and W. Chen, A review on variabl e-order fractional differ- ential equations: mathematical foundations, physical mod els, numerical methods and applications. Fract. Calc. Appl. Anal. , 22 (2019), 27–59

    2019

  16. [24]

    Sun and J

    C. Sun and J. Liu, Reconstruction of the space-dependen t source from partial Neumann data for slow diffusion system, Acta Math. Appl. Sin. (Engl. Ser.) , 36 (2020), 166–182

    2020

  17. [25]

    Sun and J

    C. Sun and J. Liu, An inverse source problem for distribu ted order time-fractional diffusion equation, Inverse Probl., 36 (2020), 055008

    2020

  18. [26]

    Umarov and S

    S. Umarov and S. Steinberg, Variable order differential equations with piecewise con- stant order-function and diffusion with changing modes. Z. Anal. Anwend. , 28 (2009), 131–150

    2009

  19. [27]

    Zhang and Z

    Z. Zhang and Z. Zhou, Backward diffusion-wave problem: St ability, regularization, and approximation. SIAM J. Sci. Comput. , 44 (2022), A3183–A3216

    2022

  20. [28]

    Zheng, Y

    X. Zheng, Y. Li and W. Qiu, Local modification of subdiffus ion by initial Fickian diffusion: Multiscale modeling, analysis and computation. Multiscale Model. Simul. , 22 (2024), 1534–1557

    2024

  21. [29]

    Zheng, Two methods addressing variable-exponent fr actional initial and boundary value problems and Abel integral equation

    X. Zheng, Two methods addressing variable-exponent fr actional initial and boundary value problems and Abel integral equation. arXiv:2404.094 21v2

  22. [30]

    Zhong, W

    M. Zhong, W. Wang and Q. Jin. Regularization of inverse p roblems by two-point gra- dient methods in banach spaces. Numer. Math. , 143 (2019):713–747. 21

    2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.