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REVIEW 2 major objections 5 minor 37 references

Carleman--Picard and time-dimensional reduction for inverse initial-data problems in nonlinear transport with memory

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Pith's one-line read For the truncated time-reduced system, the Carleman–Picard map is a strict contraction for large Carleman parameter, giving global convergence from any admissible initial guess and stable reconstruction of the initial state from noisy…

desk verdict Competent extension of Carleman-Picard to quasilinear transport with memory, but the theorems only cover the truncated N-modal system, not the original PDE problem. read the letter →

arxiv 2608.09665 v1 pith:CNDTC2OY submitted 2026-08-10 math.NA cs.NA

Navaraj Neupane , Loc H. Nguyen This is my paper · ORCID
classification math.NAcs.NA MSC 35R3035F2535R0935B4565M32
keywords inverseinitial-dataproblemquasilineartransportequationVolterramemorytermCarlemanestimateCarleman–Picardmethodtime-dimensionalreductionLegendre–exponentialbasisTikhonovregularization
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

The paper claims that the ill-posed inverse problem of recovering the initial state of a quasilinear transport equation with a Volterra memory term from outflow boundary measurements can be recast, after expanding time in a Legendre–exponential basis and truncating, as a finite nonlinear spatial system solved by a Carleman-weighted Picard iteration. The central theoretical result is that, for a sufficiently large Carleman parameter, the iteration map is a strict contraction on a fixed admissible ball, with contraction factor of order $\lambda^{-1/2}$, so it converges from any initial guess in that ball and depends Lipschitz-continuously on noisy outflow data. The authors are explicit that these theorems apply to the truncated and Tikhonov-regularized reduced problem, not to the original infinite-dimensional equation. Two-dimensional numerical experiments recover disk, two-Gaussian, and Y-shaped inclusions and show that the iteration converges in a few steps at moderate noise levels.

What carries the argument

The argument rests on two objects. The Legendre–exponential basis $\Psi_n(t)=\sqrt{(2n+1)/T}\,e^{t}P_n(2t/T-1)$ is orthonormal in $L^2_{e^{-2t}}(0,T)$ and has the property that no basis function has an identically vanishing derivative, so no spatial modal coefficient disappears from the time-derivative term after truncation. The other is the Carleman estimate (2.6) for the principal transport operator $H\cdot\nabla$, which, under the weight condition $H\cdot\nabla\varphi_*\ge\mu_0>0$, yields $\lambda\int_\Omega e^{2\lambda\varphi_*}|v|^2\le C\int_\Omega e^{2\lambda\varphi_*}|H\cdot\nabla v|^2+C\lambda\int_{\partial\Omega} e^{2\lambda\varphi_*}|v|^2$. This estimate supplies the coercivity that, together with the Lipschitz bound on the frozen nonlinear terms, makes the Picard map contractive for large $\lambda$.

What would settle it

Take a smooth exact solution of (1.1), generate exact outflow data, run the truncated Carleman–Picard method with increasing $N$ and suitably chosen $\varepsilon$, and compare $u_{0,N}$ with the true $u_0$ in a fixed norm; if the error does not tend to zero as $N\to\infty$, the assumed bridge between the truncated and full problems fails. A second test would check whether a field $H$ violating the nontrapping or weight condition (2.5) destroys the contraction property numerically.

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Extended reading notes

Core claim

For the reduced system (3.14) of spatial modal coefficients obtained by truncating the Legendre–exponential expansion at $N+1$ terms, the paper proves that the Carleman–Picard map $T_{\lambda,\varepsilon}$ defined by minimizing the weighted functional (4.2) is a strict contraction in the weighted norm (4.5) once $\lambda$ exceeds a threshold. Consequently the fixed point is unique in $B_M$ and the iteration $U^{(k+1)}=T_{\lambda,\varepsilon}(U^{(k)})$ converges geometrically from every $U^{(0)}\in B_M$. Theorem 5.1 quantifies stability: the weighted reconstruction error is bounded by the noisy outflow-data misfit plus a Tikhonov term of order $\sqrt{\varepsilon/\lambda}$. The reconstructed initial state is then $u_{0,N}(x)=\sum_{n=0}^{N}u_n(x)\Psi_n(0)$.

Load-bearing premise

The load-bearing premise is that retaining $N$ temporal modes and evaluating the nonlinear coefficients on the truncated sum produces a system whose fixed point approximates the true initial data; no $N$-dependent error estimate is given, and the theorems also require the geometric weight condition (2.5) to hold.

Editorial extensions

If this is right

  • For fixed truncation index $N$ and regularization parameter $\varepsilon$, the reconstruction error depends Lipschitz-continuously on the relative noise level $\delta$, so increased outflow noise degrades the weighted reconstruction only proportionally.
  • Because convergence holds for every initial guess in the admissible ball, the iteration can be started from the zero guess rather than from a near-solution approximation.
  • The truncation index $N$ serves as a regularization device: suppressing high-order temporal modes filters noise-sensitive components of the data.
  • Numerically, moderate positive values of the Carleman parameter improve reconstructions over the unweighted Picard scheme, while excessively large $\lambda$ severely worsens the conditioning of the discrete least-squares system.
  • The method extends the Carleman contraction principle to quasilinear transport with memory, provided the geometric propagation condition $T>\tau_H^*/\kappa_0$ and the weight condition (2.5) hold.

Reading between the lines

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

  • Beyond the paper: the missing $N$-dependent error estimate means the theorems do not by themselves justify reconstructing the true initial data of (1.1); a quantitative bound on $\|u_{0,N}-u_0\|$ in terms of the truncated tail would close the bridge between the reduced and original problems.
  • Beyond the paper: because the stability constant may depend on $N$ and no estimate uniform in $N$ is given, the simultaneous limits $N\to\infty$, $\varepsilon\to0$ are not covered; a testable extension would compute reconstruction error as $N$ grows on a smooth exact solution.
  • Beyond the paper: the plateau criterion used to select $N$ from projected noisy data is heuristic; a discrepancy-principle-type rule would make the choice data-adaptive and provable.
  • Beyond the paper: the discontinuous disk and Y-shaped tests violate the assumed $C^1$ regularity, and the paper itself calls them stress tests; a weak-solution analysis would be needed to make the theorems cover such profiles.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper treats Problem 1.1: recover the initial state u0 of the quasilinear transport equation (1.1) with a Volterra memory term from outflow boundary observations. The method has three components: a Legendre-exponential basis expansion in time truncated at N+1 modes, yielding the spatial system (3.14) for modal coefficients; a Carleman-weighted, Tikhonov-regularized least-squares minimization that defines a Picard map T_{lambda,epsilon} on an admissible ball in H^s; and convergence and stability theorems. Theorem 4.1 proves that T_{lambda,epsilon} is a strict contraction in a weighted norm for sufficiently large Carleman parameter, with contraction factor of order lambda^{-1/2}, hence global convergence of the iteration to a fixed point of the truncated system. Theorem 5.1 gives a weighted Lipschitz stability estimate with respect to noisy outflow data. Section 6 reports two-dimensional reconstructions of a disk, two Gaussian inclusions, and a Y-shaped inclusion at 5% and 10% noise, together with a study of the role of the Carleman parameter. The paper is explicit that all analytical results concern the fixed-N, Tikhonov-regularized reduced problem.

Significance. The fixed-N analysis is a useful and, on inspection, internally consistent contribution: the Carleman estimate for H dot grad is elementary and correctly proven, the variational-inequality subtraction in Theorem 4.1 is legitimate, and the stability estimate in Theorem 5.1 is clearly derived. The authors are also commendably transparent: Remark 5.1 and the concluding section state that no uniformity in N is claimed, and Remark 6.2 acknowledges that the discontinuous numerical profiles lie outside the regularity assumptions. The availability of reproducible code on Zenodo is a further strength. The main limitation is that the bridge from the truncated system (3.14) to the original Problem 1.1 is not established: no N-dependent error estimate links the fixed point of T_{lambda,epsilon} to u0, and no N-convergence experiment is reported. Thus the analytical results prove convergence of an algorithm for a surrogate finite-dimensional problem, but not yet reconstruction for the original infinite-dimensional inverse problem. With an added consistency analysis or a carefully reframed scope, the paper would be a solid contribution to numerical inverse problems for transport equations.

major comments (2)
  1. [Section 3, after Eq. (3.8); see also Remark 5.1 and Section 7] The central claim of the paper is reconstruction of u0 for Problem 1.1, but Theorems 4.1 and 5.1 establish convergence and stability only for the fixed-N truncated system (3.14). The passage from (3.2) to (3.8) is justified by saying that the tail of the Legendre-exponential expansion tends to zero in the weighted L^2 norm; this is not enough, because the coefficients S_mn, M_mn, and F_m in (3.5)-(3.7) are evaluated at the truncated sum, so smallness of the tail does not imply smallness of the truncated residual in (3.8). No estimate of the form ||u0 - u_{0,N}|| <= rho(N,epsilon,lambda) plus a data-noise term is stated, and Remark 5.1 explicitly disclaims uniformity in N. Therefore the global convergence and stability theorems do not by themselves justify reconstruction for equation (1.1). The revision should either add such an N-dependent consistency estimate, even a qualitative one, or carefully reframe the advertised scope as the reduced finite-dimensional model.
  2. [Section 6, in particular Section 6.2 and the three tests of Section 6.3] Because the theory is silent on N, the numerical experiments are the only evidence for consistency of the truncation, but they do not supply that evidence. The plateau criterion selects a single N per test and the figures report only the selected N; there is no systematic study of the reconstruction error as a function of N for fixed noise and a fixed smooth initial profile. In addition, the disk and Y-shaped profiles are discontinuous and therefore violate the smoothness assumptions of Problem 1.1, as Remark 6.2 acknowledges. A systematic N-convergence study using smooth profiles would at least provide numerical evidence for the missing bridge and would be straightforward to add.
minor comments (5)
  1. [Section 6.3] The quantity E_max is reported as a percentage but never defined; please define it explicitly, for example as 100 times the relative error of the maximum reconstructed value.
  2. [Remark 6.1] The claim that the constrained and unconstrained minimizers coincide for M sufficiently large requires the unconstrained minimizer to lie in the interior of B_M; state this explicitly and, if possible, report the H^s norm of the computed iterates to show they remain inside the ball.
  3. [Algorithm 1 and Eq. (6.4)] The stopping criterion uses an ell^2 norm on vectors of nodal values without specifying how the modal components and grid points are vectorized; please state the convention.
  4. [Theorem 4.1 and Section 6.4] The theorem guarantees contraction only for lambda larger than an unspecified lambda_1 and constant C; the numerical choice lambda=4 is therefore not certified by the theory. The lambda study is informative, but it would be helpful to state explicitly that the correspondence between the theoretical threshold and the practical choice is not quantified.
  5. [Section 3, sentence after Eq. (3.8)] The sentence 'Since the series in (3.2) converges, the neglected tail tends to zero...' is mathematically true but potentially misleading as a justification of (3.8); consider replacing it with a precise statement of what the convergence does and does not imply for the truncated residual.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the contraction and stability theorems are derived in-paper from a Carleman estimate proved in the paper, and the cited prior results are independent support.

full rationale

The derivation chain is not circular. The Carleman estimate for H·∇ (Proposition 2.1) is proved in the paper from the weight assumption (2.5) by elementary integration by parts; Theorem 4.1's contraction estimate then follows from that estimate, Lemma 4.1, and convexity arguments, all self-contained. Theorem 5.1's stability estimate is likewise a direct consequence of the same Carleman estimate plus Young's inequality. The only cited results that enter the derivation are the Legendre–exponential basis and its differentiation theorem from [11]; these are parameter-free mathematical statements with stated smoothness assumptions not containing the target inverse problem, so they are independent support rather than circular premises. The many other self-citations appear in the introduction as related-work context and are not load-bearing. The paper repeatedly and explicitly disclaims any theorem for the original untruncated problem: Section 2.1 says uniqueness 'lies outside the scope of this paper,' Remark 5.1 states 'no estimate uniform with respect to N is established here,' and Section 7 limits conclusions to the 'truncated and regularized reduced problem.' Consequently, the gap between the finite-modal fixed point and the true initial data u0 is an acknowledged consistency limitation, not a case where a fitted parameter or a self-citation is renamed as a prediction. No equation reduces to its own input by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central derivation introduces no new physical entities. All tunable constants (lambda, epsilon, N, M, phi*) are method parameters, not physical fitted values. The key uncharged assumption is that truncating the temporal expansion preserves the relation between the initial state and outflow data; this is the largest unproved item in the ledger.

free parameters (5)
  • Carleman parameter lambda = 4 in computations; up to 7 in the parameter study
    Manual tuning parameter controlling the exponential weight and the contraction factor. Theory requires lambda sufficiently large but does not specify the value. Selected using Test 1.
  • Tikhonov parameter epsilon = 1e-6
    Regularization strength chosen manually using Test 1; keeps the minimization strictly convex.
  • Truncation index N = 13 (Test 1), 10 and 8 (Test 2), 14 (Test 3)
    Selected automatically by a plateau criterion applied to the noisy outflow data. Data-dependent and controls temporal resolution.
  • Admissible ball radius M = not specified; assumed large
    Theoretical constant bounding the solution and iterates. All Lipschitz and contraction constants depend on M. Not imposed in computation.
  • Carleman weight phi* = x + 0.5y in numerics
    Chosen to satisfy H dot grad phi* >= 0.9. Affects conditioning and reconstruction quality; a modeling choice for the weight.
assumptions (6)
  • standard math Sobolev embedding H^s(Omega) into W^{1,infty}(Omega) for s > d/2 + 1 is used to bound nonlinear terms in L^infty.
    Section 4, definition of B_M and Lemma 4.1.
  • standard math Banach fixed point theorem applied in a complete equivalent weighted norm.
    Corollary 4.1.
  • domain assumption Well-posedness and regularity of the forward quasilinear transport equation with memory are assumed: u in C^1 cap H^3((0,T);H^p(Omega)), c,f in C^1, alpha in L^infty, c >= kappa0 > 0.
    Problem 1.1 and Section 3.
  • domain assumption Geometric propagation: H is nontrapping and observation time T exceeds tau_H^*/kappa0 so that all initial information reaches the outflow boundary by time T.
    Section 2.1, equations (2.3)-(2.4).
  • domain assumption Existence of a global C^2 weight phi* with H dot grad phi* >= mu0 > 0 on Omega.
    Section 2.2, equation (2.5).
  • ad hoc to paper The infinite temporal expansion can be truncated, and the neglected tail in the nonlinear coefficients is small enough that the fixed point of the reduced system approximates the true initial data; no error bound in N is established.
    Section 3 after (3.7), Remark 5.1, and Section 7.

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Pith. "Pith review of Carleman--Picard and time-dimensional reduction for inverse initial-data problems in nonlinear transport with memory." pith.science (2026). https://pith.science/paper/CNDTC2OY

@misc{pith2026260809665,
  author       = {Pith},
  title        = {Pith review of: Carleman--Picard and time-dimensional reduction for inverse initial-data problems in nonlinear transport with memory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CNDTC2OY}},
  note         = {Machine review of arXiv:2608.09665}
}
read the original abstract

We study an inverse initial-data problem for a quasilinear transport equation with nonlinear and memory effects. The unknown initial state is reconstructed from time-dependent measurements on the outflow boundary, with prescribed inflow data. We first apply a Legendre--exponential time-dimensional reduction to transform the governing equation into a finite nonlinear system in space. We then develop a Carleman-weighted and Tikhonov-regularized Picard method. At each iteration, the nonlinear terms are evaluated using the previous iterate, leading to a linear minimization problem with a unique solution. A Carleman estimate for the principal transport operator is used to prove that, for a sufficiently large Carleman parameter, the resulting Picard map is contractive on a prescribed admissible set. Consequently, the method converges from an arbitrary initial guess in that set. We also establish stability with respect to noisy outflow data. These analytical results concern the truncated and regularized reduced problem. Two-dimensional numerical experiments demonstrate accurate reconstruction of single and multiple inclusions, robustness with respect to noise, and rapid convergence of the Picard iteration.

Figures

Figures reproduced from arXiv: 2608.09665 by the authors.

Figure 1
Figure 1. Test 1. The true disk inclusion, the reconstructions, the selection of [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Test 2. The true two-Gaussian profile, the reconstructions, the selection of [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Test 3. The true Y-shaped inclusion, the reconstructions, the selection of [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Reconstructions of the two-Gaussian initial state from 5% noisy data for several values [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

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Reference graph

Works this paper leans on

37 extracted references · 33 canonical work pages

  1. [1]

    Abhishek, T

    A. Abhishek, T. T. Le, L. H. Nguyen, and T. Khan. The Carleman-Newton method to globally reconstruct the initial condition for nonlinear parabolic equations.Journal of Computational and Applied Mathematics, 445:115827, 2024

    work page 2024

  2. [2]

    G. Bal. Inverse transport theory and applications.Inverse Problems, 25(5):053001, 2009

    work page 2009

  3. [3]

    Bal and A

    G. Bal and A. Jollivet. Stability estimates in stationary inverse transport.Inverse Problems and Imaging, 2(4):427–454, 2008

    work page 2008

  4. [4]

    Bal and A

    G. Bal and A. Jollivet. Generalized stability estimates in inverse transport theory.Inverse Problems and Imaging, 12(1):59–90, 2018

    work page 2018

  5. [5]

    Bal and A

    G. Bal and A. Tamasan. Inverse source problems in transport equations.SIAM Journal on Mathematical Analysis, 39(1):57–76, 2007

    work page 2007

  6. [6]

    Camilli and R

    F. Camilli and R. De Maio. Memory effects in measure transport equations.Kinetic and Related Models, 12(6):1229–1245, 2019

    work page 2019

  7. [7]

    Cannarsa, G

    P. Cannarsa, G. Floridia, F. G¨ olgeleyen, and M. Yamamoto. Inverse coefficient problems for a transport equation by local Carleman estimate.Inverse Problems, 35(10):105013, 2019

    work page 2019

  8. [8]

    Chen and C

    G.-Q. Chen and C. Christoforou. Solutions for a nonlocal conservation law with fading memory. Proceedings of the American Mathematical Society, 135(12):3905–3915, 2007

    work page 2007

Show all 37 references
  1. [9]

    F. Colombo. An inverse problem for the strongly damped wave equation with memory.Non- linearity, 20(3):659–683, 2007

    2007

  2. [10]

    Colombo and D

    F. Colombo and D. Guidetti. Identification of the memory kernel in the strongly damped wave equation by a flux condition.Communications on Pure and Applied Analysis, 8(2):601–620, 2009

    2009

  3. [11]

    T. D. Dang, C. V. Le, K. D. Luu, and L. H. Nguyen. Recovery of initial displacement and velocity in anisotropic elastic systems by the time dimensional reduction method.Journal of Computational Physics, 542:114371, 2025

    2025

  4. [12]

    T. D. Dang, L. H. Nguyen, and H. T. T. Vu. Determining initial conditions for nonlinear hyperbolic equations with time dimensional reduction and the Carleman contraction principle. Inverse Problems, 40:125021, 2024

    2024

  5. [13]

    Gaitan and H

    P. Gaitan and H. Ouzzane. Inverse problem for a free transport equation using carleman estimates.Applicable Analysis, 93(5):1073–1086, 2014. 25

    2014

  6. [14]

    Graf and C

    T. Graf and C. T. Simmons. Variable-density groundwater flow and solute transport in frac- tured rock: Applicability of the Tang et al. [1981] analytical solution.Water Resources Re- search, 45:W02425, 2009

    1981

  7. [15]

    Janno and L

    J. Janno and L. von Wolfersdorf. An inverse problem for identification of a time- and space- dependent memory kernel in viscoelasticity.Inverse Problems, 17(1):13–24, 2001

    2001

  8. [16]

    M. V. Klibanov and O. V. Ioussoupova. Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem.SIAM J. Math. Anal., 26:147–179, 1995

    1995

  9. [17]

    M. V. Klibanov and S. E. Pamyatnykh. Lipschitz stability of a non-standard problem for the non-stationary transport equation via a Carleman estimate.Inverse Problems, 22(3):881–890, 2006

    2006

  10. [18]

    M. V. Klibanov and S. E. Pamyatnykh. Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate.Journal of Mathematical Analysis and Applications, 343(1):352–365, 2008

    2008

  11. [19]

    Lai and Q

    R.-Y. Lai and Q. Li. Parameter reconstruction for general transport equation.SIAM Journal on Mathematical Analysis, 52(3):2734–2758, 2020

    2020

  12. [20]

    Lai and H

    R.-Y. Lai and H. Zhou. Inverse problems for time-dependent nonlinear transport equations. preprint available at arXiv:2410.00369, 2024

    2024 arXiv

  13. [21]

    Lattes and J.-L

    R. Lattes and J.-L. Lions.The Method of Quasi-Reversibility: Applications to Partial Differ- ential Equations. Elsevier, New York, 1969

    1969

  14. [22]

    T. T. Le. Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method. In D.-L. Nguyen, L. H. Nguyen, and T.-P. Nguyen, editors, Advances in Inverse Problems for Partial Differential Equations, volume 784 ofContemporary Math...

    2023

  15. [23]

    T. T. Le and L. H. Nguyen. A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data.Journal of Inverse and Ill-Posed Problems, 30(2):265–286, 2022

    2022

  16. [24]

    T. T. Le, L. H. Nguyen, and H. V. Tran. A Carleman-based numerical method for quasilin- ear elliptic equations with over-determined boundary data and applications.Computers and Mathematics with Applications, 125:13–24, 2022

    2022

  17. [25]

    T. T. Le, L. V. Nguyen, L. H. Nguyen, and H. Park. The time dimensional reduction method to determine the initial conditions without the knowledge of damping coefficients.Computers & Mathematics with Applications, 166:77–90, 2024

    2024

  18. [26]

    T. T. Le, C. B. Van, T. D. Dang, and L. H. Nguyen. Inverse initial data reconstruc- tion for Maxwell’s equations via time-dimensional reduction method.preprint available at arXiv:2506.20777, 2025

    2025 arXiv

  19. [27]

    M. J. Lighthill and G. B. Whitham. On kinematic waves. II. a theory of traffic flow on long crowded roads.Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229(1178):317–345, 1955. 26

    1955

  20. [28]

    Neupane and L

    N. Neupane and L. H. Nguyen. Inverse initial data for nonlinear Schr¨ odinger equation via Carleman estimates and the contraction principle.preprint available at arXiv:2605.11409, 2026

    2026 arXiv

  21. [29]

    L. H. Nguyen. The Carleman contraction mapping method for quasilinear elliptic equations with over-determined boundary data.Acta Mathematica Vietnamica, 48:401–422, 2023

    2023

  22. [30]

    P. M. Nguyen and L. H. Nguyen. A Carleman contraction method for inverse initial data recovery in the Navier–Stokes equations with unknown body force.preprint available at arXiv:2604.09934, 2026

    2026 arXiv

  23. [31]

    S. P. Pudasaini and K. Hutter.Avalanche Dynamics: Dynamics of Rapid Flows of Dense Granular Avalanches. Springer, Berlin, 2007

    2007

  24. [32]

    P. I. Richards. Shock waves on the highway.Operations Research, 4(1):42–51, 1956

    1956

  25. [33]

    S. B. Savage and K. Hutter. The motion of a finite mass of granular material down a rough incline.Journal of Fluid Mechanics, 199:177–215, 1989

    1989

  26. [34]

    A. N. Tikhonov and V. Y. Arsenin.Solutions of Ill-Posed Problems. Winston and Sons, Washington, DC, 1977

    1977

  27. [35]

    C. B. Van, T. T. Le, and L. H. Nguyen. The inverse initial data problem for anisotropic Navier–Stokes equations via Legendre time reduction method.Communications in Nonlinear Science and Numerical Simulation, 161:110074, 2026

    2026

  28. [36]

    C. B. Van, T. P. B. Nguyen, M.-B. Tran, and L. H. Nguyen. Inverse initial data reconstruc- tion for a memory convection-diffusion equation via Legendre spatial reduction and Tikhonov regularization.preprint available at arXiv:2606.20875, 2026

    2026 arXiv

  29. [37]

    M. Th. van Genuchten and R. J. Wagenet. Two-site/two-region models for pesticide transport and degradation: Theoretical development and analytical solutions.Soil Science Society of America Journal, 53(5):1303–1310, 1989. 27

    1989

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