Forward and inverse problems for a time-fractional pseudo-parabolic equation with variable coefficients

Elbek Husanov; Ravshan Ashurov

arxiv: 2605.13285 · v1 · pith:3VGKUCOAnew · submitted 2026-05-13 · 🧮 math.AP

Forward and inverse problems for a time-fractional pseudo-parabolic equation with variable coefficients

Ravshan Ashurov , Elbek Husanov This is my paper

Pith reviewed 2026-05-14 18:21 UTC · model grok-4.3

classification 🧮 math.AP

keywords time-fractional equationpseudo-parabolic equationinverse source problemSchauder fixed point theoremFourier methodvariable coefficientsglobal existenceCaputo derivative

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The pith

Global existence and uniqueness hold for the forward problem of a time-fractional pseudo-parabolic equation with time-dependent coefficient, and global existence holds for the associated inverse source problem.

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

The paper proves that the forward problem for the equation involving a Caputo time-fractional derivative of order rho acting on u plus mu A u, plus sigma(t) A u equal to r(t) g, admits a unique global solution even when sigma varies with time. This extends prior results restricted to constant sigma by applying the Fourier method in a Hilbert space setting with positive self-adjoint operator A. For the inverse problem of recovering the time-dependent source r(t) from the general overdetermination condition F applied to u equals Phi, global existence of a solution follows from Schauder's fixed point theorem while uniqueness is also shown. When A is realized as a second-order differential operator a numerical scheme is constructed. The results matter because they enable modeling of anomalous diffusion and related processes where coefficients change over time.

Core claim

The central claim is that global existence and uniqueness of the solution to the forward problem is proved for time-dependent sigma(t) via the Fourier method, while global existence of the solution to the inverse problem of determining the source is proved by applying Schauder's fixed point theorem to a mapping induced by the general functional F under the overdetermination condition.

What carries the argument

The Fourier eigenfunction expansion for the forward problem together with Schauder's fixed point theorem applied to the operator mapping induced by the general functional F on the solution of the time-fractional pseudo-parabolic equation.

If this is right

The solution of the forward problem can be written explicitly as a series in the eigenfunctions of A with coefficients determined by a fractional ordinary differential equation.
The inverse problem admits a global-in-time solution for any functional F meeting the fixed-point hypotheses, including common choices such as point evaluation or integral averages.
When A is a second-order differential operator an explicit finite-difference or spectral algorithm can be implemented to compute the solution numerically.
Uniqueness of the inverse solution guarantees that the recovered source r(t) is the only one consistent with the given overdetermination.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same fixed-point argument could be tested on related inverse problems where the fractional order rho itself is unknown.
Stability estimates with respect to perturbations in sigma(t) or in the data Phi would follow from minor modifications of the contraction-mapping estimates already used.
Numerical experiments on bounded domains with the Laplacian could quantify how the computational cost scales with the fractional order rho.

Load-bearing premise

The functional F must be such that the induced mapping on the solution space satisfies the continuity and compactness conditions needed for Schauder's fixed point theorem to apply.

What would settle it

An explicit choice of the functional F and data Phi for which the induced mapping fails to be compact on the convex set, or for which two distinct solutions satisfy the same overdetermination, would falsify the existence or uniqueness claims.

Figures

Figures reproduced from arXiv: 2605.13285 by Elbek Husanov, Ravshan Ashurov.

read the original abstract

In this work, forward and inverse problems for a time-fractional pseudo-parabolic equation $D_t^{\rho} [u(t) + \mu Au(t)] + \sigma(t) Au(t) = r(t)g$ are investigated in a Hilbert space, where $A$ is an unbounded, positive, self-adjoint operator. According to the known papers, the forward problem has been studied only in the case $\sigma(t) = const$. The main novelty of the forward problem in this work is that the model is further generalized and investigated for a time-dependent coefficient $\sigma(t)$. To determine the solution of the forward problem, the Fourier method is employed, and the global existence and uniqueness of the solution are proved. Moreover, when the operator $A$ is a second-order differential operator, a numerical scheme and an efficient computational algorithm are developed. The inverse problem of determining a time-dependent source function is considered under the overdetermination condition of the form $F[u(t)] = \Phi(t)$. The functional $F$ is taken in a general form, and such an inverse problem has not been considered before. The global existence of the solution to the inverse problem is proved by applying Schauder's fixed point theorem, and its uniqueness is established. Furthermore, several examples related to the operator $A$ and the functional $F$ are provided.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends the forward problem to time-dependent sigma(t) and applies Schauder to a general inverse source problem, but the compactness needed for the fixed-point map is not obviously verified.

read the letter

The main things to know are that the paper generalizes the forward problem for this time-fractional pseudo-parabolic equation to a time-dependent coefficient sigma(t) in front of the Au term, and it sets up an inverse problem to recover the source r(t) from a general functional condition F[u] = Phi, proving existence by Schauder's fixed point theorem after reducing to a map on the solution space. They use the Fourier method on the eigenbasis of A for the forward part and claim global existence and uniqueness. When A is a second-order differential operator they also sketch a numerical scheme and algorithm. The inverse part treats F in general form, which goes beyond the more specific functionals in earlier constant-coefficient papers. That is the actual novelty they deliver. The forward extension looks workable under the standing assumptions that A has discrete positive spectrum and suitable regularity; the Fourier expansion carries over once sigma(t) is positive and integrable enough. The numerical part adds a bit of concrete value for people who want to compute examples. The soft spot is the inverse problem. Schauder's theorem requires the constructed map to send a closed convex bounded set into a compact subset of itself, so you need continuity plus compactness of the image. With variable sigma(t) and arbitrary F, boundedness alone does not automatically give equicontinuity or compact embedding from the fractional evolution operator. The abstract states that they apply the theorem after assuming the right conditions on A and F, but without seeing the explicit estimates that close the compactness step it is hard to be sure the argument holds for truly general F. If those estimates are only sketched or rely on extra hidden regularity, the fixed-point conclusion would need tightening. This is aimed at specialists already working on fractional evolution equations and abstract inverse problems in Hilbert spaces. A reader who knows the constant-sigma literature will see the incremental step right away and can judge whether the general-F setup is useful for their own models. I would send it to peer review. The forward results are on standard ground and the inverse claim is worth recording, but the compactness verification for Schauder is the part that referees should check in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates forward and inverse problems for the time-fractional pseudo-parabolic equation D_t^ρ [u(t) + μ A u(t)] + σ(t) A u(t) = r(t) g in a Hilbert space, where A is an unbounded positive self-adjoint operator. For the forward problem with time-dependent σ(t), the Fourier method is used to prove global existence and uniqueness of the solution. When A is a second-order differential operator, a numerical scheme is developed. For the inverse problem of recovering the source r(t) from the general overdetermination condition F[u(t)] = Φ(t), global existence is proved by applying Schauder's fixed point theorem and uniqueness is established, with examples for A and F provided.

Significance. If the central claims hold, the work extends existing results on constant-coefficient cases to time-dependent σ(t) and introduces a general functional F for the inverse problem, which has not been considered previously. This contributes to the theory of fractional evolution equations and could support applications in inverse source identification for models with memory effects.

major comments (2)

[Inverse problem section] Inverse problem section: The proof of global existence invokes Schauder's fixed point theorem on the map T that sends u to the forward solution with source determined by F[u]=Φ. However, the manuscript does not explicitly verify that T maps a closed convex bounded set into a compact subset (e.g., via equicontinuity of the fractional evolution family or compact embedding), which is required for arbitrary F and time-dependent σ(t). This step is load-bearing for the existence claim.
[Forward problem section] Forward problem section: The Fourier expansion is applied to obtain the solution series, but the global existence proof must explicitly derive uniform bounds on the coefficients that account for the time-variation of σ(t) (e.g., in the resulting Volterra-type integral equations or Gronwall estimates). Without these details, convergence of the series for general σ(t) is not fully supported.

minor comments (2)

[Abstract] The abstract states that uniqueness for the inverse problem is established, but the main text should clarify whether this holds for general F or requires additional assumptions on F.
[Preliminaries] Notation for the fractional derivative D_t^ρ and the precise domain of A should be introduced with a dedicated preliminary section or subsection to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which will help improve the clarity and rigor of the proofs. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

Referee: [Inverse problem section] Inverse problem section: The proof of global existence invokes Schauder's fixed point theorem on the map T that sends u to the forward solution with source determined by F[u]=Φ. However, the manuscript does not explicitly verify that T maps a closed convex bounded set into a compact subset (e.g., via equicontinuity of the fractional evolution family or compact embedding), which is required for arbitrary F and time-dependent σ(t). This step is load-bearing for the existence claim.

Authors: We agree that the compactness condition for Schauder's fixed point theorem must be verified explicitly. In the revised version, we will add a dedicated paragraph (or subsection) detailing the choice of the closed convex bounded set in the appropriate function space and proving that T maps it into a compact subset. This will rely on the equicontinuity properties of the mild solution operator for the time-fractional pseudo-parabolic equation with variable σ(t), combined with the compact embedding of the domain of A into the Hilbert space, under the standing assumptions on F and the data. We will also clarify the continuity of T. revision: yes
Referee: [Forward problem section] Forward problem section: The Fourier expansion is applied to obtain the solution series, but the global existence proof must explicitly derive uniform bounds on the coefficients that account for the time-variation of σ(t) (e.g., in the resulting Volterra-type integral equations or Gronwall estimates). Without these details, convergence of the series for general σ(t) is not fully supported.

Authors: We acknowledge that the uniform bounds on the Fourier coefficients require more explicit derivation to cover arbitrary continuous σ(t). In the revision, we will expand the estimates section by deriving a priori bounds for the coefficients via Gronwall-type inequalities applied directly to the resulting Volterra integral equations, showing that the bounds are independent of the mode index and of the truncation parameter. This will establish convergence of the series in the target space for general σ(t) satisfying the paper's assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity: forward solution via Fourier and inverse via external Schauder theorem

full rationale

The forward problem derives global existence/uniqueness from the Fourier expansion in the eigenbasis of A (discrete positive spectrum assumed) combined with standard estimates for the fractional evolution; this is independent of the target result. The inverse problem constructs a map from the forward solution and invokes Schauder's fixed-point theorem under the explicit hypothesis that F satisfies the required continuity/compactness conditions. No equation reduces to itself by definition, no fitted parameter is relabeled as a prediction, and no load-bearing step rests on a self-citation chain. The cited theorems (Fourier method, Schauder) are external and their applicability is stated as independent of the specific solution being proved.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard functional-analysis assumptions about the operator A and the applicability of Schauder's theorem; no free parameters or new entities are introduced.

axioms (2)

domain assumption A is an unbounded, positive, self-adjoint operator with discrete spectrum
Invoked to justify the Fourier method and spectral expansion.
standard math Schauder's fixed point theorem applies to the operator mapping constructed for the inverse problem
Used directly to obtain existence of the source function r(t).

pith-pipeline@v0.9.0 · 5549 in / 1231 out tokens · 90886 ms · 2026-05-14T18:21:40.928286+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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work page 2019

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A. V. Pskhu,Fractional Differential Equations(Moscow, Russia, Nauka, 2005)

work page 2005

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P. J. Chen and M. E. Gurtin, “On a theory of heat conduction involving two temperatures,” Z. Angew. Math. Phys. (19), 614–627 (1968)

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G. I. Barenblatt, I. P. Zheltov and I. N. Kochina, “Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks,” Journal of Applied Mathematics and Mechanics (24), 1286–1303 (1960)

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W. Rundell, “Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data,” Appl. Anal. (10) 231–242 (1980)

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D. K. Durdiev, H. B. Elmuradova, “Kernel determination problem in the fractional pseudo-integro-differential equation,” Rend. Circ. Mat. Palermo, II. (73), 3313–3326 (2024)

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An undetermined coefficient problem for a fractional diffusion equation,

Z. Zhang, “An undetermined coefficient problem for a fractional diffusion equation,” Inverse Problems (1), 1–21 (2016). 18R. Ashurov and E. Husanov

work page 2016

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An undetermined time-dependent coefficient in a fractional diffusion equation,

Z. Zhang, “An undetermined time-dependent coefficient in a fractional diffusion equation,” Inverse Probl. Imaging (5), 875–900 (2017)

work page 2017

[10] [10]

Inverse problem of determining time-dependent leading coeffi- cient in the time-fractional heat equation,

D. Serikbaev, M. Ruzhansky, N. Tokmagambetov, “Inverse problem of determining time-dependent leading coeffi- cient in the time-fractional heat equation,” Fract. Calc. Appl. Anal. (28), 2918–2968 (2025)

work page 2025

[11] [11]

Heat source determining inverse problem for nonlocal in time equation,

D. Serikbaev, M. Ruzhansky and N. Tokmagambetov, “Heat source determining inverse problem for nonlocal in time equation,” Math. Meth. Appl. Sci. (48), 1768–1791 (2025)

work page 2025

[12] [12]

M. M. Dzhrbashyan,Integral Transforms and Representation of Functions in the Complex Domain( Nauka, Moscow, 1966)

work page 1966

[13] [13]

On the Nonlocal Problems in time for time-fractional subdiffusion equations,

R. R. Ashurov, Yu. E. Fayziev, “On the Nonlocal Problems in time for time-fractional subdiffusion equations,” Fractal Fract.6(1) 41, (2022)

work page 2022

[14] [14]

Generalized Taylor’s formula,

Z. M. Odibat, N. T. Shawagfeh, “Generalized Taylor’s formula,” Appl. Math. Comput. (186), 286–293 (2007)

work page 2007

[15] [15]

A. A. Kilbas, H. M. Srivstava, J. J. Trujillo,Theory and applications of fractional differential equations(Elsevier Science, 2006)

work page 2006

[16] [16]

Henry,Geometric theory of semilinear parabolic equations(Springer-Verlag, Berlin, 1981)

D. Henry,Geometric theory of semilinear parabolic equations(Springer-Verlag, Berlin, 1981)

work page 1981

[17] [17]

A. N. Kolmogorov and S. V. Fomin,Elements of Theory of Functions and Functional Analysis(Graylock press, USA, 1957). Ravshan Ashurov: V. I. Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences, University street 9, Tashkent, 100174, Uzbekistan. Central Asian University, 264 Milliy Bog Street, Tashkent, 111221, Uzbekistan. Email address:ashu...

work page 1957