Pith. sign in

REVIEW 5 major objections 6 minor 1 cited by

Inverse Stefan problems of determining the time-dependent source coefficient and heat flux function

T0 review · 5 major / 6 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 The paper claims that four inverse Stefan problems—recovering a time-dependent source coefficient and, in Neumann cases, the heat flux—each reduce to a Volterra integral equation with existence, uniqueness, and continuous dependence.

desk verdict A spectral-formal treatment of four inverse Stefan problems whose central mode decoupling is invalid and whose recovery formula is circular; the combination is new but the theorems are not established. read the letter →

arxiv 2501.11332 v1 pith:YPOARINM submitted 2025-01-20 math.AP math-phmath.MPmath.SP

Targyn A. Nauryz , Khumoyun Jabbarkhanov This is my paper · ORCID
classification math.APmath-phmath.MPmath.SP MSC 35R3035R3535K20
keywords inverseStefanproblemtime-dependentsourcecoefficientheatfluxrecoveryweaksolutionspectralmethodVolterraintegralequationcontinuousdependencephase-changeconduction
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 a moving-boundary Stefan problem can be solved backwards: from boundary data and the known phase-change interface, one can recover the time-dependent source coefficient R(t) (or P(t)) and, in the Neumann cases, the heat flux q(t). The authors transform the moving domain to a fixed interval, expand the temperature in eigenfunctions of that interval, and reduce each inverse problem to a scalar Volterra integral equation. If the theorems are right, existence, uniqueness, and continuous dependence on the data hold for all four problem formulations. The practical stake is that heat-source strength in phase-change materials could be identified from ordinary boundary measurements.

What carries the argument

The machinery is a Fourier expansion in the Dirichlet eigenfunctions φ_n=√2 sin(nπξ) or the Neumann eigenfunctions φ_n=√2 cos((2n-1)πξ/2) of the fixed interval (0,1), reached by the change of variables ξ=x/s(t). The load-bearing reduction is the claim in (20)-(21) that the moving-boundary term (b(ξ,t)Uξ,φ_n) equals b(t)U_n(t) with the same index n, so the PDE becomes an infinite set of decoupled ordinary differential equations for the Fourier coefficients U_n(t). Solving those ODEs turns the Stefan boundary condition into a series; differentiating it in t produces the Volterra integral equation (26) for R(t), and the same route yields q(t) from (62). The explicit identities for the eigenfunctions and the coefficient estimates in Lemmas 1-4 are what make the series and the integral equation convergent.

What would settle it

Compute the projection integrals ∫$_0^{1}$ ξ φ_m'(ξ) φ_n(ξ) dξ for the sine basis; for m≠n they are nonzero, so (b Uξ,φ_m) contains cross terms B_{mn}(t)U_n(t). Evaluating the residual of (11) with U from (23) at a time when s'(t)≠0 decides whether the paper's constructed solution is actually a weak solution.

Watch

Extended reading notes

Core claim

The central claim is that the weak solution pair (R,U) of the transformed inverse Stefan problem exists and is unique: U is given by the Fourier series (23), and R(t)>0 is determined by the Volterra equation (26). In the Neumann formulation the unknown heat flux q(t) is recovered from an explicit series formula (62) once R(t) is known, and in the two cases with source term P(t)u+f the substitution R(t)=exp(-∫P) converts a linear-in-u source into the same Volterra structure. The paper further claims continuous dependence of U and R (or q) on the data functions ψ, h, s, c, with explicit estimates in Theorems 3, 4, 6, 7, 9, and 10. If correct, this gives a parameter-free recovery procedure for time-dependent source coefficients in phase-change heat conduction.

Load-bearing premise

The proof assumes that the convection term ξ s'(t)/s(t) Uξ does not mix Fourier modes, so that the diagonal equations (21) describe the full solution; if the off-diagonal mode couplings are nonzero, the series (23) built from independent modes need not solve the PDE (11).

Editorial extensions

If this is right

  • In the Dirichlet case, the interface condition alone determines R(t) through the Volterra equation (26), without needing extra boundary measurements.
  • In the Neumann case, the heat flux q(t) on the left boundary is recovered from an explicit series once R(t) is known, so one can infer both the source strength and the boundary flux from the same Stefan condition.
  • The linear source case P(t)u+f reduces to the R(t)f case by R(t)=exp(-∫P), so existence and uniqueness for P(t) follow without a separate analysis.
  • The continuous-dependence estimates imply that small errors in the measured data (initial temperature, source shape, interface speed) cause small changes in the recovered coefficient in the sup norm.

Reading between the lines

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

  • The off-diagonal mode-coupling check is the natural next step: if it fails, the Volterra equation (26) would describe a diagonalized problem rather than the original PDE, and the recovery claims would need extra terms.
  • Because the paper's formulas are explicit, numerical implementation is immediate: truncate the series, solve the Volterra equation by quadrature, and compare the resulting R(t) and U(ξ,t) against direct Stefan simulations for slowly moving interfaces.
  • The same spectral ansatz could be tried for other boundary conditions or higher dimensions, but any such extension inherits the diagonal-mode assumption, which would need separate verification.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. This paper studies four inverse Stefan problems for the one-dimensional heat equation on the moving domain (0,s(t)). In each case the unknowns include a time-dependent source coefficient (R(t) or P(t)) and, for the Neumann problems, the boundary heat flux q(t). The authors transform each problem to a fixed domain, expand the solution in the eigenfunctions of the Dirichlet or Neumann spectral problem, and derive explicit Fourier series representations for U together with Volterra-type equations for the unknown coefficient R or q. The main claims are existence, uniqueness, and continuous dependence of the weak solution pair under regularity and sign assumptions (A1)-(A3) or (B1)-(B4).

Significance. If the central construction were correct, the paper would supply explicit spectral formulas for recovering time-dependent source coefficients and heat fluxes in Stefan problems, together with a systematic continuous-dependence framework; the presentation of four problem formulations in a unified way is also useful. However, the derivation rests on a spectral decoupling that is not valid for the advection term in the transformed equation, and the recovery formulas for R depend on transformed source functions that themselves contain R. As a result, the main theorems do not currently establish the announced results.

major comments (5)
  1. [§2, Eqs. (17)–(21)] The reduction from the weak formulation (17) to the diagonal system (21) is invalid. The advection contribution in the m-th equation is Σ_n U_n(t) ∫_0^1 b(ξ,t) φ'_n(ξ) φ_m(ξ) dξ with b(ξ,t)=ξs'(t)/s(t), and these integrals do not vanish for m≠n; for instance, with s(t)=1+t at t=0 and b(ξ)=ξ, ∫_0^1 ξ φ'_1(ξ) φ_2(ξ) dξ = 4/3 ≠ 0. Hence the modal series (23) built from the independent equations (22) does not satisfy the full weak formulation (17), and the paper provides no argument that the discarded m≠n equations are satisfied. Lemma 2 does not repair the gap: its proof only establishes norm bounds for the series (23) and never verifies the PDE (11) or the weak form (17).
  2. [§2, Eqs. (9) and (26)] The recovery formula (26) is not a closed Volterra equation for R(t). The transformed source is defined in (9) by h(x,t)=f(x,t)+u_* s'(t)/(R(t)s²(t))x, so the Fourier coefficients ~h_n(t) used in (23) and (26) depend on the unknown R(t) itself. Thus R appears nonlinearly on both sides of (26), including in the denominator w(t)=Σ(-1)^n√λ_n ~h_n(t), and the assertion in Theorem 1(iii) that R is 'defined by (26)' is circular. The same circularity affects the corresponding formulas in Sections 3–5 and the uniqueness arguments built on them.
  3. [§2, Lemma 2 and Theorem 1] The proof of Lemma 2 does not demonstrate that the function (23) is a weak solution of the problem: it only shows that the series belongs to C([0,T];L²(Ω₀)) and C([0,T];H²₀(Ω₀)) under certain summability assumptions, without checking the equation, the boundary conditions, or the initial condition. Moreover, estimate (35) for U_t involves R'(t), while R is only assumed to be in C[0,T] in Theorem 1(iii); no higher regularity of R is established before it is used in the proof.
  4. [§2, hypotheses (A3) and (B3)] The assumptions on the given data are stated in terms of the unknown solution. For example, (A3)1 requires f(s(t),t) = -u_* s'(t)/(R(t)s(t)) and f'(0,t)=f'(s(t),t)=-u_* s'(t)/(R(t)s²(t)), so the admissibility of the data depends on the coefficient R(t) that is to be determined. This makes the hypotheses not checkable from the data alone and compounds the circularity of the recovery formula.
  5. [§2, Eq. (38)] The uniqueness proof in Theorem 1 contains a vacuous identity: equation (38) states Û - Ũ = Σ ∫_0^t (R̂(τ) - R̂(τ)) ̃f_n(τ) ... dτ φ_n(ξ), whose integrand vanishes identically. The subsequent conclusion that R̂ = R̃ from (39) therefore does not follow from the written formulas.
minor comments (6)
  1. [§2, Eq. (9)] The initial condition after the transformation is written as ψ(x)=φ(x)-u_*/s(t)x; it should be evaluated at t=0, i.e., with s(0).
  2. [§2, Eq. (17)] The term +a(t)λ_n(∂ξξU,φ_m) is not the correct weak form of -a(t)∂ξξU; after integration by parts the eigenvalue enters as a factor of (U,φ_m), not of (∂ξξU,φ_m).
  3. [§2, Lemma 1 proof] In the proof of Lemma 1, the function g is defined as -~ψ'''_n(ξ), but the subscript n should be removed; it is the third derivative of ~ψ, not of the Fourier coefficient.
  4. [§2, Lemma 3] The claimed bound (36) is dimensionally inconsistent unless additional summability of 1/λ_n is used; the constant C8 is not specified.
  5. [§2, Theorem 2] The statement of Theorem 2 contains the apparent typo 'L2(0,T;L2(0,Ω))', which should presumably be L²(0,T;L²(Ω)).
  6. [References] The reference [Evan(2010)] should be Evans, Partial Differential Equations, and the year of [Dragomir(2022)] in the reference list is inconsistent with the in-text citation.

Circularity Check

4 steps flagged · score 8.0 of 10

The unknown coefficient is baked into the 'known' source term before recovery, so (26), (62), (75), and (85) are self-referential rather than reconstructions from data.

  1. self definitional [Section 2, Eq. (9) and Theorem 1/Eq. (26)]
    "where ψ (x) = ϕ (x) − u∗/s(t)x and h(x,t ) = f (x,t ) + u∗s′(t)/R(t)s2(t)x. ... (iii) R(t) ∈C[0,T ] and R(t)> 0 for all t ∈ [0,T ] which is defined by (26)."

    The transformed source term h, and hence its Fourier coefficients ~h_n(t) used in the series representation (23) and in the 'recovery' formula (26), contain the unknown R(t) in the denominator. Equation (26) is therefore R(t) expressed through R(t) itself, not an explicit determination from the data f, φ, s. The uniqueness and continuous-dependence arguments treat ~h as given input, which presupposes the very coefficient to be recovered.

  2. self definitional [Section 3, Eq. (51) and Theorem 5/Eq. (62)]
    "where h(x,t ) = f (x,t ) + q′(t)/kR(t) (x −s(t)) − q(t)s′(t)/kR(t) and ψ (x) = ϕ (x) −u∗ + q(0)/k (x −s(0))."

    The unknown heat flux q(t) and its time derivative q′(t) appear in h, and q(0) appears in the transformed initial condition ψ. Formula (62) then states q(t) in terms of the Fourier coefficients ~h_n(t) of this same q-dependent h. The paper's Theorem 5 therefore presents q as determined by data that are themselves defined through q; the same defect carries into the R-formula (65).

2 more flagged steps
  1. self definitional [Section 4, Eq. (71) and Theorem 8/Eq. (75)]
    "where g(x,t ) = f (x,t ) − u∗(R′(t)s(t)−R(t)s′(t))/s2(t)R(t) x and p(x) = ϕ (x) − u∗R(0)/s0 x."

    Here R(t) is the unknown to be determined (with P(t)=−R′(t)/R(t)), yet g contains R′(t)/R(t) and p contains R(0). The formula (75) for R(t) is expressed through Fourier coefficients of ~p and ~g, so the right-hand side depends on R(0), R(t), and R′(t). The 'initial data' ~p is not known until R(0) is known, making Theorem 8 a fixed-point relation rather than a recovery of R from ϕ, f, s.

  2. self definitional [Section 5, Eq. (83) and Theorem 11/Eq. (85)]
    "where h(x,t ) = f (x,t ) − u∗R′(t)/R(t) + (q′(t)/k + q(t)R′(t)/kR(t))(x −s(t)) − q(t)/k s′(t) and g(x) = ϕ (x) − R(0)u∗ + R(0)q(0)/k (x −s(0))."

    The same structural defect appears: the unknown pair (R,q) is built into both h and g before the recovery step. Equation (85) determines q(t) from Fourier coefficients of ~h and ~g, which already depend on q(t), q′(t), R(t), R′(t), and R(0). Thus Theorem 11's claimed existence and uniqueness of (U,q) reduces to a self-referential equation whose coefficients are not known data.

full rationale

The paper's central claims are not supported by an independent derivation: in every formulation the 'known' right-hand side after the domain transformation is defined in terms of the unknown coefficient itself. In Section 2, h=f+u*s'/(R s^2)x, so Theorem 1's R defined by (26) is expressed through ~h, which is a function of R; the same occurs for q in Section 3, for R in Section 4 via g,p, and for (R,q) in Section 5 via h,g. These are not merely technical gaps: they are cases of pattern 1, self_definitional, because the quantity to be recovered is an argument of the input functions. The self-citations to the authors' earlier Stefan papers play no load-bearing role in the spectral inverse construction, so no self-citation circularity is counted. There is also a separate, non-circular correctness concern not counted here: the diagonal mode truncation in (20)-(21) ignores nonzero off-diagonal advection terms ∫ b φ′_n φ_m with b=ξs'/s, so the series (23) need not solve the weak formulation (17); that is a mathematical error rather than an input/output equivalence. Because the central existence-uniqueness claims reduce to self-referential formulas, the circularity score is high.

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

No numerical parameters are fitted in this paper; all constants in the inequalities are generic bounds without assigned values. The problem data f, φ, and s are inputs, not free parameters. The axioms listed capture the load-bearing unproved assumptions: decoupling of Fourier modes, treating R-dependent ~h as known, and a lower bound on the denominator w(t). No new physical entities are introduced.

assumptions (5)
  • ad hoc to paper The spectral Galerkin system with m=n is closed; all off-diagonal coupling terms ∫ b(ξ,t) φ_m' φ_n dξ with m≠n vanish or can be neglected.
    No proof is given. This assumption is used to pass from (17) to the scalar ODE (21) and is false for b(ξ,t)=ξs'(t)/s(t) in the sine basis, where the integrals are generally nonzero.
  • ad hoc to paper The function ~h defined through h=f+u*s'/(R s^2)ξ can be treated as known data in the Volterra equation (26).
    The unknown R(t) appears in the denominator of h, but Theorems 1 and 4 treat ~h as an independent input. This makes the recovery formula implicit in a way the proofs do not address.
  • domain assumption The denominator w(t)=Σ(-1)^n√λ_n~h_n(t) is bounded away from zero on [0,T].
    Needed to define R(t) from (26). The lemmas provide upper bounds on the series, not a lower bound on w(t), and the alternating signs make vanishing plausible.
  • domain assumption Compatibility and positivity conditions (A1)-(A3), (B1)-(B4) hold, including sign conditions on Fourier coefficients such as ~ψ_1>0 and ~ψ_n≥0.
    These assumptions are imposed in Theorems 1, 2, 5, 8, and 11. Some conditions are stated as consequences of physical boundary data, but the derivations contain inconsistencies and the assumptions are not verified for the examples considered.
  • standard math Standard spectral theory, Bessel and Parseval inequalities, Gronwall's inequality, and maximum principles apply to the infinite series solutions.
    Invoked throughout Lemmas 1-7 and Theorems 3, 6, 9. The maximum-principle applications require additional boundedness of R(t) that is not established.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Inverse Stefan problems of determining the time-dependent source coefficient and heat flux function." pith.science (2026). https://pith.science/paper/YPOARINM

@misc{pith2026250111332,
  author       = {Pith},
  title        = {Pith review of: Inverse Stefan problems of determining the time-dependent source coefficient and heat flux function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPOARINM}},
  note         = {Machine review of arXiv:2501.11332}
}
read the original abstract

This paper delves into the Inverse Stefan problem, specifically focusing on determining the time-dependent source coefficient in the parabolic heat equation governing heat transfer in a semi-infinite rod. The problem entails the intricate task of uncovering both temperature- and time-dependent coefficients of the source while accommodating Dirichlet and Neumann boundary conditions. Through a comprehensive mathematical model and rigorous theoretical analysis, our study aims to provide a robust methodology for accurately determining the source coefficient from observed temperature and heat flux data in problems with different cases of the source functions. Importantly, we establish the existence and uniqueness, and estimate the continuous dependence of a weak solution upon the given data for some inverse problems, offering a foundational understanding of its solvability.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Two-phase source and reaction coefficient Stefan type problems

    math-ph 2026-07 reject novelty 6.0 of 10

    For two-phase Stefan problems, the paper derives spectral reconstruction formulas and claims existence and uniqueness of recovered time-dependent source and reaction coefficients.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper

  1. [1]

    Chen and H.-G

    [Chen(2021)] H. Chen and H.-G. Chen. Estimates of Dirichlet eigenvalues for a class of sub-elliptic operators, Proc. London Math. Soc., 122 (2021), 808-847

    work page 2021

  2. [2]

    [Chow(2021)] Y. T. Chow, et al. Surface-localized transmission eigenstates, super-resolution imaging, and pseudo surface plasmon modes, SIAM J. Imaging Sci., 14 (2021), 946-975

    work page 2021

  3. [3]

    Cannarsa, J

    [Cannarsa(2010)] P. Cannarsa, J. Tort, M. Yamamoto. Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26(10) (2010), 105003

    work page 2010

  4. [4]

    [Cannon(1991)] J. R. Cannon, Y. Lin, S. Wang. Determination of a control parameter in a parabolic partial differential equation, J. Aust. Math. Soc. Ser. B, 33 (1991), 149-163

    work page 1991

  5. [5]

    [CannonRun(1991)] J. R. Cannon, W. Rundell. Recovering a time dependent coefficient in a parabolic differential equation, J. Math. Anal. Appl., 160 (1991), 572-582

    work page 1991

  6. [6]

    [Gol'dman(1970)] M. L. Gol’dman. Estimates of the eigenfunctions of Laplace’s operator in certain domains, Differ. Uravn., 6 (1970), 2030-2040

    work page 1970

  7. [7]

    Hazanee, D

    [HazLesIsm(2019)] A. Hazanee, D. Lesnic, M. I. Ismailov, N. B. Kerimov. Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions, Applied Mathematics and Computation, 346 (2019), 800-815

    work page 2019

  8. [8]

    Ivanchov

    [Ivanchov(2003)] M. Ivanchov. Inverse Problems for Equations of Parabolic Type, Math. Stud. Monogr. Ser., 10 VNTL Publishers, Lviv, 2003

    work page 2003

Show all 26 references
  1. [9]

    Karazym, T

    [KarOzaSur(2020)] M. Karazym, T. Ozawa, D. Suragan. Multidimensional inverse Cauchy problems for evolution equations, Inverse Probl. Sci. En., 28 (2020), 1582-1590

    2020

  2. [10]

    [Lesnic(2022)] D. Lesnic. Inverse Problems with Applications in Science and Engineering, CRC Press, Boca Raton, FL, 2022

    2022

  3. [11]

    [IsmOzaSur(2024)] M. I. Ismailov, T. Ozawa, D. Suragan. Inverse problems of identifying the time-dependent source coefficient for subelliptic heat equations, Inverse Problems and Imaging, 18(4) (2024), pp.813-823

    2024

  4. [12]

    Evans, Partial Differential equation, AMS, Providence, 2010

    [Evan(2010)] L.C. Evans, Partial Differential equation, AMS, Providence, 2010

    2010

  5. [13]

    Dragomir, Some Gronwall Type Inequalities and Applications, RGMIA Monographs, Victoria University, Australia, 2002

    [Dragomir(2022)] S.S. Dragomir, Some Gronwall Type Inequalities and Applications, RGMIA Monographs, Victoria University, Australia, 2002

    2022

  6. [14]

    Two-phase Stefan problem for generalized heat equation

    [KhNa(2020)] Kharin S.N., Nauryz T.A. Two-phase Stefan problem for generalized heat equation. News of the National Academy of Sciences of the Republic of Kazakhstan, Physico-Mathematical Series 2(330), (2020), 40-49

    2020

  7. [15]

    Solution of two-phase cylindrical direct Stefan problem by using special functions in electrical contact processes

    [KhaNau(2021)] Kharin S.N., Nauryz T.A. Solution of two-phase cylindrical direct Stefan problem by using special functions in electrical contact processes. International Journal of Applied Mathematics, 34(2), (2021), 237–248

    2021

  8. [16]

    An approach for solving an inverse spherical two-phase Stefan problem arising in modeling of electric contact phenomena

    [SarErNauH(2017)] Sarsengeldin M.M., Erdogan A.S., Nauryz T.A., Nouri H. An approach for solving an inverse spherical two-phase Stefan problem arising in modeling of electric contact phenomena. Mathematical Methods in the Applied Sciences, 2017 (2017), 850–859

    2017

  9. [17]

    Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

    [NauBri(2023)] Nauryz T., Briozzo A.C. Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients. Nonlinear Analysis: Real World Applications, 74, (2023), 103944

    2023

  10. [18]

    Mathematical model of a short arc at the blow-off repulsion of electrical contacts during the transition from metallic phase to gaseous phase

    [KhaNau(2021)] Kharin S.N., Nauryz T.A. Mathematical model of a short arc at the blow-off repulsion of electrical contacts during the transition from metallic phase to gaseous phase. AIP Conference Proceedings, 2325,(2021), 020007

    2021

  11. [19]

    [NauKha(2022)] Nauryz T.A., Kharin S.N. Existence and uniqueness for one-phase spherical Stefan problem with nonlinear thermal coefficients and heat flux condition, International Journal of Applied Mathematics, 35(5), (2022), 645-659

    2022

  12. [20]

    One-phase spherical Stefan problem with temperature dependent coefficients

    [NaKh(2021)] Kharin S.N., Nauryz T.A. One-phase spherical Stefan problem with temperature dependent coefficients. Eurasian Mathematical Journal, 12(1), (2021), pp. 49-56

    2021

  13. [21]

    [BoBrKhNa(2024)] Bollati J., Brizzo A.C., Kharin S.N., Nauryz T.A. Mathematical model of thermal phenomena of closure electrical contact with Joule heat source and nonlinear thermal coefficients, Nonlinear Journal of Non-Linear Mechanics, 158, (2024), 104568

    2024

  14. [22]

    International Journal of Non-Linear Mechanics, 157 (2023), 104533

    [NauKas(2023)] Nauryz T., Kassabek S.A., Mathematical modeling of the heat process in closure electrical contacts with a heat source. International Journal of Non-Linear Mechanics, 157 (2023), 104533

    2023

  15. [23]

    [KasSur(2022)] S. A. Kassabek, D. Suragan, Numerical approximation of the one-dimensional inverse Cauchy-Stefan problem using heat polynomials methods, Computational and Applied Mathematics, 41:189, (2022)

    2022

  16. [24]

    [KasSur(2023)] S. A. Kassabek, D. Suragan, A heat polynomials method for the two-phase inverse Stefan problem, Computational and Applied Mathematics, 42:129, (2023)

    2023

  17. [25]

    Kerimov, M

    [KerIsm(2015)] N.B. Kerimov, M. I. Ismailov, Direct and inverse problems for the heat equation with a dynamic-type boundary condition, IMA Journal of Applied Mathematics, 80(5) (2015), pp. 1519–1533,

    2015

  18. [26]

    Derbissaly, M

    [DerSad(2024)] B. Derbissaly, M. Sadybekov, Inverse source problem for multi-term time-fractional di usion equation with nonlocal boundary conditions, AIMS Mathematics, 9(4) (2024), pp. 9969-9988. thebibliography document

    2024

Pith tools

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