{"id":"a468a88f-2de6-4b5f-89d2-62273cfbe2cd","arxiv_id":"2501.10398","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A claimed resolution of Adomian's expansion-parameter dilemma via dimensionless perturbation theory that rests on invalid commutativity and a degenerate normalization.","lead":"This paper attempts to reformulate the Adomian decomposition method using dimensionless perturbation theory to fix an expansion-parameter ordering mismatch. The proposed fix relies on arbitrary scale choices and on commutativity assumptions between integration and nonlinear polynomials that do not hold, so the central resolution fails.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (27) asserts L^{-1}A_k = A_k L^{-1}, which is false for nonlinear polynomials; the recurrence (25) therefore does not follow, and the paper's central resolution rests on an unproven commutativity.","rationale":"The reader's weakest assumption is exactly Eq. (27), and I agree that this is the most load-bearing point: without it, Eq. (25) cannot be derived from Eq. (24). I add that the derivation also contains an index mismatch—Eq. (24) is an equation for u_k, while Eq. (25) claims a recurrence for u_{k+1}—so the step is doubly unsupported. A direct one-dimensional example shows the false commutativity changes even the first Adomian correction, so the proposed reformulation fails on the simplest nonlinear case. The paper provides no formal verification or numerical evidence to mitigate this. Therefore the reader's REJECT verdict is appropriate; my read does not move it.","tokens_in":9979,"tokens_out":7363,"duration_ms":62925,"concrete_test":"For u' = 1 - u^2, u(0)=0, set u_0(τ)=τ. Standard ADM gives u_1 = -L^{-1}(u_0^2) = -∫_0^τ s^2 ds = -τ^3/3. The paper's Eq. (25), using Eq. (27), gives u_1 = -λ A_0(L^{-1}u_0) = -λ(∫_0^τ s ds)^2 = -λτ^4/4. Setting λ=1, compare with the exact solution tanh τ = τ - τ^3/3 + 2τ^5/15 - ...; the first correction is wrong, directly showing that the commutativity asserted in Eq. (27) fails and that Eq. (25) does not reproduce the correct ADM series.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Eq. (25) resolves Adomian's order-mismatch dilemma. Eq. (25) is obtained from Eq. (24) only by moving L^{-1} past the nonlinear Adomian polynomials, via the asserted identities (26) and (27). Identity (27) is false. For example, for N(u)=u^2, A_1(u_0,u_1)=2u_0u_1, so with L=d/dτ and L^{-1}f=∫_0^τ f, L^{-1}A_1 = 2∫_0^τ u_0u_1, whereas A_1(L^{-1}u_0,L^{-1}u_1)=2(∫_0^τ u_0)(∫_0^τ u_1). These are generically unequal. Since every A_k for k≥1 is a homogeneous polynomial of degree k+1 in the components, the identity fails for all k≥1. Identity (26) is also not generally valid when R contains lower-order derivatives, because L^{-1}∂_τ u = u-u(0) while ∂_τ L^{-1}u = u. Moreover, even granting (27), applying L^{-1} to Eq. (24), ∂_τ^s u_k = -λR[u_k]-λA_k, yields an equation for u_k, not u_{k+1}; the index shift in Eq. (25) is not derived. No proof, numerical validation, or convergence check of identities (26)-(27) is supplied, so the load-bearing recurrence is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a reformulation of the Adomian decomposition method (ADM) using dimensionless nonlinear perturbation theory. The author identifies an 'Adomian dilemma': the recurrence derived from the decomposition (Eq. 6) is not reproduced if a perturbation-style series with an expansion parameter λ is substituted directly into the governing equation. The paper attempts to resolve this by nondimensionalizing the equation, introducing representative scales Ū and t0, assuming a scaling law for the Adomian polynomials (Eq. 13), and deriving a new recurrence (Eq. 25) via commutativity of the inverse operator with the remainder operator and the Adomian polynomials (Eqs. 26-27). The paper also claims connections to Lyapunov's stability theory and concludes that the order mismatch is resolved. No worked example or numerical validation is provided.","tokens_in":10427,"tokens_out":6300,"duration_ms":58567,"significance":"If the derivation were correct, the paper would offer a useful clarification of the relationship between ADM and perturbation theory. The literature review is reasonable, and the identification of the order-mismatch issue is a genuine topic in the ADM literature. However, the central derivation rests on several unproven and, in general, false identities, and the normalization steps eliminate the physical content they claim to introduce. The paper makes falsifiable claims but supplies no numerical or analytical validation. The strength of the paper is its clear statement of the problem; the weakness is that the proposed resolution is not supported by the mathematics presented.","major_comments":[{"comment":"The step from Eq. (24) to Eq. (25) is not justified. Eq. (24) states ∂τ^s u_k = -λ R[u_k] - λ A_k({u_j≤k}), so applying L^{-1} to both sides yields an expression for u_k, not u_{k+1}. To obtain u_{k+1}, one would need an equation involving ∂τ^s u_{k+1} on the left, but Eq. (24) has u_k on the right and the same index on the left. The index shift from k to k+1 is asserted, not derived, and this is load-bearing for the claimed resolution.","section":"§3.2, Eqs. (24)-(25)"},{"comment":"The commutativity identity L^{-1}A_k[{u_j≤k}] = A_k[L^{-1}{u_j≤k}] is false for general nonlinear operators. For example, take N(u)=u^2, so A_1=2u_0u_1. With L^{-1}f=∫_0^τ f, the left side is 2∫_0^τ u_0u_1, while the right side is 2(∫_0^τ u_0)(∫_0^τ u_1); these are generically unequal. Since A_k for k≥1 generally contains products of components, the identity fails for essentially all nonlinearities. Eq. (26) is also false when R contains lower-order derivatives, e.g., R=∂τ gives L^{-1}∂τ u = u-u(0), whereas ∂τ L^{-1}u = u. These identities are central to Eq. (25), so the recurrence is unsupported.","section":"§3.2, Eq. (27)"},{"comment":"The scaling law in Eq. (13) is not generally valid. The notation in Eq. (12) uses p, q, r as component indices in the arguments of A_n, but Eq. (13) turns these same symbols into exponents of the representative scales Ū, V̇, and U′. There is no reason that a sum of terms of different degrees, such as A_2 = f'(u0)u2 + (1/2)f''(u0)u1^2, should scale as a single monomial with p+q+r=n. For a general analytic nonlinearity, A_n contains terms of differing homogeneity, so a unique α_n^* does not exist. This undermines the subsequent nondimensionalization and the claim of a parameter-free resolution.","section":"§3.1, Eq. (13)"},{"comment":"The normalizations Ū=1 and t0^s=λ are dimensionally inconsistent and eliminate the physical content the paper claims to introduce. t0^s has dimensions T^s while λ is declared dimensionless, so Eq. (21) cannot hold as an equality of physical quantities. Setting Ū=1 is just a choice of units, and the subsequent λ=1 in Eq. (19) fixes t0=1, removing the representative time scale. The procedure recovers, by construction, the standard perturbation assumption that the nonlinear and remainder terms are multiplied by a small parameter λ; it does not derive this assumption from physical scaling.","section":"§3.2, Eqs. (20)-(22)"},{"comment":"No numerical example or convergence check is provided. Since the central claim is that Eq. (25) resolves the order mismatch and leads to faster convergence and higher accuracy, the absence of even a single worked nonlinear problem (compared with an exact or numerical solution) leaves the practical claims untested. This is particularly important given that the derivation of Eq. (25) is invalid.","section":"Whole paper"}],"minor_comments":[{"comment":"The abstract contains grammatical errors, e.g., 'unresolved issue regarding the mismatch' should be 'an unresolved issue regarding the mismatch', and the affiliation contains a typo: 'Unive rsity'.","section":"Abstract and affiliation"},{"comment":"The factor t0^s/U^{k-1} preceding A_k in Eq. (18) is not clearly defined for k=0, and its derivation from the nondimensionalization of A_k is not shown; please clarify the notation.","section":"Eq. (18)"},{"comment":"The paper uses 'Lyapunov's stability theory' and 'Lyapunov's artificial small parameter method' as if they were the same thing, but the connection between Lyapunov exponents (Refs. [42,43]) and the formal perturbation parameter is not established; this conflation should be addressed explicitly.","section":"Section 2 and Appendix"}],"recommendation":"reject","confidential_remarks":"The manuscript addresses a real problem in the ADM literature, but the proposed resolution is built on identities that are false for generic nonlinear operators and on an unjustified index shift. These are load-bearing errors that cannot be repaired by local revision; the paper would need a fundamentally different derivation and at least one validated example. I therefore recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Jordan — the stress-test note holds up on reading. This paper claims to resolve Adomian's expansion-parameter mismatch via a dimensionless perturbation reformulation, but the derivation fails at the load-bearing step: identities (26)–(27), used to turn Eq. (24) into the new recurrence (25), are false for nonlinear operators. For s=1, L=d/dτ, N(u)=u²: A₁=2u₀u₁, so L⁻¹A₁=2∫u₀u₁, whereas A₁(L⁻¹u₀,L⁻¹u₁)=2(∫u₀)(∫u₁); these differ generically, even for u₀=1. Identity (26) fails whenever R contains a lower-order derivative, since L⁻¹∂_τu=u−u(0) but ∂_τL⁻¹u=u. Neither identity is proved, and (25) depends on both.\n\nWhat is good: the paper is readable, frames the λ-mismatch (the parameter used to generate Adomian polynomials versus the λ-free recurrence) accurately, cites Zhang–Liang (2015) honestly, and the appendix on the combinatorics of A_n is a competent review. A reader wanting a compact statement of the ordering issue will find useful orientation here.\n\nThe soft spots are all load-bearing. The index shift in (25) is not derived: inverting (24) gives an equation for u_k, not u_{k+1}. The scaling law (13) is not general because A_n is a sum of products of varying degree in the components, not a fixed power of Ū; the claim that t₀^s/Ū^{k−1}A_k is unconditionally dimensionless is consequently wrong. The normalization is circular and dimensionally incoherent: Ū=1 strips the nondimensionalization, t₀^s=λ identifies a time scale with a dimensionless parameter, and λ=1 at the end removes the ordering the paper says it introduces. Eq. (23) even mixes powers of λ, since ∂^s u₀ is order 0 while gλ is order 1. There is no worked example, no numerics, and no convergence check.\n\nWho benefits: someone tracing the ADM/perturbation-theory literature gets a decent survey paragraph, not a usable result. My recommendation is to reject. If I were the editor, a desk reject pointing at (27) and the unjustified index shift is defensible; a single ADM-literate referee could document the flaw quickly, but a full review is more than this paper needs. I would not cite it for the resolution, and I would not bring it to the group.","headline":"The paper's proposed resolution of Adomian's order-mismatch dilemma fails because the commutativity identities (26)–(27) behind the new recurrence (25) are false for nonlinear operators, and the remaining normalization argument is circular.","tokens_in":10877,"tokens_out":19905,"would_cite":false,"duration_ms":163189,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that a dimensionless perturbation reformulation of the Adomian decomposition method, centered on the recurrence (25), resolves the long-standing mismatch in the order of the expansion parameter.","keywords":["Adomian decomposition method","nonlinear perturbation theory","expansion parameter order","dimensionless analysis","recurrence formula","Lyapunov stability theory","commutative operators","nonlinear differential equations"],"falsifier":"Take $f(u)=u^2$ with $u_0=t$ and $u_1=t$. The first Adomian polynomial is $A_1=2u_0u_1=2t^2$, and the claimed identity (27) would require $L^{-1}(2t^2)=2(\\int_0^t u_0\\,dt)(\\int_0^t u_1\\,dt)$, i.e. $\\frac{2}{3}t^3=\\frac{1}{2}t^4$, which is false. Checking this one equality for any simple nonlinearity settles the central claim.","tokens_in":9771,"feed_emoji":"📐","tokens_out":5255,"duration_ms":44701,"temperature":0.7,"pith_summary":"The paper tries to resolve the long-standing inconsistency in the Adomian decomposition method (ADM) between the order of the expansion parameter used to build Adomian polynomials and the order of the recurrence that generates the series terms. It reformulates the method as a dimensionless nonlinear perturbation theory, setting the dimensionless time scale equal to the expansion parameter $\\lambda$, and derives a new recurrence, Eq. (25), in which $\\lambda$ appears explicitly and uniformly. If the derivation holds, the expansion parameter acquires a physical meaning and the mismatch identified in earlier work disappears, opening a cleaner route to higher-order ADM terms. The argument stands or falls on a commutation claim about the inverse operator and the nonlinear polynomial, which is stated but not proved.","feed_headline":"New recurrence resolves Adomian method's parameter-order mismatch","feed_subtitle":"A dimensionless perturbative treatment gives λ a physical time scale and synchronizes every term's order.","key_machinery":"The load-bearing object is the dimensionless perturbation series $u(\\lambda)=\\sum_{n=0}^\\infty \\lambda^n u_n$, combined with the normalizations $\\bar U=1$ and $t_0^s=\\lambda$, which converts the governing equation into a standard perturbation hierarchy. The recurrence (25) is produced by applying the inverse operator $L^{-1}$ termwise and assuming the commutation identities $L^{-1}R[u_k]=R[L^{-1}u_k]$ and $L^{-1}A_k[\\{u_{j\\le k}\\}]=A_k[L^{-1}\\{u_{j\\le k}\\}]$ (Eqs. (26)-(27)); the second identity is what lets the nonlinear Adomian polynomial be evaluated after integration. This step is essential because without it the new recurrence does not follow from the perturbation hierarchy.","core_discovery":"On the paper's own terms, the central discovery is that Adomian's dilemma comes from using $\\lambda$ only as a temporary bookkeeping device when constructing the Adomian polynomials, while deriving the recurrence without any $\\lambda$. After nondimensionalizing the governing equation and choosing the dimensionless time scale as $t_0^s = \\lambda$, the paper obtains the recurrence $$u_{k+1} = -\\$\\lambda$ R[$L^{{-1}}$(u_k)] - \\$\\lambda$ A_k($L^{{-1}}$\\{u_{j\\le k}\\}), \\tag{25}$$ in which every term is visibly of matched order in $\\lambda$. The paper claims this recurrence, together with the commutation relations (26)-(27), resolves the order mismatch and ties ADM to standard perturbation theory with a physically dimensioned parameter.","pith_inferences":["The commutation identity (27) is not generally valid for polynomial nonlinearities, so the new recurrence (25) is not equivalent to the standard ADM recurrence unless the class of nonlinearities is restricted; the paper does not state such a restriction.","If $L^{-1}$ is kept outside the Adomian polynomial, the standard recurrence (6) follows instead, suggesting the 'dilemma' may be a bookkeeping artifact of where $L^{-1}$ is placed rather than a physical inconsistency.","A direct numerical test would compare the series generated by (25) with the standard ADM series for a nonlinear oscillator or a convection-diffusion-reaction equation with a known exact solution; the two series already differ at the $u_1$ term whenever the nonlinearity is not linear."],"forward_implications":["If Eq. (25) is correct, every component $u_{k+1}$ carries exactly one more power of $\\lambda$ than $u_k$, so the final series is order-consistent in the expansion parameter.","The expansion parameter acquires a physical dimension as $t_0^s$, connecting ADM's abstract $\\lambda$ to a real time scale in transport and wave problems.","The scaling property (13) extends the construction to nonlinearities involving $U$, $\\dot U$, and $U'$, so derivative-coupled nonlinear terms are covered by the same dimensionless ordering.","Truncating the series and setting $\\lambda=1$ gives an analytical approximation whose terms are synchronized, which the paper argues improves the basis for convergence and accuracy of ADM solutions."],"supporting_citations":[{"why":"Identifies the parameter-order mismatch and frames ADM as a special case of Lyapunov's artificial small parameter method; this is the paper's target.","marker":"[25]"},{"why":"Introduced $\\lambda$ as a term-collecting device in constructing Adomian polynomials.","marker":"[26]"},{"why":"Provides the convenient computational form of Adomian polynomials and asserts $\\lambda$ is not a true perturbation parameter.","marker":"[27]"},{"why":"Defines the decomposed series $U=\\sum U_n$ that underlies the ADM recurrence.","marker":"[9]"},{"why":"Serves as the canonical book-length source for Adomian polynomials and frontier problems.","marker":"[10]"},{"why":"Documents convergence limitations of ADM for initial-value problems, motivating the reformulation.","marker":"[16]"},{"why":"Supplies the convection-diffusion-reaction example used to instantiate the general operator form.","marker":"[44]"}],"fun_headline_variants":["ADM fix: dimensionless λ recurrence matches all orders","λ gains a time scale, Adomian orders sync up","Nondimensional reformulation resolves ADM's λ-order flaw","Dimensionless λ aligns Adomian terms' order"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation requires that the inverse integral operator $L^{-1}$ can be pushed inside the nonlinear Adomian polynomial, so that integrating the polynomial equals evaluating the polynomial on integrated components; this equality does not hold for ordinary polynomial nonlinearities such as $u^2$.","fun_headline_variants_meta":{"raw":{"variants":["ADM fix: dimensionless λ recurrence matches all orders","λ gains a time scale, Adomian orders sync up","Nondimensional reformulation resolves ADM's λ-order flaw","Dimensionless λ aligns Adomian terms' order"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00117,"raw_usage":{"total_tokens":4762,"prompt_tokens":791,"completion_tokens":3971,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":407,"completion_tokens_details":{"reasoning_tokens":3904}},"tokens_in":407,"tokens_out":3971,"duration_ms":27796,"temperature":1.0,"reasoning_tokens":3904,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:38:12.159468+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $f(u)=u^2$ with $u_0=t$ and $u_1=t$. The first Adomian polynomial is $A_1=2u_0u_1=2t^2$, and the claimed identity (27) would require $L^{-1}(2t^2)=2(\\int_0^t u_0\\,dt)(\\int_0^t u_1\\,dt)$, i.e. $\\frac{2}{3}t^3=\\frac{1}{2}t^4$, which is false. Checking this one equality for any simple nonlinearity settles the central claim.","supporting_citations":[{"cited_title":"Applied Mathematics Letters 48, 177–179 (2015) https://doi.org/10","cited_arxiv_id":null,"evidence_quote":"Identifies the parameter-order mismatch and frames ADM as a special case of Lyapunov's artificial small parameter method; this is the paper's target."},{"cited_title":"Journal of Mathematical Analysis and Applications 91(1), 39–46 (1983) https://doi.org/10.1016/0022-247X(83)90090-2","cited_arxiv_id":null,"evidence_quote":"Introduced $\\lambda$ as a term-collecting device in constructing Adomian polynomials."},{"cited_title":"Computers & Mathematics with Applications 22(8), 91–94 (1991) https://doi.org/10.1016/ 0898-1221(91)90017-X","cited_arxiv_id":null,"evidence_quote":"Defines the decomposed series $U=\\sum U_n$ that underlies the ADM recurrence."},{"cited_title":"Springer, Dordrecht (1994)","cited_arxiv_id":null,"evidence_quote":"Serves as the canonical book-length source for Adomian polynomials and frontier problems."},{"cited_title":"Numerical Methods for Partial Differential Equatio ns 27(4), 749–766 (2011) https://doi.org/10.1002/num.20549","cited_arxiv_id":null,"evidence_quote":"Documents convergence limitations of ADM for initial-value problems, motivating the reformulation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the convection-diffusion-reaction example used to instantiate the general operator form."}],"review_version":1}