{"id":"23973c96-42d1-4ec8-a617-ae3047401f04","arxiv_id":"1908.02306","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Two new non-classical Lagrange basis families are introduced for Müntz pseudo-spectral methods, with error bounds, Erdélyi-Kober fractional differentiation matrices, and numerical tests.","lead":"This paper introduces two new families of basis functions for spectral methods, based on Müntz and Jacobi functions, and derives error bounds and fractional differentiation matrices. A generalist reader might care because the methods target fractional differential equations with singular solutions, where standard smooth bases converge slowly.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 3.9 uses interpolation spaces whose exponents differ from the actual LMF basis spans; this makes the central interpolant construction internally inconsistent as written.","rationale":"The reader's weakest_assumption identifies the same two issues: the internal inconsistency in Definition 3.9 and the heavy dependence on unverified results from the companion preprint [13]. My stress-test confirms that the inconsistency is real and located precisely in the exponent mismatch between the LMF basis (3.1) and the spaces (2.15)–(2.16). This matters because Definition 3.9 is the foundation for the two new interpolants and their discrete orthogonality, which are in turn used in the error bounds and differentiation-matrix derivations. I do not see a fatal flaw that would force rejection: the error-bound arguments can likely be recovered by separating the approximation space from the product space, and the numerical experiments are consistent with the claimed behavior. However, the manuscript is not yet self-contained or internally consistent enough to accept as is. The need to fix the space definitions, add the missing parameter checks, and either include proofs or clearly verify the companion results supports the existing CONDITIONAL verdict rather than a stronger one. I therefore recommend no change to the reader's verdict.","tokens_in":27548,"tokens_out":37965,"duration_ms":372427,"concrete_test":"Check membership of the basis functions (3.1) in the spaces (2.15)–(2.16): compute the leading exponent of 1L_r and compare it with the exponents in P_N^{(β,μ,σ,η)}, and repeat for 2L_r with P_N^{(α,σ,η)}. If membership fails, replace the codomains in Definition 3.9 by the spans in (3.26)–(3.27), introduce a separate symbol for the product spaces used in (3.4), and re-verify Theorems 3.12, 3.16, and 3.17 under the corrected definitions, including the parameter restrictions β-μ > -1 and α-μ > -1 from Remark 2.10.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing soft spot is the mismatch between the spaces P_N^{(β,μ,σ,η)} and P_N^{(α,σ,η)} defined in (2.15)–(2.16) and the LMFs-1/2 basis functions in (3.1). The basis 1L_r^{(β,μ,σ,η)}(x) = (x/x_r)^{σ(β-η-μ)} h_r^σ(x) lies in span{x^{σ(β-η-μ)+kσ}}, while (2.15) defines P_N^{(β,μ,σ,η)} as span{x^{2σ(β-η-μ)+kσ}}. Similarly, 2L_r has the factor (b^σ-x^σ)^α x^{ση}, while (2.16) uses (b^σ-x^σ)^{2α} x^{2ση}. Definition 3.9 states that the new interpolants map C(Λ) into these P_N spaces, and the discrete orthogonality statements immediately after quantify over ψ ∈ P_N. As written, the nodal interpolants built from (3.1) are not elements of the stated codomains, so the definition of the interpolants and the claimed discrete orthogonality are not justified. The quadrature-exactness identities (3.3)–(3.4) use P_N in a different sense, as a product space containing products of two basis functions, so the same symbol carries two incompatible meanings. This can likely be repaired by introducing separate notation for the LMF span and the product space, but the central construction as stated is non-rigorous. A compounding issue is that the fractional differentiation formulas in Remark 2.10 and the Gauss-Jacobi-Müntz quadrature rules in Theorem 2.13 are imported from the companion preprint [13] without proof or explicit parameter checks such as β-μ > -1 and α-μ > -1, so Theorems 3.18–3.21 inherit any error or omitted restriction in [13].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces two new families of non-classical Lagrange basis functions (LMFs-1 and LMFs-2, Eq. (3.1)) built from Jacobi-Müntz functions, defines two associated interpolants, proves weighted error bounds of order O(N^{-m}) (Theorems 3.12, 3.16 and 3.17), and derives left- and right-sided Erdélyi-Kober fractional differentiation matrices both by direct nodal expansion (Theorems 3.18–3.19) and by a collocation-matrix inversion approach (Theorems 3.20–3.21 and 3.23). The numerical section applies the bases to fractional ODEs and PDEs, including a nonlinear Burgers example with endpoint singularities. The theoretical development relies on orthogonality, fractional derivative and quadrature results imported without proof from the companion preprint [13].","tokens_in":28049,"tokens_out":25831,"duration_ms":238939,"significance":"If the technical results hold, the construction unifies and generalizes the four Lagrange basis types in Table 1 and provides a spectral method tailored to endpoint singularities and Erdélyi-Kober fractional operators. The explicit closed form for the inverse of the collocation matrices in Theorem 3.23 is a practically useful contribution, and the numerical experiments in Section 4 cover a broad range of problems, with Examples 4.1–4.2 providing direct verification against exact fractional derivatives. The significance is conditional, however, on the correctness of the companion preprint [13] and on repairing the space-mismatch in Definition 3.9; the central claims are plausible but not yet fully rigorous as written.","major_comments":[{"comment":"Definition 3.9 declares the two interpolants to take values in P_N^{(β,μ,σ,η)} and P_N^{(α,σ,η)} as defined in (2.15)–(2.16), but those spaces contain exponents 2σ(β−η−μ), 2ση and 2α, whereas the LMFs in (3.1) and the spaces 1F_N and 2F_N in (3.26)–(3.27) contain exponents σ(β−η−μ), ση and α. Consequently the interpolants built from the LMFs are not elements of the stated codomains, and the discrete orthogonality identities immediately following Definition 3.9 are not justified as written. The factor 2 is correct for the product spaces used in the quadrature exactness statements (3.4), so the fix is to introduce separate notation for the interpolation spaces (e.g., 1F_N and 2F_N) and to use it consistently in Definition 3.9, in (3.38)–(3.39), and in Theorems 3.12 and 3.17.","section":"Section 3, Definition 3.9"},{"comment":"The reduction of Equation (4.19) to the Riccati equation (4.20) is incorrect. For μ=1, Remark 2.4 gives aD^1_{x,σ,η} y = (1/σ)x y' + (η+1)y; substituting into (4.19) and multiplying by σ/x yields y' − 2y/x^2 = 1 − y^2, not y' − 2y = 1 − y^2. Hence the exact solution (4.21) does not solve the stated equation at μ=1, and the convergence reported in Figure 7 does not validate the method for this example. The equation, the exact solution, or both need to be corrected.","section":"Section 4, Example 4.5"},{"comment":"Theorems 3.18–3.21 apply the Jacobi-Müntz differentiation formulas of Remark 2.10 and the Gauss-Jacobi-Müntz quadrature rules of Theorem 2.13, all imported from the companion preprint [13] without proof or restatement of the required parameter restrictions. In particular, formula (3.40) contains Γ(j+β−μ+1) in the denominator and (3.48) contains Γ(j+α−μ+1), so the restrictions β−μ>−1 and α−μ>−1 are needed to avoid poles; these conditions are not stated in the theorems and are not checked in the numerical experiments. The paper should either state and verify these hypotheses explicitly or include proofs of the imported formulas in an appendix, since the differentiation matrices are a central claim of the paper.","section":"Sections 2.1–3.1"},{"comment":"The matrices L_SD^μ and R_SD^μ are defined as LU LV^{-1} and RU RV^{-1}, but no proof is given that the collocation matrices LV and RV are nonsingular for the stated parameters and nodes. Since the 'stable' differentiation matrices are obtained by inverting these dense matrices, the paper should prove, or at least state with a proof sketch, that the LMF basis is unisolvent at the Gauss-Jacobi-Müntz nodes. The numerical evidence in Figures 1–2 shows that even the second approach breaks down for sufficiently large N, so the terminology 'stable' in Theorems 3.20–3.21 should also be calibrated accordingly.","section":"Section 3.1, Theorems 3.20–3.21"}],"minor_comments":[{"comment":"The approximation y_N(x) is written as a sum over s=0,...,N of y(x_s) 1L_k(x); the index in the basis function should be s rather than k.","section":"Section 4.2.1, Eq. (4.5)"},{"comment":"The second interpolant is denoted with the superscript (α,β,μ,σ,η) in (3.23) and (3.37); the superscript should be (α,β,σ,η) to match Definition 3.9 and the right-sided basis in (3.1).","section":"Equations (3.23) and (3.37)"},{"comment":"In the row for Type 1, the entry g(x)=1 cannot be correct, because the cardinal basis in (1.4) would have denominator zero; the intended value is presumably g(x)=x.","section":"Table 1"},{"comment":"The parameter line 'η = −µ = 1.75' is self-contradictory, since it implies μ=−1.75 and violates the stated condition 1<μ<2; it should read η=−1.75, μ=1.75.","section":"Section 4.2.2, Example 4.6"},{"comment":"The text refers to the 'left-sided EK fractional derivative' in Example 4.2, but the example uses the right-sided basis 2J and the right-sided matrices of Theorems 3.19 and 3.21; the wording should be 'right-sided'.","section":"Section 4.1, Example 4.2"},{"comment":"The displayed equality N^{-1}‖x^{-σ(β−η−μ)−1}u‖_{w(α+1,β+1,σ)} = N^{-1}‖x^{σ−2}(b^σ−x^σ)u‖_{w1^σ} has exponents that differ by x^2; the power σ−2 appears to be a misprint for σ−1 based on the preceding expressions.","section":"Remark 3.13"},{"comment":"References [29] and [30] are duplicates of the same paper, and reference [26] is missing author/editor information; these should be cleaned up.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The core technical machinery of this manuscript — Jacobi-Müntz orthogonality, fractional differentiation formulas, and quadrature rules — is taken from the authors' companion arXiv preprint [13]. I recommend that the editor verify the status of [13] before final acceptance and that the authors be asked to state explicitly which results are assumed from [13] and which are proven here. The space-mismatch in Definition 3.9 and the incorrect reduction in Example 4.5 should also be addressed before the paper is reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper introduces two new Lagrange-type basis families on [0,b] with endpoint-singularity weights, proves error bounds by reducing to mapped-Jacobi interpolation, and derives Erdélyi-Kober fractional differentiation matrices in two ways. The construction is plausible and the numerics look good. It deserves a serious referee, but there is an inconsistency in the interpolation-space definitions that must be fixed.\n\nWhat's genuinely new: the LMFs-1 and LMFs-2 bases in (3.1) generalize the four Lagrange basis types in Table 1, and the associated differentiation matrices (Theorems 3.18–3.21) are new for these bases. The error analysis is sensible: Remark 3.10 shows the interpolants are weights times mapped-Jacobi interpolants, so Theorem 3.17 cleanly inherits the standard N^{-m} bounds. The explicit inverse in Theorem 3.23 is a nice practical contribution, and the numerical experiments, including the Burgers equation with endpoint singularities, support the claims.\n\nSoft spots, in order of severity. First, the spaces P_N in (2.15)–(2.16) have exponents 2σ(β-η-μ) and 2ση, but the basis functions in (3.1) span σ(β-η-μ)+kσ and ση+kσ. Definition 3.9 maps into the wrong spaces, so the discrete orthogonality statement after it does not follow as written. This looks like a factor-2 typo, but it has to be fixed and re-checked. Second, the paper imports orthogonality, quadrature exactness, and fractional differentiation formulas from the companion preprint [13] without proof or parameter restrictions (β-μ > -1, α-μ > -1). Since [13] is also an arXiv preprint, Theorems 3.18–3.21 are only as good as [13]. Third, there is no code, no data, and no comparison against existing fractional spectral methods. Fourth, Example 4.6 has a sign typo (η=-μ is written as 1.75, should be -1.75).\n\nNone of these are fatal. The central idea holds up, and the inconsistency is likely a notation slip rather than a deep flaw. If the authors fix the spaces, state the inherited parameter conditions, and provide a reproducible example, this would be a useful incremental paper. I'd send it to peer review, not desk reject.\n\nWho it's for: people interested in spectral methods for fractional differential equations with non-polynomial bases and endpoint singularities. A referee should check that [13] actually proves what is cited.","headline":"New Lagrange-Müntz bases with plausible theory and numerics, but the interpolation-space definitions don't match the basis spans and the paper leans on an unverified companion preprint.","tokens_in":28504,"tokens_out":6783,"would_cite":false,"duration_ms":56883,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26A33","33C45","41A55","34L10","65M70","58C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces two Lagrange–Müntz basis families, proves their interpolants achieve weighted error $O(N^{-m})$ for functions with endpoint singularities, and derives explicit Erdélyi–Kober fractional differentiation matrices in two…","keywords":["Lagrange–Müntz basis functions","Jacobi–Müntz functions","Erdélyi–Kober fractional derivatives","fractional differentiation matrices","pseudo-spectral method","non-classical interpolants","Müntz quadrature rules","endpoint singularities"],"falsifier":"Two concrete checks settle the matter: compare the exponents in the span definitions against the space $P_N$ used in Definition 3.9 for one parameter set, and compute a known Erdélyi–Kober derivative both through the matrix of Theorem 3.20 and through direct high-precision quadrature of the integral definition; a span mismatch or a disagreement beyond roundoff would show the claim as written is wrong.","tokens_in":27328,"feed_emoji":"🧮","tokens_out":12545,"duration_ms":117228,"temperature":0.7,"pith_summary":"Two new Lagrange-type basis functions are introduced, both built from the authors' Jacobi–Müntz functions by multiplying a mapped cardinal function with a weight factor. The paper proves that the associated interpolants approximate any function in the appropriate mapped weighted space at rate $O(N^{-m})$, and it derives explicit left- and right-sided Erdélyi–Kober fractional differentiation matrices in two different ways. If the proofs are right, one parameterized family of bases covers the four classical Lagrange basis types and gives spectral-like accuracy for solutions with endpoint singularities, including fractional ordinary and partial differential equations. The second, matrix-based variant of the differentiation matrices is reported to remain accurate at much larger $N$ than the direct expansion formula.","feed_headline":"New basis functions tame singular solutions of fractional PDEs","feed_subtitle":"Two new bases fold the classic Lagrange families into one scheme with fast, stable fractional differentiation.","key_machinery":"The load-bearing object is the pair of Lagrange–Müntz basis families: the mapped cardinal function $h_r^{\\sigma}$ (the product over all $j\\ne r$ of the mapped-node ratios) is multiplied, respectively, by a power of $x/x_r$ and by a power of $x/x_r$ times a power of $(b^\\sigma-x^\\sigma)$, so that each basis function carries the same weight structure as the Jacobi–Müntz functions it generalizes. Four supporting pieces carry the argument: the Gauss–Jacobi–Müntz quadrature rules make discrete inner products exact on the appropriate spans; the Jacobi–Müntz differentiation identity converts a fractional derivative of a basis element into another Jacobi–Müntz function with shifted parameters; the mapped-Jacobi interpolation error estimates provide the $O(N^{-m})$ rates; and the stable differentiation matrices are assembled as $U V^{-1}$, replacing a direct expansion that loses accuracy at large $N$.","core_discovery":"The paper's central claim is that the first basis function $(x/x_r)^{\\sigma(\\beta-\\eta-\\mu)}h_r^{\\sigma}(x)$ and the second basis function $(x/x_r)^{\\sigma\\eta}((b^\\sigma-x^\\sigma)/(b^\\sigma-x_r^\\sigma))^{\\alpha}h_r^{\\sigma}(x)$ are genuine generalizations of the four known Lagrange basis families, where $h_r^{\\sigma}$ is the cardinal product over mapped nodes. Theorems 3.12 and 3.17 establish stability and weighted error bounds of size $O(N^{-m})$ for the two non-classical Jacobi–Müntz interpolants. Theorems 3.18–3.21 then deliver the Erdélyi–Kober fractional differentiation matrices in two forms: one by expanding the cardinal functions and applying the Jacobi–Müntz differentiation identity term by term, and one as the product $U V^{-1}$ of the matrix of fractional derivatives of the Jacobi–Müntz basis and the inverse of its collocation matrix. The numerical experiments apply these matrices to linear, nonlinear, and partial fractional equations, and report that the stable variant keeps errors bounded with condition numbers growing like $O(N^{2\\mu})$.","pith_inferences":["This construction should extend to Gauss–Radau and Gauss–Lobatto nodes, which the paper mentions in passing; that would let the collocation method handle boundary and initial conditions without eliminating unknowns by hand.","One could automate parameter selection by estimating the leading singular exponent of the solution from the data and then choosing $\\alpha,\\beta,\\eta,\\mu,\\sigma$ to match it, making the method an adaptive spectral scheme.","Composing the fractional differentiation matrices for several orders may give a route to variable- or distributed-order Erdélyi–Kober equations, although no composition rule is proved in the paper."],"forward_implications":["Fractional equations whose solutions have endpoint singularities can be collocated directly in these bases, and the paper proves weighted errors of size $O(N^{-m})$ for solutions in the appropriate mapped weighted spaces.","The parameter choices recover the four standard Lagrange basis families, so the new bases are a common umbrella for existing Lagrange-collocation schemes rather than a separate method.","Linear multi-term fractional equations reduce to a single dense linear system built from the differentiation matrices; nonlinear and time-dependent problems reduce to systems solved by Newton iteration or standard ODE solvers.","In the tested examples the stable matrix variant of the differentiation matrices remains accurate at substantially larger $N$ than the direct expansion formula, with condition numbers growing like $O(N^{2\\mu})$.","For integer-order problems the same construction yields first-order differentiation matrices, and the Burgers example indicates that choosing $\\sigma=1/2$ approximates singular solutions far better than the smooth case $\\sigma=1$."],"supporting_citations":[{"why":"Supplies the Jacobi–Müntz functions, their fractional differentiation identity, and the Gauss–Jacobi–Müntz quadrature rules that the basis and matrices are built on.","marker":"[13]"},{"why":"Provides the mapped-Jacobi interpolation error and stability theorems that the new interpolant error bounds are reduced to.","marker":"[24]"},{"why":"Defines the left and right Erdélyi–Kober fractional integrals and derivatives that the differentiation matrices discretize.","marker":"[14]"},{"why":"One of the classical Lagrange basis types, with $g(x)=x^\\sigma$, that the new basis generalizes.","marker":"[6]"},{"why":"A pseudo-spectral scheme for fractional equations, another Lagrange-basis type (Type 3) contained in the new family.","marker":"[7]"},{"why":"A spectral collocation method for fractional equations with endpoint singularities, the problem class the new bases target.","marker":"[34]"}],"fun_headline_variants":["New Jacobi-Müntz bases generalize Lagrange for fractional PDEs","Two non-classical bases unify Lagrange, with proven error bounds","Generalized Lagrange bases yield stable fractional matrices","Müntz-type bases fold Lagrange families into one scheme","New basis functions extend Lagrange to fractional differentiation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on the companion paper's formulas for differentiating and integrating Jacobi–Müntz functions holding in the parameter ranges used here, including restrictions such as $\\beta-\\mu>-1$ and $\\alpha-\\mu>-1$ that this paper does not restate or verify; the written identification of the interpolation space with $P_N$ must also match the actual span of the new bases.","fun_headline_variants_meta":{"raw":{"variants":["New Jacobi-Müntz bases generalize Lagrange for fractional PDEs","Two non-classical bases unify Lagrange, with proven error bounds","Generalized Lagrange bases yield stable fractional matrices","Müntz-type bases fold Lagrange families into one scheme","New basis functions extend Lagrange to fractional differentiation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000298,"raw_usage":{"total_tokens":1693,"prompt_tokens":879,"completion_tokens":814,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":735}},"tokens_in":495,"tokens_out":814,"duration_ms":8694,"temperature":1.0,"reasoning_tokens":735,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:48:53.799728+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Two concrete checks settle the matter: compare the exponents in the span definitions against the space $P_N$ used in Definition 3.9 for one parameter set, and compute a known Erdélyi–Kober derivative both through the matrix of Theorem 3.20 and through direct high-precision quadrature of the integral definition; a span mismatch or a disagreement beyond roundoff would show the claim as written is wrong.","supporting_citations":[{"cited_title":"Esmaeili, M","cited_arxiv_id":null,"evidence_quote":"Supplies the Jacobi–Müntz functions, their fractional differentiation identity, and the Gauss–Jacobi–Müntz quadrature rules that the basis and matrices are built on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the mapped-Jacobi interpolation error and stability theorems that the new interpolant error bounds are reduced to."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the left and right Erdélyi–Kober fractional integrals and derivatives that the differentiation matrices discretize."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"One of the classical Lagrange basis types, with $g(x)=x^\\sigma$, that the new basis generalizes."},{"cited_title":"Baleanu, K","cited_arxiv_id":null,"evidence_quote":"A pseudo-spectral scheme for fractional equations, another Lagrange-basis type (Type 3) contained in the new family."}],"review_version":1}