{"id":"5dcacd5a-3bb9-4772-8669-c5611d2e6ea8","arxiv_id":"2411.17122","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"This paper applies Helmholtz-equation particular solutions in a method-of-fundamental-solutions framework to diffusion and wave initial boundary value problems, with accuracy shown only on toy examples.","lead":"A numerical paper combines established meshless techniques (method of fundamental solutions and dual reciprocity) with approximate particular solutions of Helmholtz equations to solve diffusion and wave problems. The method is demonstrated on simple benchmarks, but algebraic errors in the time-stepping derivations make the written scheme inconsistent with the reported results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The displayed Laplace and time-difference transformations contain algebraic errors that prevent the reported results from being reproduced; central claim unsupported as written.","rationale":"In good faith, the paper aims to provide a meshless solver for diffusion and wave IBVPs by reducing each time step to a Helmholtz BVP and solving it with MFS/MPS. For that central claim to hold, two things must be true: the transformed Helmholtz problems must faithfully represent the original equations, and the approximate particular solutions used in MPS must be accurately computable. The second condition is indeed unaddressed: Eqs. (13)–(14) involve infinite integrals with Bessel/Hankel kernels, and the paper never states the quadrature rule, truncation criterion, or error control. The reader's weakest-assumption analysis correctly identifies this as a serious gap. However, the first condition fails on the face of the manuscript. I verified the Laplace transforms explicitly: for Eq. (47) with k = 5.8e−7, the correct transformed equation is ΔV − (s_l/k)V = 0, not ΔV − s_l V = 0 as in Eq. (48). For the wave example, L[u_tt] with u0 = 0 and v0 = xy gives s^2 U − xy, so ΔU − s^2 U = −xy, the opposite sign from Eq. (52). The diffusion backward-Euler scheme similarly has a reciprocal-k error in Eq. (31), and the definition of λ in Eq. (35) is dimensionally mismatched to the coefficient in Eq. (34). These are internal inconsistency findings, not disagreements with an external consensus, so they directly undermine the stated derivation. The reported numerical errors are plausible in magnitude, which suggests the examples may have been generated by a corrected version of the formulas, but without code or precise algorithms the manuscript as written is not reproducible. A careful revision that fixes the transformations, specifies the quadrature for the particular solutions, and provides code or detailed pseudocode could change this assessment. Because the central derivation is currently unreliable and the numerical claims cannot be verified from the text, the rejection verdict is appropriate.","tokens_in":13539,"tokens_out":5588,"duration_ms":55134,"concrete_test":"Recompute Example 5.6 at the points and times in Tables 16–17 using the same MFS/MPS setup but with the correct Laplace-transformed equation ΔU − s_l^2 U = −xy and with the paper's displayed Eq. (52) ΔU − s_l^2 U = +xy, applying Stehfest inversion (25)–(26). If the +xy version reproduces the reported errors, then Eq. (52) is a typo and the method works as implemented; if it produces errors near 2|ue| or otherwise qualitatively different values, the written algorithm is not the one that generated the tables and the central claim fails as stated.","verdict_should_be":"REJECT","load_bearing_attack":"The most immediate load-bearing defect is that the equations connecting the time-dependent PDEs to the Helmholtz problems are internally inconsistent. For Example 5.4, Eq. (48) writes ΔV − s_l V = 0, but the Laplace transform of Eq. (47) with k = 5.8e−7 gives ΔV − (s_l/k)V = 0, a factor of roughly 1.7e6 in the Helmholtz coefficient. For Example 5.6, the wave equation u_tt = Δu with u0 = 0 and v0 = xy transforms to ΔU − s^2 U = −xy, not +xy as in Eq. (52). The diffusion time-stepping in Eq. (31) has k/τ where the correct coefficient is 1/(kτ), and Eq. (35) sets λ = k/τ while the preceding equation contains a k/τ term, so λ^2 is also inconsistent. These are not cosmetic typos: they change the Helmholtz operator and forcing that MFS/MPS must solve. Since no code, pseudocode, or additional implementation details are supplied, the reader cannot determine which equations produced Tables 12–20. The quadrature of the particular solutions in Eqs. (13)–(14) is a real secondary gap, but these algebraic errors are the more immediate barrier to accepting the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a meshless method for diffusion and wave initial-boundary-value problems by converting them into sequences of Helmholtz boundary value problems, using either Laplace transforms or finite-difference time stepping, and solving each Helmholtz problem with a combination of the method of fundamental solutions (MFS) and the method of particular solutions (MPS). The particular solutions are taken from the integral representations of Li [9] rather than from RBF collocation. Numerical experiments cover homogeneous and non-homogeneous Helmholtz problems in two and three dimensions, a diffusion problem with very small diffusivity, and a vibrating membrane problem, with error tables comparing against analytical solutions.","tokens_in":94,"tokens_out":6760,"duration_ms":117443,"significance":"If the derivations and numerical results were correct, the paper would offer a useful meshless alternative for time-dependent problems: it avoids meshing, uses the MFS for homogeneous parts, and replaces RBF collocation for particular solutions by the closed-sum approximations of [9]. The comparison against independent exact solutions in Examples 5.4-5.7 is a genuine strength, and the paper is not circular. However, several of the central algebraic transformations from the time-dependent PDEs to Helmholtz equations are incorrect as printed, and the numerical experiments therefore cannot be reproduced from the displayed equations. The paper also omits the quadrature details needed to evaluate the particular solutions in Eqs. (13)-(14), so its numerical claims are not independently checkable without code. The underlying idea is plausible, but the manuscript in its current form does not support the claimed conclusions.","major_comments":[{"comment":"The backward Euler discretization of the diffusion equation (1/k)u_t = Δu is not what is written. With u_t(x,nτ) ≈ (u_n - u_{n-1})/τ, the correct implicit scheme is Δu_n - (1/(kτ))u_n = -(1/(kτ))u_{n-1}. Instead, Eq. (31) contains k(u_n-u_{n-1})/τ, and Eq. (34) becomes Δu_n - (k/τ)u_n = -(k/τ)u_{n-1} + ..., with λ = k/τ in Eq. (35). For k = 5.8e-7 in Example 5.5, this changes the Helmholtz coefficient by the factor k^2 ≈ 3.4e-13, so the scheme actually implemented cannot be the discretization of Eq. (45). This is a load-bearing error for the diffusion time-difference examples.","section":"§4, Eqs. (30)-(35)"},{"comment":"The Laplace transform of the wave equation (1/c^2)u_tt = Δu is (s^2/c^2)U - (s/c^2)u_0 - (1/c^2)v_0 = ΔU. Eq. (28) is missing the factors 1/c^2 on the initial-condition terms. For Example 5.6, where c=1, u_0=0 and v_0=xy, the correct transformed equation is ΔU - s_l^2 U = -xy, but Eq. (52) has +xy on the right-hand side. This sign error changes the forcing term in every Helmholtz solve used for the reported wave results.","section":"§3, Eq. (28) and §5, Example 5.6, Eq. (52)"},{"comment":"The reduction from Eq. (45) to Eq. (47) is internally inconsistent. Since v = u - u_0 and u_0 = 1 is harmonic, v satisfies the same diffusion equation as u, namely (1/k)v_t = Δv, not v_t = Δv as printed in Eq. (47). Consequently, the Laplace-transformed Helmholtz problem should be ΔV - (s_l/k)V = 0, not ΔV - s_l V = 0 as in Eq. (48). For k = 5.8e-7 the coefficient s_l/k is roughly 1.72e6 times larger than s_l, so the example as written cannot reproduce the analytical solution (46), which explicitly contains k. This is a load-bearing inconsistency for the Laplace-transform diffusion example.","section":"§5, Example 5.4, Eqs. (47)-(48)"},{"comment":"The approximate particular solutions un(x) are defined by sums of integrals over [0,∞) involving the RBF φ and Bessel/Hankel functions. The manuscript does not state how these infinite integrals are evaluated numerically, what quadrature is used, or what error control is imposed on the truncation. Since un(x) is added into every reported numerical solution through Eq. (17), an inaccurate or unstable evaluation of these integrals would invalidate all examples that use MPS. The paper provides no code, pseudocode, or implementation details, so this gap directly affects reproducibility of Tables 5-20.","section":"§2, Eqs. (13)-(14)"}],"minor_comments":[{"comment":"The affiliation line contains a typo: \"Portland State Univerisity\" should be \"Portland State University\".","section":"General"},{"comment":"The left-hand side of Eq. (25) is written as eu(x,s), but the formula is an approximation to u(x,t) after inverse Laplace transformation; the argument should be t, not s.","section":"Eq. (25)"},{"comment":"Equation (31) introduces f(eun-1(x)) even though the diffusion equation (18) has no source term and the examples all set f = 0. The role of f in the time-difference formulation should be clarified or removed.","section":"Eq. (31)"},{"comment":"For the record, Eq. (36) is algebraically consistent with the central-difference discretization of (1/c^2)u_tt = Δu when c=1: moving the diagonal term to the left gives Δu_n - (1/(c^2τ^2))u_n = (u_{n-2} - 2u_{n-1})/(c^2τ^2), which matches the printed equation. No sign error appears there.","section":"Eq. (36)"},{"comment":"The error tables report errors only at a small number of interior points and do not report convergence rates, condition numbers of the MFS linear systems, or dependence on the MFS source radius in the time-dependent examples. Such information would help the reader assess the reliability of the method.","section":"§5, Examples 5.4-5.7"},{"comment":"The boundary condition in Eq. (8) is written with x ∈ Ω, but it should be x ∈ ∂Ω to describe a boundary condition.","section":"Eq. (8)"}],"recommendation":"major_revision","confidential_remarks":"The algebraic errors in Eqs. (28), (31)-(35), (47)-(48), and (52) are serious and would normally push toward rejection, but they are the kind of local, fixable defects that a careful revision can address: the coefficients and signs must be corrected and all affected numerical examples recomputed. The larger concern is that the paper provides no implementation details for the quadrature in Eqs. (13)-(14), so even a corrected derivation would be difficult to verify. I recommend major revision rather than acceptance or outright rejection, provided the authors can supply a corrected derivation, reproducible experiments, and a clear quadrature description."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know this paper before pointing a student at it: the time-discretization algebra is wrong in several load-bearing places, so the reported tables do not follow from the displayed equations. Equation (31) treats the diffusion equation (1/k)u_t = ∆u but then uses k/τ where the backward Euler step gives 1/(kτ). Equation (28) drops the 1/c^2 factor on the initial terms in the wave Laplace transform, and Eq. (52) has the wrong sign on the forcing term (should be −xy, not +xy). Example 5.4 similarly drops the 1/k in equation (47), which is why (48) is missing the 1.7e6 factor. These are not typos; they change the Helmholtz operator and the forcing that the MFS/MPS solve.\n\nWhat the paper does well: it gives a clean statement of the DRM pipeline, uses Li's approximate particular solutions [9] rather than the more common RBF collocation, and the MFS-only example (Example 5.1) is carefully done. The 2D/3D Helmholtz examples show reasonable accuracy when the equations are self-consistent. The author is honest about the sources of the ingredients, and the paper is not circular—the tests are against independent exact solutions.\n\nThe soft spots are proportionate to the central claim. The algebraic errors alone make the method non-reproducible from the text; no code or pseudocode is included, and the quadrature of the integrals in (13)–(14) is never described, despite being the step on which the whole MPS accuracy rests. There is also no error analysis and no comparison against existing meshless or classical solvers, so even with corrected algebra the contribution would be incremental.\n\nI would not send this to a referee as-is. It deserves a major revision with the equations fixed, the quadrature spelled out, and a reproducible example. If the author does that, the paper could be a useful, if modest, addition to the meshless literature.\n\nBest, [Your name]","headline":"The displayed time-discretization equations have load-bearing algebraic errors that prevent reproduction; the paper needs major revision before it can be evaluated.","tokens_in":14356,"tokens_out":4893,"would_cite":false,"duration_ms":41900,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N80","65M06","65M70"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Laplace transform or time differences turn diffusion and wave equations into a sequence of Helmholtz solves, and an explicit radial-basis particular-solution formula keeps the whole solver meshless.","keywords":["meshless methods","method of fundamental solutions","method of particular solutions","Helmholtz equation","diffusion equation","wave equation","Laplace transform","radial basis functions"],"falsifier":"A reader could fix a Helmholtz problem with a known exact solution, compute the particular solution $u_n$ by two independent high-precision evaluations of the improper integrals in (13), and compare the final MFS+MPS solution as $n$ and $N$ grow; if the error does not decrease toward the MFS boundary error, the particular-solution quadrature is the bottleneck.","tokens_in":13282,"feed_emoji":"🧮","tokens_out":5763,"duration_ms":49013,"temperature":0.7,"pith_summary":"The paper claims that diffusion and wave initial-boundary value problems can be solved without constructing a mesh by converting each problem, either through the Laplace transform or through time differences, into a sequence of Helmholtz boundary value problems. Each Helmholtz problem is then solved by combining the method of fundamental solutions with the method of particular solutions, where the particular solution is the explicit approximate formula of reference [9] built from radial basis functions and Bessel/Hankel integrals. If correct, the recipe gives a simple, point-based numerical method for time-dependent problems on irregular domains, using relatively few points; the paper reports numerical errors for each worked example in support of this claim.","feed_headline":"Diffusion and wave equations solved meshlessly via Helmholtz steps","feed_subtitle":"Laplace transform or time differences turn each time step into a Helmholtz solve that needs only scattered points, no mesh.","key_machinery":"The load-bearing object is the approximate particular solution $u_n(x)$ for the Helmholtz operator, given in equations (13) for two dimensions and (14) for three dimensions. It is a finite sum over grid points $j/n$ in a thickened domain $\\Omega_\\delta$ of values $f(j/n)$ multiplied by products of Bessel and Hankel functions and one-dimensional integrals of the radial basis function $\\phi$ against Bessel functions, with one integral extending to infinity. This explicit formula replaces the expensive exact singular potential integral (11) and avoids the ill-conditioned linear systems of RBF collocation. The MFS part supplies the homogeneous correction, and the stated $O(1/\\sqrt{n})$ approximation order of $u_n$ to the exact particular solution is the accuracy estimate that the final method inherits.","core_discovery":"At the core is the reduction: applying the Laplace transform to the diffusion equation produces $\\Delta U - \\lambda^2 U = -u_0/k$, and to the wave equation produces $\\Delta U - \\lambda^2 U = su_0 + v_0$, both with complex Helmholtz parameter $\\lambda$; similarly, backward differences in time produce a Helmholtz equation at each step. The paper's claim is that solving these Helmholtz problems with the MFS/MPS combination, using the approximate particular solutions (13) and (14), and then either inverting the Laplace transform with the Stehfest weights (25)-(26) or iterating the time steps, yields the solution of the original diffusion or wave problem. The numerical examples, including a square diffusion problem, a rectangular membrane wave problem, and 2D and 3D Helmholtz problems, are presented as evidence of the construction's efficiency.","pith_inferences":["The paper does not describe how the improper integrals in (13) and (14) are evaluated; a natural next step is a careful quadrature and convergence study of these integrals, since their error propagates directly into every Helmholtz solve (this is the editor's inference, not a paper claim).","The same reduction would apply to other time-dependent linear PDEs whose transform is a Helmholtz-type operator, such as the telegraph equation or damped wave equations.","One testable extension is to replace the Gaussian and compactly supported RBFs used in the examples with other kernels and compare whether the accuracy plateaus; the formulas (13) and (14) are written for a general RBF $\\phi$."],"forward_implications":["For any fixed Laplace parameter $s_l$, one only needs to solve a modified Helmholtz boundary value problem, so the same MFS/MPS solver core applies at every quadrature point.","The time-difference variant produces a sequence of Helmholtz problems whose right-hand sides depend on the previous numerical solution, so the overall method is a marching scheme with no mesh at any stage.","Because the particular solutions are given by explicit sums over grid points rather than by solving a collocation linear system, the resolution parameter $n$ can be increased without the usual ill-conditioning that limits RBF collocation.","In the worked examples the reported errors range roughly from $10^{-8}$ to $10^{-2}$ depending on the problem and parameters, suggesting the method is viable but its accuracy depends on the choice of RBF, $n$, $\\delta$, and the number of boundary and source points."],"supporting_citations":[{"why":"Supplies the explicit approximate particular solutions of the Helmholtz equation used throughout, equations (13)-(14), and the approximation order result.","marker":"[9]"},{"why":"Supplies the Stehfest numerical inversion of Laplace transforms and the weights $\\alpha_l$ used to recover time-domain solutions.","marker":"[8]"},{"why":"Supplies the radial-basis approximation of the function $f$ and the $O(1/\\sqrt{n})$ error estimate inherited by the particular solutions.","marker":"[2]"},{"why":"Supplies the MFS framework for potential, Helmholtz, and diffusion problems, including the Laplace-transform route used here.","marker":"[1]"},{"why":"Provides analytic particular solutions for inhomogeneous Helmholtz-type equations, used as background and comparison for the particular-solution approach.","marker":"[20]"}],"fun_headline_variants":["Helmholtz meshless solves for diffusion and wave equations","Meshless Helmholtz method tackles diffusion and wave equations","Diffusion and wave equations solved with Helmholtz, no mesh","Helmholtz steps unlock meshless diffusion and wave equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method presupposes that the improper integrals in equations (13) and (14), involving products of the radial basis function with Bessel and Hankel functions, can be evaluated accurately and stably with controlled error, but the paper does not state how these integrals are computed.","fun_headline_variants_meta":{"raw":{"variants":["Helmholtz meshless solves for diffusion and wave equations","Meshless Helmholtz method tackles diffusion and wave equations","Diffusion and wave equations solved with Helmholtz, no mesh","Helmholtz steps unlock meshless diffusion and wave equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000834,"raw_usage":{"total_tokens":3580,"prompt_tokens":828,"completion_tokens":2752,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":444,"completion_tokens_details":{"reasoning_tokens":2685}},"tokens_in":444,"tokens_out":2752,"duration_ms":16432,"temperature":1.0,"reasoning_tokens":2685,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:30:14.059101+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A reader could fix a Helmholtz problem with a known exact solution, compute the particular solution $u_n$ by two independent high-precision evaluations of the improper integrals in (13), and compare the final MFS+MPS solution as $n$ and $N$ grow; if the error does not decrease toward the MFS boundary error, the particular-solution quadrature is the bottleneck.","supporting_citations":[{"cited_title":"Approximation of potential integral by radial bases for solutions of Helmholtz equa- tion,","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit approximate particular solutions of the Helmholtz equation used throughout, equations (13)-(14), and the approximation order result."},{"cited_title":"Algorithm 368: Numerical inversion of Laplace transforms [D5],","cited_arxiv_id":null,"evidence_quote":"Supplies the Stehfest numerical inversion of Laplace transforms and the weights $\\alpha_l$ used to recover time-domain solutions."},{"cited_title":"Radial Basis Approximation for Newtonian Potentials,","cited_arxiv_id":null,"evidence_quote":"Supplies the radial-basis approximation of the function $f$ and the $O(1/\\sqrt{n})$ error estimate inherited by the particular solutions."},{"cited_title":"The Method of Fundamental Solutions for Potential, Helmholtz andDiffusionProblems,","cited_arxiv_id":null,"evidence_quote":"Supplies the MFS framework for potential, Helmholtz, and diffusion problems, including the Laplace-transform route used here."},{"cited_title":"Analytic Particular Solutions for Inhomogeneous Helmholtz-Type Equations,","cited_arxiv_id":null,"evidence_quote":"Provides analytic particular solutions for inhomogeneous Helmholtz-type equations, used as background and comparison for the particular-solution approach."}],"review_version":1}