pith. sign in

arxiv: 2606.16578 · v2 · pith:VORU767Qnew · submitted 2026-06-15 · 💻 cs.GR · cs.NA· math.NA

Walking on Heat Stars for Parabolic Heat Equations with Neumann Boundary Conditions

Anchang Bao , Enya Shen , Yongjun Zhang , Zhongwei Liu , Jianmin Wang This is my paper

Pith reviewed 2026-06-30 11:29 UTC · model grok-4.3

classification 💻 cs.GR cs.NAmath.NA
keywords Monte Carlo methodsparabolic equationsheat equationWalk on StarsNeumann boundary conditionsimportance samplinggrid-free solversboundary integral methods
0
0 comments X

The pith

A logarithmic time and direction parameterization of heat balls factorizes the double-layer kernel into independent Gamma and uniform parts, enabling exact importance sampling for unbiased Monte Carlo solutions of parabolic heat equations.

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

The paper develops Walk on Heat Stars as a grid-free Monte Carlo solver for transient parabolic equations with Neumann boundary conditions. It extends the boundary integral approach of prior Walk on Stars methods by using a non-cylindrical formulation that handles the time-varying domains created by heat-ball sampling. The central step reparameterizes each heat ball with a logarithmic time coordinate and a spatial direction, which separates the double-layer kernel into a Gamma distribution for time and a uniform distribution for direction. This separation produces unbiased estimators for the recursive walk position, the Neumann flux term, and any volumetric source without requiring volumetric meshes or global solves. The work also supplies a gradient estimator expressed as weighted boundary integrals and adapts a space-time denoiser, with validation showing expected Monte Carlo convergence rates on both analytic test cases and practical cooling geometries.

Core claim

Walk on Heat Stars closes the gap for transient parabolic equations by introducing a parameterization of the heat ball geometry using a logarithmic time coordinate and a spatial direction that reveals the double-layer kernel factorizes into independent Gamma and uniform components, enabling exact directional importance sampling of the recursive next walk position, the Neumann flux contribution, and the volumetric source term.

What carries the argument

The logarithmic-time and directional parameterization of the heat ball geometry that factorizes the double-layer kernel into independent Gamma and uniform components.

If this is right

  • Unbiased Monte Carlo estimators become available for the solution value, Neumann flux, and volumetric source at any query point.
  • The method handles both pure and mixed Neumann boundary conditions without requiring Dirichlet data.
  • Spatial derivatives of the solution can be estimated directly as weighted boundary integrals without recursive differentiation.
  • A heteroscedastic regression denoiser adapted to space-time reduces variance while preserving the unbiased property.
  • The approach converges at the standard Monte Carlo rate on analytic solutions across varied geometries and frequencies.

Where Pith is reading between the lines

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

  • The same factorization technique could be tested on other parabolic operators whose fundamental solutions admit analogous scaling.
  • Practical thermal simulations in graphics could avoid meshing overhead for dynamic heat-sink and cooling problems.
  • The separation into Gamma time and uniform direction suggests possible extensions to anisotropic diffusion or moving-boundary problems.
  • Gradient estimators that avoid recursion may reduce variance in optimization loops that rely on solution sensitivities.

Load-bearing premise

The double-layer kernel on a heat ball separates cleanly into independent Gamma and uniform factors once the ball is parameterized by logarithmic time and spatial direction.

What would settle it

A direct numerical quadrature check on a simple heat ball showing that the proposed Gamma-uniform sampling distribution does not reproduce the exact integral of the double-layer kernel.

Figures

Figures reproduced from arXiv: 2606.16578 by Anchang Bao, Enya Shen, Jianmin Wang, Yongjun Zhang, Zhongwei Liu.

Figure 1
Figure 1. Figure 1: Monte Carlo estimation of transient heat conduction in a turbine housing geometry with pure Neumann boundary conditions. A uniform initial [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Heat ball geometry. Left: Euclidean ball with straight radial rays 𝑥 = 𝑥0 + 𝑅𝜔. Right: Heat ball in (1 + 1)-dimensional space-time. The boundary is the heat sphere; curved radial traces |𝑥 | = √ 2𝑛𝑐𝑢𝑒−𝑢 replace straight rays. The ball expands from a point, reaches maximum radius at 𝑠 = 𝑐/𝑒, and contracts back to a point at 𝑠 = 𝑐. region H (𝑥0, 𝑡0, 𝑐) = {(𝑦, 𝑡) | 0 < 𝑡0 − 𝑡 < 𝑐, Φ(𝑥0 − 𝑦, 𝑡0 − 𝑡) > 𝜏 (𝑐)} ,… view at source ↗
Figure 3
Figure 3. Figure 3: Walk on Heat Spheres (WoHS). The next state [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: From Walk on Stars to Walk on Heat Stars. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Directional sampling: Euclidean versus heat-ball geometry. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Equivalence of WoHSt and Deaconu and Herrmann [2018] time sampling. Left: Histogram of backward time 𝑠 for both methods vs. theory, 𝑛 = 2 (top) and 𝑛 = 3 (bottom). Right: Distribution of 𝑢 = log(𝑐/𝑠 ); both methods follow Γ(𝑛/2 + 1, 2/𝑛). and a spot) spanning a range of boundary complexity. Models and Dirichlet-Neumann partitions are shown in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Test geometries and boundary conditions. We evaluate WoHSt on [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convergence of Walk on Heat Stars at the spatial frequency [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Temporal convergence of Walk on Heat Stars at [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Effect of heteroscedastic regression denoising on Walk on Heat Stars estimates. The raw Monte Carlo estimates (left) exhibit high variance, while the [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left: visualization of ground truth quiver and gradient length (only [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: WoHSt gradient estimation (denoised) at 𝜔 = 4𝜋. The denoised gradient field (arrows) closely matches the ground truth, demonstrating the correctness of the gradient formula when combined with variance reduction. Raw gradient estimates (not shown) exhibit substantially higher variance. Base: Dirichlet boundary condition Fin: Neumann boundary condition t = 0.6s RelMSE 2.56e-4 2.78e-3 3.68e-3 65 80 t = 3.0s … view at source ↗
Figure 13
Figure 13. Figure 13: WoHSt estimates of transient heat conduction in a heat sink geometry with mixed boundary conditions. [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
read the original abstract

Monte Carlo methods have proven highly effective for elliptic partial differential equations through algorithms such as Walk on Spheres and Walk on Stars, which evaluate solutions at individual points without volumetric meshing or global linear solves. Extending these methods to the transient regime has remained an open challenge: parabolic equations couple space and time through an anisotropic scaling, requiring joint sampling of spatial displacements and backward time steps whose distribution was not previously available in a unified, exact form. We present Walk on Heat Stars, a grid-free Monte Carlo solver that closes this gap by extending the boundary integral framework of Walk on Stars to the parabolic setting. Our method introduces a non-cylindrical boundary integral formulation that accommodates the time-varying domains induced by heat-ball sampling. The heat ball geometry is parameterized by a logarithmic time coordinate and a spatial direction, revealing that the double-layer kernel factorizes into independent Gamma and uniform components. This parameterization enables exact directional importance sampling of the recursive next walk position, the Neumann flux contribution, and the volumetric source term, yielding unbiased Monte Carlo estimators for all three components. We additionally derive a preliminary gradient estimator that expresses spatial derivatives as weighted boundary integrals of the solution, requiring no recursion on the gradient, and adapt a heteroscedastic regression-based denoiser to the space-time domain for variance reduction. We validate our method on analytical solutions across a range of geometries and spatial frequencies, confirm convergence at the expected Monte Carlo rate, and demonstrate practical applicability on heat sink and cooling scenes with mixed or pure Neumann boundary conditions.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The paper introduces Walk on Heat Stars, a grid-free Monte Carlo method extending Walk on Stars to parabolic heat equations with Neumann boundary conditions. It uses a non-cylindrical boundary integral formulation to handle time-varying domains from heat-ball sampling. The heat ball is parameterized by logarithmic time coordinate and spatial direction, allowing the double-layer kernel to factorize into independent Gamma (time) and uniform (direction) components. This enables exact directional importance sampling for recursive walk positions, Neumann flux, and volumetric sources, producing unbiased estimators. Additional elements include a non-recursive gradient estimator as weighted boundary integrals and a heteroscedastic regression denoiser adapted to space-time; validation is reported on analytical solutions, convergence rates, and practical heat sink/cooling scenes.

Significance. If the central factorization and unbiasedness claims hold, the work is significant for closing a longstanding gap in extending elliptic Monte Carlo PDE solvers (Walk on Spheres/Stars) to the parabolic regime without meshing or global solves. Strengths include the exact, parameter-free factorization into Gamma and uniform factors (enabling unbiased sampling of all three components), the non-cylindrical formulation for time-varying domains, and the preliminary gradient estimator. These could enable new applications in transient heat transfer simulations in graphics and engineering.

major comments (1)
  1. [Abstract] The central claim of exact unbiased Monte Carlo estimators rests on the heat-ball parameterization and double-layer kernel factorization (abstract). Without the explicit derivation, error analysis, or validation details visible, the support for unbiasedness across the Neumann parabolic case cannot be verified at this stage.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their summary and for highlighting the potential significance of the work if the central claims hold. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] The central claim of exact unbiased Monte Carlo estimators rests on the heat-ball parameterization and double-layer kernel factorization (abstract). Without the explicit derivation, error analysis, or validation details visible, the support for unbiasedness across the Neumann parabolic case cannot be verified at this stage.

    Authors: The abstract summarizes the contributions at a high level, as is conventional. The explicit derivation of the non-cylindrical boundary integral formulation, the heat-ball parameterization by logarithmic time coordinate and spatial direction, and the resulting factorization of the double-layer kernel into independent Gamma (time) and uniform (direction) components appears in Section 3 of the full manuscript. This factorization directly enables the exact directional importance sampling for the recursive walk position, Neumann flux, and volumetric sources, establishing unbiasedness. Error analysis and convergence rates are derived in Section 4. Validation on analytical solutions (including convergence at the Monte Carlo rate) and practical scenes appears in Section 5. These sections supply the requested support for the Neumann parabolic case. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation introduces independent parameterization

full rationale

The paper's central step is introducing a logarithmic-time + direction parameterization of the heat ball, from which the double-layer kernel factorization into Gamma(time) and uniform(direction) factors is derived. This is presented as a new geometric observation enabling exact sampling, not a redefinition or fit of the target quantities. No self-citations, fitted inputs renamed as predictions, or uniqueness theorems imported from prior author work appear in the provided text. The claimed unbiased estimators follow directly from the factorization rather than reducing to the inputs by construction. This is the expected non-finding for a methods paper whose novelty lies in an explicit kernel decomposition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central claim appears to rest on the stated kernel factorization whose validity cannot be audited without the full text.

pith-pipeline@v0.9.1-grok · 5815 in / 1035 out tokens · 28129 ms · 2026-06-30T11:29:45.360860+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 6 canonical work pages · 1 internal anchor

  1. [1]

    The deal. II library, version 9.7

    The deal.II library, Version 9.7.Journal of Numerical Mathematics33, 4 (2025), 403–415. doi:10.1515/jnma-2025-0115 Ghada Bakbouk and Pieter Peers

    work page doi:10.1515/jnma-2025-0115 2025

  2. [2]

    Mean Value Caching for Walk on Spheres. (2023). Anchang Bao, Enya Shen, and Jianmin Wang

    2023

  3. [3]

    Monte Carlo PDE Solvers for Nonlinear Radiative Boundary Conditions

    Monte Carlo PDE Solvers for Nonlinear Radiative Boundary Conditions.arXiv preprint arXiv:2604.21717(2026). Anchang Bao, Jie Xu, Enya Shen, and Jianmin Wang

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  4. [4]

    InProceedings of the SIGGRAPH Asia 2025 Con- ference Papers

    Off-Centered WoS-Type Solvers with Statistical Weighting. InProceedings of the SIGGRAPH Asia 2025 Con- ference Papers. 1–11. Mégane Bati, Stéphane Blanco, Christophe Coustet, Vincent Eymet, Vincent Forest, Richard Fournier, Jacques Gautrais, Nicolas Mellado, Mathias Paulin, and Benjamin Piaud

    2025

  5. [5]

    Rahel Brügger, Helmut Harbrecht, and Johannes Tausch

    Coupling Conduction, Convection and Radiative Transfer in a Single Path-Space: Application to Infrared Rendering.ACM Transactions on Graphics (TOG) 42, 4 (2023), 1–20. Rahel Brügger, Helmut Harbrecht, and Johannes Tausch

    2023

  6. [6]

    Martin Costabel

    Boundary inte- gral operators for the heat equation in time-dependent domains.arXiv preprint arXiv:2010.04934(2020). Martin Costabel

    work page arXiv 2010

  7. [7]

    Michael Czekanski, Benjamin Faber, Margaret Fairborn, Adelle Wright, and David Bindel

    Boundary integral operators for the heat equation.Integral Equations and Operator Theory13, 4 (1990), 498–552. Michael Czekanski, Benjamin Faber, Margaret Fairborn, Adelle Wright, and David Bindel

    1990

  8. [8]

    Fernando de Goes and Mathieu Desbrun

    Walking on Spheres and Talking to Neighbors: Variance Reduction for Laplace’s Equation.arXiv preprint arXiv:2404.17692(2024). Fernando de Goes and Mathieu Desbrun

    work page arXiv 2024

  9. [9]

    Stochastic Computation of Barycentric Coordinates.ACM Transactions on Graphics43, 4 (2024),

    2024

  10. [10]

    Madalina Deaconu and Samuel Herrmann

    Hitting time for Bessel processes— Walk on moving spheres algorithm (WoMS).The Annals of Applied Probability23, 6 (2013), 2259–2289. Madalina Deaconu and Samuel Herrmann

    2013

  11. [11]

    Initial-boundary value problem for the heat equation—A stochastic algorithm.The Annals of Applied Probability28, 3 (2018), 1943–1976. M. Deaconu, S. Herrmann, and S. Maire

    2018

  12. [12]

    Yixin Hu, Teseo Schneider, Bolun Wang, Denis Zorin, and Daniele Panozzo

    The walk on moving spheres: A new tool for simulating Brownian motion’s exit time from a domain.Mathematics and Computers in Simulation135 (2017), 28–38. Yixin Hu, Teseo Schneider, Bolun Wang, Denis Zorin, and Daniele Panozzo

    2017

  13. [13]

    Tianyu Huang

    Fast tetrahedral meshing in the wild.ACM Transactions on Graphics (ToG)39, 4 (2020), 117–1. Tianyu Huang

    2020

  14. [14]

    InProceedings of the SIGGRAPH Asia 2025 Technical Communications

    Geometric Queries on Closed Implicit Surfaces for Walk on Stars. InProceedings of the SIGGRAPH Asia 2025 Technical Communications. 1–4. Tianyu Huang, Jingwang Ling, Shuang Zhao, and Feng Xu

    2025

  15. [15]

    Proceedings of the Imperial Academy20, 10 (1944), 706–714

    two-dimensional brownian motion and harmonic functions. Proceedings of the Imperial Academy20, 10 (1944), 706–714. Zilu Li, Guandao Yang, Xi Deng, Christopher De Sa, Bharath Hariharan, and Steve Marschner

    1944

  16. [16]

    InSIGGRAPH Asia 2023 Conference Papers

    Neural Caches for Monte Carlo Partial Differential Equation Solvers. InSIGGRAPH Asia 2023 Conference Papers. 1–10. Zilu Li, Guandao Yang, Qingqing Zhao, Xi Deng, Leonidas Guibas, Bharath Hariharan, and Gordon Wetzstein

    2023

  17. [17]

    InACM SIGGRAPH 2024 Conference Papers

    Neural Control Variates with Automatic Integration. InACM SIGGRAPH 2024 Conference Papers. 1–9. Bailey Miller, Rohan Sawhney, Keenan Crane, and Ioannis Gkioulekas

    2024

  18. [18]

    Bailey Miller, Rohan Sawhney, Keenan Crane, and Ioannis Gkioulekas

    Boundary Value Caching for Walk on Spheres.ACM Transactions on Graphics (TOG)42, 4 (2023), 1–11. Bailey Miller, Rohan Sawhney, Keenan Crane, and Ioannis Gkioulekas. 2024a. Differen- tial Walk on Spheres.ACM Transactions on Graphics (TOG)43, 6 (2024), 1–18. Bailey Miller, Rohan Sawhney, Keenan Crane, and Ioannis Gkioulekas. 2024b. Walkin’Robin: Walk on Sta...

    2023

  19. [19]

    Mervin E Muller

    Solving partial differential equations in participating media.ACM Transactions on Graphics (TOG)44, 4 (2025), 1–21. Mervin E Muller

    2025

  20. [20]

    The Annals of Mathematical Statistics(1956), 569–589

    Some continuous Monte Carlo methods for the Dirichlet problem. The Annals of Mathematical Statistics(1956), 569–589. Yang Qi, Dario Seyb, Benedikt Bitterli, and Wojciech Jarosz

    1956

  21. [21]

    Karl K Sabelfeld

    A monte carlo method for fluid simulation.ACM Transactions on Graphics (TOG)41, 6 (2022), 1–16. Karl K Sabelfeld

    2022

  22. [22]

    Karl K Sabelfeld

    Random walk on spheres algorithm for solving transient drift- diffusion-reaction problems.Monte Carlo Methods and Applications23, 3 (2017), 189–212. Karl K Sabelfeld

    2017

  23. [23]

    Rohan Sawhney

    A global random walk on spheres algorithm for transient heat equation and some extensions.Monte Carlo Methods and Applications25, 4 (2019), 301–316. Rohan Sawhney. 2021.FCPW: Fastest Closest Points in the West. Rohan Sawhney and Keenan Crane

    2019

  24. [24]

    Rohan Sawhney, Bailey Miller, Ioannis Gkioulekas, and Keenan Crane

    Monte Carlo geometry processing: A grid- free approach to PDE-based methods on volumetric domains.ACM Transactions on Graphics39, 4 (2020). Rohan Sawhney, Bailey Miller, Ioannis Gkioulekas, and Keenan Crane

    2020

  25. [25]

    arXiv preprint arXiv:2302.11815(2023)

    Walk on stars: A grid-free monte carlo method for pdes with neumann boundary conditions. arXiv preprint arXiv:2302.11815(2023). Rohan Sawhney, Dario Seyb, Wojciech Jarosz, and Keenan Crane

    work page arXiv 2023

  26. [26]

    Maximilian Seitzer, Arash Tavakoli, Dimitrije Antic, and Georg Martius

    Grid-free Monte Carlo for PDEs with spatially varying coefficients.ACM Transactions on Graphics (TOG)41, 4 (2022), 1–17. Maximilian Seitzer, Arash Tavakoli, Dimitrije Antic, and Georg Martius

    2022

  27. [27]

    arXiv preprint arXiv:2203.09168(2022)

    On the pitfalls of heteroscedastic uncertainty estimation with probabilistic neural networks. arXiv preprint arXiv:2203.09168(2022). Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein

    work page arXiv 2022

  28. [28]

    Advances in neural information processing systems33 (2020), 7462–7473

    Implicit neural representations with periodic activation functions. Advances in neural information processing systems33 (2020), 7462–7473. Ryusuke Sugimoto, Christopher Batty, and Toshiya Hachisuka

    2020

  29. [29]

    InACM SIGGRAPH 2024 Conference Papers

    Velocity-Based Monte Carlo Fluids. InACM SIGGRAPH 2024 Conference Papers. 1–11. Ryusuke Sugimoto, Terry Chen, Yiti Jiang, Christopher Batty, and Toshiya Hachisuka

    2024

  30. [30]

    Neil A Watson

    A practical walk-on-boundary method for boundary value problems.ACM Transactions on Graphics (TOG)42, 4 (2023), 1–16. Neil A Watson. 2012.Introduction to heat potential theory. Number

    2023

  31. [31]

    Zihan Yu, Rohan Sawhney, Bailey Miller, Lifan Wu, and Shuang Zhao

    Solving inverse PDE problems using grid-free Monte Carlo estimators.ACM Transactions on Graphics (TOG)43, 6 (2024), 1–18. Zihan Yu, Rohan Sawhney, Bailey Miller, Lifan Wu, and Shuang Zhao

    2024

  32. [32]

    Zihan Yu, Lifan Wu, Zhiqian Zhou, and Shuang Zhao

    Robust Derivative Estimation with Walk on Stars.ACM Transactions on Graphics (TOG)44, 6 (2025), 1–16. Zihan Yu, Lifan Wu, Zhiqian Zhou, and Shuang Zhao

    2025

  33. [33]

    InACM SIGGRAPH 2024 Conference Papers

    A Differential Monte Carlo Solver For the Poisson Equation. InACM SIGGRAPH 2024 Conference Papers. 1–10. Zihong Zhou, Eugene d’Eon, Rohan Sawhney, and Wojciech Jarosz

    2024

  34. [34]

    A Derivation of the Non-Cylindrical Boundary Integral Equation We derive the non-cylindrical boundary integral representation(11) from first principles

    Harmonic Caching for Walk on Spheres.ACM Transactions on Graphics (TOG)44, 6 (2025), 1–15. A Derivation of the Non-Cylindrical Boundary Integral Equation We derive the non-cylindrical boundary integral representation(11) from first principles. The proof combines three classical ingredients: Green’s second identity (spatial), the Reynolds transport theorem...

    2025

  35. [35]

    Walking on Heat Stars for Parabolic Heat Equations with Neumann Boundary Conditions•19 maps to the wall component of the caloric measure with no addi- tional correction

    When the ray strikes𝜕Ω 𝑁 at distance 𝜌 before the heat-sphere surface, the inversion 𝑐wall =𝜌 2/(2𝑛𝑢𝑒 −𝑢) Conference acronym ’XX, June 03–05, 2018, Woodstock, NY. Walking on Heat Stars for Parabolic Heat Equations with Neumann Boundary Conditions•19 maps to the wall component of the caloric measure with no addi- tional correction. B.2 Source Term Sampling...

    2018

  36. [36]

    The direction 𝜔 is reused from the directional sampling step (Section 4.5.1), requiring no ad- ditional random variate

    The temporal coordinate 𝑢 is drawn directly from the Gamma distribution. The direction 𝜔 is reused from the directional sampling step (Section 4.5.1), requiring no ad- ditional random variate. The point (𝑦, 𝑠) is then mapped via (58); if 𝑦∉H ★ the contribution is zero. The estimator isb𝐼𝑓 =𝑍·𝑓(𝑦, 𝑡−𝑠) , with no explicit evaluation of𝐺required. C Gradient ...

    2012

  37. [37]

    Conference acronym ’XX, June 03–05, 2018, Woodstock, NY

    The critical difference from the Laplace case is that 𝑤(𝑟) is non-constant, so ∇𝑦𝑤≠ 0and the volume term survives. Conference acronym ’XX, June 03–05, 2018, Woodstock, NY. 20•Bao et al. With 𝜔𝑖 =(𝑦 𝑖 −𝑥 0𝑖)/𝑟, the surface and volume terms separate as 𝑆𝑖 =𝜏(𝑐) ∫ 𝑐 0 𝑟(𝑠) 𝑛+1 4𝑠2 ∫ 𝑆𝑛−1 𝑢(𝑥 0 +𝑟(𝑠)𝜔, 𝑡 0 −𝑠)𝜔 𝑖 d𝜔d𝑠,(74) 𝑉𝑖 =−𝜏(𝑐) ∫ 𝑐 0 1 2𝑠2 ∫ 𝑟(𝑠) 0 𝑟 𝑛 ∫...

    2018