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This paper is devoted to dimensional reductions via the norm resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its…

Mathematical Physics · Physics 2018-11-26 David Krejcirik , Nicolas Raymond , Julien Royer , Petr Siegl

In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such…

Probability · Mathematics 2023-03-10 Martin Redmann

This paper deals with the large time dynamics of bounded solutions of reaction-diffusion equations with unbounded initial support in $\mathbb{R}^N$. We prove a variational formula for the spreading speeds in any direction, and we also…

Analysis of PDEs · Mathematics 2023-07-08 François Hamel , Luca Rossi

This manuscript aims to provide a self-contained introduction to the regularity theory for elliptic PDE, focusing on the main ideas rather than proving all results in their greatest generality. It can be seen as a bridge between an…

Analysis of PDEs · Mathematics 2023-01-05 Xavier Fernández-Real , Xavier Ros-Oton

This article provides techniques of raising the regularity of fractional order equations and resolves fundamental questions on the one-dimensional homogeneous boundary-value problem of skewed (double-sided) fractional diffusion advection…

Classical Analysis and ODEs · Mathematics 2020-05-12 Yulong Li

The problem of diffusion in a time-dependent (and generally inhomogeneous) external field is considered on the basis of a generalized master equation with two times, introduced in [1,2]. We consider the case of the quasi Fokker-Planck…

Statistical Mechanics · Physics 2015-05-18 S. A. Trigger , G. J. F. van Heijst , O. F. Petrov , P. P. J. M. Schram

To analyze possible generalizations of reaction-diffusion schemes for the case of subdiffusion we discuss a simple monomolecular conversion A --> B. We derive the corresponding kinetic equations for local A and B concentrations. Their form…

Statistical Mechanics · Physics 2009-11-11 I. M. Sokolov , M. G. W. Schmidt , F. Sagues

This work presents a physics-conditioned latent diffusion model tailored for dynamical downscaling of atmospheric data, with a focus on reconstructing high-resolution 2-m temperature fields. Building upon a pre-existing diffusion…

Machine Learning · Computer Science 2025-10-29 Paul Rosu , Muchang Bahng , Erick Jiang , Rico Zhu , Vahid Tarokh

Open biochemical systems of interacting molecules are ubiquitous in life-related processes. However, established computational methodologies, like molecular dynamics, are still mostly constrained to closed systems and timescales too small…

Quantitative Methods · Quantitative Biology 2025-10-15 Margarita Kostré , Christof Schütte , Frank Noé , Mauricio J. del Razo

We provide explicit classical solutions and stochastic analogues for distributed-order space-time fractional diffusion equations on bounded domains with zero exterior boundary conditions. We also show that our results still hold when the…

Analysis of PDEs · Mathematics 2022-10-11 Ngartelbaye Guerngar , James McCormick

This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually…

Numerical Analysis · Mathematics 2017-09-26 Federico Fuentes , Brendan Keith , Leszek Demkowicz , Patrick Le Tallec

This paper was originally published in June 2013 in JFM (vol. 725). There was one result concerning an expression of viscous dissipation in convective flows and a second result on the condition of applicability of the anelastic liquid…

Fluid Dynamics · Physics 2019-11-05 Thierry Alboussiere , Yanick Ricard

We establish the existence of solutions to a class of non-linear stochastic differential equation of reaction-diffusion type in an infinite-dimensional space, with diffusion corresponding to a given transition kernel. The solution obtained…

Probability · Mathematics 2021-08-10 Conrado da Costa , Bernardo Freitas Paulo da Costa , Daniel Valesin

In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also…

We provide the proof of convergence of the directional diffusion splitting scheme for two-dimensional parabolic and elliptic advection-diffusion-reaction problems with certain restrictions on problem data

Numerical Analysis · Mathematics 2024-11-20 R. Drebotiy , H. Shynkarenko

Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental…

Numerical Analysis · Mathematics 2016-12-21 Romana Boiger , Jan Hasenauer , Sabrina Hross , Barbara Kaltenbacher

In this paper, a new variational formulation based on discontinuous Galerkin technique for a reaction-diffusion problem is introduced, and the discontinuous Galerkin technique of this work is different from the general discontinuous…

Numerical Analysis · Mathematics 2012-04-19 Zhihao Ge , Jiwei Cao

Based on easy-to-follow considerations it is not difficult to be vehemently opposed not only the solutions found in that paper but also the conclusions manifested there.

Quantum Physics · Physics 2009-11-11 Antonio S. de Castro

I would like to thank the authors for their interesting and very clearly presented paper discussing probabilistic solvers for ODEs and PDEs.

Methodology · Statistics 2016-10-27 Jon Cockayne

In this comment we report that a recent paper [Eur. Phys. J. Plus 138, 195 (2023)] contains significant errors, omissions and inaccuracies.

Exactly Solvable and Integrable Systems · Physics 2023-08-02 Cihangir Özemir , Faruk Güngör
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