Tootfinder

A note on Tricomi-type partial differential equations with white noise initial condition
Enrico Bernardi, Alberto Lanconelli
https://arxiv.org/abs/2509.11753 https://

A note on Tricomi-type partial differential equations with white noise initial condition
We study a class of Tricomi-type partial differential equations previously investigated in [28]. Firstly, we generalize the representation formula for the solution obtained there by allowing the coefficient in front of the second-order partial derivative with respect to $x$ to be any non integer power of $t$. Then, we analyze the robustness of that solution by taking the initial data to be Gaussian white noise and we discover that the existence of a well-defined random field solution is lost up…

Adaptive least-squares space-time finite element methods for convection-diffusion problems
Christian K\"othe, Olaf Steinbach
https://arxiv.org/abs/2509.11955 https://

Adaptive least-squares space-time finite element methods for convection-diffusion problems
In this paper we formulate and analyse adaptive (space-time) least-squares finite element methods for the solution of convection-diffusion equations. The convective derivative $\mathbf{v} \cdot \nabla u$ is considered as part of the total time derivative $\frac{d}{dt}u = \partial_t u + \mathbf{v} \cdot \nabla u$, and therefore we can use a rather standard stability and error analysis for related space-time finite element methods. For stationary problems we restrict the ansatz space $H^1_0(Ω)$ …

On the Characterization of gH-partial derivatives and gH-Product for Interval-Valued Functions
Amir Suhail, Tauheed, Akhlad Iqbal
https://arxiv.org/abs/2509.07565 https://

On the Characterization of gH-partial derivatives and gH-Product for Interval-Valued Functions
In this paper, we show by a counterexample that the gH-partial derivative of interval-valued functions (IVFs) may exist even when the partial derivative of the end point functions do not. Next, we introduce the gH-partial derivative in terms of gH-derivative and discuss its complete characterization. Furthermore, we introduce the gH-product of a vector with an n-tuples of intervals and illustrate by a suitable example that our definition refines the definition existing in the literature. To ill…

Had a great time presenting our paper “Domain-specific tensor languages” today at #ICFP2025 in Singapore 🇸🇬
The images attached are screenshots from the paper as teasers — diagrams for derivative rules, the Riemann curvature 4-tensor in Einstein’s index notation, and the same in diagram form.
📄 Slides:

Estimation for the logarithmic partial derivative and Wiman-Valiron theory in several complex variables
Junfeng Xu, Sujoy Majumder, Nabadwip Sarkar
https://arxiv.org/abs/2509.07113

Estimation for the logarithmic partial derivative and Wiman-Valiron theory in several complex variables
The first objective of the paper is to estimate logarithmic partial derivative for meromorphic functions in several complex variables. Our estimations for logarithmic partial derivatives extend the results of Gundersen \cite{GG2} to the higher dimensions. The second objective of the paper is to study Wiman-Valiron theory into higher dimensions. The third objective of this paper is to study the growth of solutions of a class of complex partial differential equations and our obtained result is th…

$L^2$ normal velocity implies strong solution for graphical Brakke flows
Kotaro Motegi
https://arxiv.org/abs/2510.11377 https://arxiv.org/pdf/2510.11377

$L^2$ normal velocity implies strong solution for graphical Brakke flows
We prove that if a one-parameter family of varifolds has an $L^2$ normal velocity $v$ in the sense of Brakke, and if the family is represented as the graph of a continuous function $f$ with continuous spatial derivative $\nabla f$, then $f$ has weak derivatives $\partial_t f, \nabla^2 f \in L^2$, and $v$ coincides with the usual normal velocity of the graph. Moreover, by combining this result with parabolic regularity theory, we show that graphical Brakke flows with forcing term in $L^{p,q}$ an…

Rational functions that share finite values with their first derivative
Andreas Sauer, Andreas Schweizer
https://arxiv.org/abs/2509.08517 https://arxiv.org…

Rational functions that share finite values with their first derivative
We treat shared value problems for rational functions $R(z)$ and their derivative $R'(z)$ in the plane and on the sphere. We also consider shared values for the pair $R(w)$ and $\partial_{z} R = λw \cdot R'(w)$ on ${\mathbb C} \setminus \{ 0 \}$ and $\widehat{\mathbb C}$, again with rational functions $R$. In ${\mathbb C} \setminus \{ 0 \}$ this is related to shared values of meromorphic functions $f : {\mathbb C} \to \widehat{\mathbb C}$ and $f'$ through $f(z)=R(w)$ with $w=\exp(λz)$, while …

The homotopy type of the Poincar\'e cobordism category for surfaces
Az\'elie Picot
https://arxiv.org/abs/2510.07659 https://arxiv.org/pdf/2510.0765…

The homotopy type of the Poincaré cobordism category for surfaces
We define a version of the surface cobordism category $\mathrm{Cob}_2^{\mathrm{SG}}(X)$ over a base space $X$ where surfaces are considered up to self homotopy equivalences instead of diffeomorphisms. We prove the induced functor $B\mathrm{Cob}_2^{\mathrm{SG}}(-): \mathcal{S} \rightarrow \mathcal{S}$ is not $1$-excisive. We show its first derivative $\partial_1 B\mathrm{Cob}_2^{\mathrm{SG}}(-)$ in the Goodwillie sense is equivalent to a Thom spectrum over $B\mathrm{haut}_*^+(S^2)$.

Space Mapping Optimization using Neural Networks for Efficient Parameter Estimation
Dhruvil Kamleshkumar Kotecha
https://arxiv.org/abs/2509.06899 https://a…

Space Mapping Optimization using Neural Networks for Efficient Parameter Estimation
This project focuses on optimizing input parameters of a partial derivative function of a fine model using Neural network-based Space Mapping Optimization (SMO). The fine model is known for its high accuracy but is computationally expensive. On the other hand, the coarse model is represented by a neural network, which is much faster but less accurate. The SMO approach is applied to bridge the gap between these two models and estimate the optimal input parameters for the fine model. Additionally…

A control-based spatial source reconstruction in fractional heat equations
Galina Garc\'ia, Joaqu\'in Vidal, Sebasti\'an Zamorano
https://arxiv.org/abs/2510.07528 ht…

A control-based spatial source reconstruction in fractional heat equations
This article addresses the inverse source problem for a nonlocal heat equation involving the fractional Laplacian. The primary goal is to reconstruct the spatial component of the source term from partial observations of the system's state and its time derivative over a subset of the domain. A reconstruction formula for the Fourier coefficients of the unknown source is derived, leveraging the null controllability property of the fractional heat equation when the fractional order lies in the inte…

A probabilistic representation for the gradient in a linear parabolic PDE with Neumann boundary condition
Abdelatif Bench\'erif Madani
https://arxiv.org/abs/2510.01898 https…

A probabilistic representation for the gradient in a linear parabolic PDE with Neumann boundary condition
We give a probabilistic representation for the gradient of a 2nd order linear parabolic PDE $\partial_{t}u(t,x)=(1/2)a^{ij}\partial_{ij}u(t,x)+b^{i}\partial_{i}u(t,x)$ with Cauchy initial condition $u(0,x)=f(x)$ and Neumann boundary condition in a (closed) convex bounded smooth domain $D$ in $\mathbb{R}^{d}$, $d\geq 1$. The idea is to start from a penalized version of the associated reflecting diffusion $X^{x}$, proceed with a pathwise derivative, show that the resulting family of $ν$-directio…

A Gutzwiller trace formula for singular potentials
Jared Wunsch, Mengxuan Yang, Yuzhou Joey Zou
https://arxiv.org/abs/2509.05220 https://arxiv.org/pdf/2509…

A Gutzwiller trace formula for singular potentials
The Gutzwiller trace formula relates the asymptotic spacing of quantum-mechanical energy levels in the semiclassical limit to the dynamics of periodic classical particle trajectories. We generalize this result to the case of non-smooth potentials, for which there is partial reflection of energy from derivative discontinuities of the potential. It is the periodic trajectories of an associated branching dynamics that contribute to the trace asymptotics in this more general setting; we obtain a pr…

D3PINNs: A Novel Physics-Informed Neural Network Framework for Staged Solving of Time-Dependent Partial Differential Equations
Xun Yang, Guanqiu Ma
https://arxiv.org/abs/2508.20440

D3PINNs: A Novel Physics-Informed Neural Network Framework for Staged Solving of Time-Dependent Partial Differential Equations
In this paper, we propose a novel framework, Dynamic Domain Decomposition Physics-Informed Neural Networks (D3PINNs), for solving time-dependent partial differential equations (PDEs). In this framework, solutions of time-dependent PDEs are dynamically captured. First, an approximate solution is obtained by the Physics-Informed Neural Networks (PINNs) containing the domain decomposition, then the time derivative terms in the PDE will be retained and the other terms associated with the solution…

Forward and backward problems for abstract time-fractional Schr\"odinger equations
S. E. Chorfi, F. Et-tahri, L. Maniar, M. Yamamoto
https://arxiv.org/abs/2510.03600 https:…

Forward and backward problems for abstract time-fractional Schrödinger equations
We investigate forward and backward problems associated with abstract time-fractional Schrödinger equations $\mathrm{i}^ν\partial_t^αu(t) + A u(t)=0$, $α\in (0,1)\cup (1,2)$ and $ν\in\{1,α\}$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. This kind of equation, which incorporates the Caputo time-fractional derivative of order $α$, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the f…

Partial sharing and cross sharing of entire function with its derivative
Sujoy Majumder, Nabadwip Sarkar, Debabrata Pramanik
https://arxiv.org/abs/2509.20273 https://

Partial sharing and cross sharing of entire function with its derivative
In the paper, we investigate the uniqueness problem of a power of an entire function that share one value partially with it's linear differential polynomial and obtain a result, which improves several previous results in a large scale. Also in the paper we include some applications of our main result. Moreover, we solve the question raised by Wang and Liu (Value cross-sharing of meromorphic functions, Comput. Methods Funct. Theory (2023). https://doi.org/10.1007/s40315-023-00481-9) for a specia…

A kernel compression method for distributed-order fractional partial differential equations
Jonas Beddrich, Barbara Wohlmuth
https://arxiv.org/abs/2508.13631 https://

A kernel compression method for distributed-order fractional partial differential equations
We propose a kernel compression method for solving Distributed-Order (DO) Fractional Partial Differential Equations (DOFPDEs) at the cost of solving corresponding local-in-time PDEs. The key concepts are (1) discretization of the integral over the order of the fractional derivative and (2) approximation of linear combinations of integral kernels with exponential sums, expressing the non-local history term as a sum of auxiliary variables that solve a weakly coupled, local in time system of PDEs.…

The pluricomplex Poisson kernel for convex finite type domains
Leandro Arosio, Filippo Bracci, Matteo Fiacchi
https://arxiv.org/abs/2509.26230 https://arxi…

The pluricomplex Poisson kernel for convex finite type domains
Given a bounded convex domain $D\subset \mathbb C^n$ of finite D'Angelo type and a boundary point $ξ\in \partial D$, we prove that the homogeneous complex Monge-Ampère equation $(dd^cu)^n=0$ possesses a continuous strictly negative solution $Ω_ξ$ that vanishes on $\partial D\setminus \{ξ\}$ and has a simple pole at $ξ$. We establish that $Ω_ξ(z)$ equals (up to sign) the normal derivative at $ξ$ of the pluricomplex Green function $G_z$, and its sublevel sets are the horospheres centered…

A stability-enhanced nonstandard finite difference framework for solving one and two-dimensional nonlocal differential equations
Shweta Kumari, Mani Mehra
https://arxiv.org/abs/2508.13542

A stability-enhanced nonstandard finite difference framework for solving one and two-dimensional nonlocal differential equations
Standard finite difference (SFD) schemes often suffer from limited stability regions, especially when applied in explicit setup to partial differential equations. To address this challenge, this study investigates the efficacy of nonstandard finite difference (NSFD) schemes in enhancing stability of explicit SFD schemes for 1D and 2D Caputo-type time-fractional diffusion equations (TFDEs). A nonstandard L1approximation is proposed for the Caputo fractional derivative, and its local truncation e…