16 papers · ranked by Valyu relevance
Yuxin Wang, Yueyang Shen, Daxuan Deng, Ivo D. Dinov
Historically, PDEs have been classified as ‘‘elliptic’’, ‘‘parabolic’’, or ‘‘hyperbolic’’.9,11,12 Naturally, most of the fundamental PDEs, namely the Laplace equation, heat equation, and wave equation, fall into these types of categories. Contrasting the differences between separate partial differential equations may…
Leticia Mattos Da Silva, Oded Stein, Justin Solomon
We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To…
Martin Dindoš
In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial _t u -{{\,\textrm{div}\,}}A \nabla u+B\cdot \nabla u=0$ on cylindrical domains of the form $\Omega ={\mathcal{O}}\times{\mathbb{R}}$ where ${\mathcal{O}}\subset{\mathbb{R}}^n$ is a uniform domain…
Yurij Salmaniw, Alexander P Browning
Parameter identifiability is often requisite to the effective application of mathematical models in the interpretation of biological data, however theory applicable to the study of partial differential equations remains limited. We present a new approach to structural identifiability analysis of fully observed…
Martin Dindoš, Linhan Li, Jill Pipher
In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE −∂tu + div(A∇u) = 0 on a Lipschitz cylinder O × R is solvable for some p ∈ (1, ∞) under the assumption that the matrix A is elliptic, has bounded and measurable coefficients and its coefficients satisfy a natural…
Paolo Bonicatto, Gennaro Ciampa, Gianluca Crippa
We study the Cauchy problem for the advection-diffusion equation $\partial _t u +{{\,\mathrm{\textrm{div}}\,}}u\varvec{b}= \Delta u$ associated with a merely integrable divergence-free vector field $\varvec{b}$ defined on the torus. We discuss existence, regularity and uniqueness results for distributional and…
Antonio Bueno, Rafael López
in a domain Ω ⊂ R 2 , where φ ∈ C 1 ([−1, 1]) and a, b ∈ R. We approach the existence of radial solutions when Ω is a disk of small radius, giving an affirmative answer when the PDE is of elliptic type. In the hyperbolic case we show that no radial solution exists, while in the parabolic case we find explicitly all the…
Luke Bhan, Yuanyuan Shi, Iasson Karafyllis, Miroslav Krstić + 1 more
'James B. Rawlings'] Abstract— Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output is the state estimate, while their inputs are the…
Matthias Ehrhardt, Jochen Glück, Pavel Petrov, Stefan Tappe
Pseudodifferential parabolic equations with an operator square root arise in wave propagation problems as a one-way counterpart of the Helmholtz equation. The expression under the square root usually involves a differential operator and a known function. We discuss a rigorous definition of such operator square roots…
Tian Jing
In this paper, we study the mixed-type equation uu x = uyy, which behaves as forward and backward parabolic equations depending on the sign of u. The equation arises from the study of boundary layers with separation. We seek solutions that change their type smoothly to better understand the equation. We simplify the…
M. V. George, Bo Guan
Over many decades fully nonlinear PDEs, and the complex Monge-Amp`ere equation in particular played a central role in the study of complex manifolds. Most previous works focused on problems that can be expressed through equations involving real (1, 1) forms. As many important questions, especially those linked to…
Ray Zirui Zhang, Christopher E. Miles, Xiaohui Xie, John S. Lowengrub
We propose a new neural network based method for solving inverse problems for partial differential equations (PDEs) by formulating the PDE inverse problem as a bilevel optimization problem. At the upper level, we minimize the data loss with respect to the PDE parameters. At the lower level, we train a neural network to…
Dimitrios Katsaounis, Mark A.J. Chaplain, Nikolaos Sfakianakis
Invasion of the surrounding tissue is a key aspect of cancer growth and spread involving a coordinated effort between cell migration and matrix degradation, and has been the subject of mathematical modelling for almost 30 years. In this current paper we address a long-standing question in the field of cancer cell…
Mátyás Paczkó, Eörs Szathmáry, András Szilágyi
The RNA world hypothesis proposes that during the early evolution of life, primordial genomes of the first self-propagating evolutionary units existed in the form of RNA-like polymers. Autonomous, non-enzymatic and sustained replication of such information carriers presents a problem, because product formation and…
Tess Bonnard, Emilie Doat, Jean-René Cazalets, Clément Morgat + 2 more
Motion sickness (MS) is commonly hypothesized to arise from sensory conflicts between incongruent sources of sensory information. Different types of sensory conflicts can induce MS, yet it remains unclear whether distinct contexts produce different physiological responses. Moreover, there is a lack of reliable…
Authors not listed
Real-world datasets in chemical engineering and bioengineering processes--such as those from catalytic reactors, multiphase flows, polymerization reactors, bioreactors, and clinical trials--can often be unlabelled or disorganized, rendering the training of existing supervised learning models ineffective at learning the…