16 papers · ranked by Valyu relevance
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…
Per Kristen Jakobsen
We will start our investigations by considering second order equations. More specifically we will consider scalar linear second order PDEs with two independent variables. The most general such equation is of the form (9-1)$Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu=G,\tag{85}$ where u = u(x, y), A = A(x, y) etc. The…
William G. Litvinov, Eugene Lytvynov
We show that infinitely differentiable solutions to parabolic and hyperbolic equations, whose right-hand sides are analytical in time, are also analytical in time at each fixed point of the space. These solutions are given in the form of the Taylor expansion with respect to time t with coefficients depending on x. The…
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…
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…
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…
Iasson Karafyllis, Miroslav Krstić
This work provides stability results in the spatial sup norm for hyperbolic-parabolic loops in one spatial dimension. The results are obtained by an application of the small-gain stability analysis. Two particular cases are selected for the study because they contain challenges typical of more general systems (to which…
Peter Gangl, Kevin Sturm, Michael Neunteufel, Joachim Schöberl
In this paper, we present a framework for automated shape differentiation in the finite element software NGSolve. Our approach combines the mathematical Lagrangian approach for differentiating PDE-constrained shape functions with the automated differentiation capabilities of NGSolve. The user can decide which degree of…
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…
Klaus Deckelnick, Vanessa Styles
In this paper we analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain…
F. Hamel, F. Lavigne, G. Martin, L. Roques
We study the dynamics of adaptation of a large asexual population in a n-dimensional phenotypic space, under anisotropic mutation and selection effects. When n = 1 or under isotropy assumptions, the ‘replicator-mutator’ equation is a standard model to describe these dynamics. However, the n-dimensional anisotropic case…
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…
Axel G. R. Turnquist, Horacio G. Rotstein
Quadratization of biophysical (conductance-based) models having a parabolic-like voltage nullcline in the subthreshold voltage regime refers to the process by which these models are substituted by “caricature” models having a strictly parabolic voltage nullcline and a linear nullcline for the recovery variable. We…
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…
David Thompson, Johan Gielis
Our understanding of quantum phenomena often begins with simple particle-in-a-box style problems, the solutions of which introduce the student to foundational quantum concepts such as degeneracy and quantization. Simple model geometries of confinement afford analytic solutions, which are readily derivable, easily…