•We study degenerate second order operators with “Euler-type” degeneracy.•We show boundary continuity assuming only local boundedness.•We strongly constrain behaviors near the boundary and near ...infinity.•We prove a Liouville theorem for global solutions assuming just local boundedness.
We study a class of second-order boundary-degenerate elliptic equations in two dimensions with minimal regularity assumptions. We prove a maximum principle and a Harnack inequality at the degenerate boundary and, assuming local boundedness, continuity. For globally defined non-negative solutions we provide strong constraints on behavior at infinity, and prove a Liouville-type theorem for entire solutions on the closed half-plane. The class of PDE in question includes many from mathematical finance, Keldysh- and Tricomi-type PDE, and the 2nd order reduction of the fully non-linear 4th order Abreu equation from Kähler geometry. We present some possible future research directions.
A New Family of the Local Fractional PDEs Yang, Xiao-Jun; Machado, J.A. Tenreiro; Nieto, Juan J.
Fundamenta informaticae,
01/2017, Letnik:
151, Številka:
1-4
Journal Article
Recenzirano
A new family of the local fractional PDEs is investigated in this article. The linear, quasilinear, semilinear and nonlinear local fractional PDEs are presented. Furthermore, three types of the local ...fractional PDEs are discussed, namely, parabolic, hyperbolic and elliptic. Several examples illustrate the corresponding models in nonlinear mathematical physics.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
We consider a system of reaction-advection-diffusion partial differential equation (PDE) with a distributed input subject to an unknown and arbitrarily large time delay. Using Lyapunov technique, we ...derive a delay-adaptive predictor feedback controller to ensure the global stability of the closed-loop system in the <inline-formula><tex-math notation="LaTeX">L^2</tex-math></inline-formula> sense. More precisely, we express the input delay as a 1-D transport PDE with a spatial argument leading to the transformation of the time delay into a spatially distributed shift. For the resulting mixed transport and reaction-advection-diffusion PDE system, we employ a PDE backstepping design and certainty equivalence principle to derive the suitable adaptive control law that compensates for the effect of the unknown time delay. Our controller ensures the global stabilization in the <inline-formula><tex-math notation="LaTeX">L^2</tex-math></inline-formula> sense. Our result is the first delay-adaptive predictor feedback controller with a PDE plant subject to a delayed distributed input. The feasibility of the proposed approach is illustrated by considering a mobile robot that spread a neutralizer over a polluted surface to achieve efficient decontamination with an unknown actuator delay arising from the noncollocation of the contaminant diffusive process and the moving neutralizer source. Consistent simulation results are presented to prove the effectiveness of the proposed approach.
Partial Differential Equations (PDE) appear in multiple Physic and Engineering applications. Normally, when modeling an application, the use of well-known and already solved PDE is considered. But ...what happens if a new PDE is used? Solving a new PDE is not an easy task. In this paper, we use a Computer Algebra System (Cas) in order to find the solution of PDE of first order.
Specifically, we deal with Pfaff Equations, Quasilinear PDE and general first order PDE (using Lagrange–Charpit Method).
To solve these PDE, we combine the power of a Cas with the flexibility of programming with it. Furthermore, the developed programs do not only provide the final result but also display all the intermediate steps which lead to find the solution of the PDE. This way, we introduce SFOPDES, a new Stepwise First Order PDE Solver which serves as a tutorial showing, step by step, the way to deal with PDE.
We investigate the stability of statistically stationary conductive states for Rayleigh–Bénard convection that arise due to a bulk stochastic internal heating. Our results indicate that stochastic ...forcing at small magnitude has little to no effect, while strong stochastic forcing has a destabilizing effect. The methodology put forth in this article, which combines rigorous analysis with careful computation, provides an approach to hydrodynamic stability which is applicable to a variety of systems subject to a large scale stochastic forcing.
•Nonlinear stability is defined for stochastic hydrodynamic systems with a stationary distribution.•Partially rigorous justification is provided.•Monte Carlo simulations imply that stochastic forcing has a destabilizing effect.•The marginally stable setting has rare events that are strongly destabilizing.
The paper presents a unique reaction-diffusion coupled system with p(u(t,x))-growth, employing the decomposition approach of H−1 norm. This method is designed to address image denoising by ...considering both texture and smooth components during the recovery process. The existence and uniqueness of weak solutions for the coupled system are established using Galerkin's method within a suitable space framework. Experimental analysis demonstrates the model's effectiveness in image restoration, and it is compared with other competitive models to showcase its performance.
Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of available results in this direction are based on microscopic approaches, where ODEs describe the ...evolution of regular cars and AVs. In this paper, we propose a multiscale approach, based on recently developed models for moving bottlenecks. Our main result is the proof of existence of solutions for time-varying bottleneck speed, which corresponds to open-loop controls with bounded variation.
•We present a general method to represent tangential tensor fields on surfaces.•It enables the finite element approximation of vector/tensor-valued partial differential equations on surfaces.•It ...describes tensor fields intrinsically, avoiding Cartesian representations and tangency constraints.
We introduce a new method, the Local Monge Parametrizations (LMP) method, to approximate tensor fields on general surfaces given by a collection of local parametrizations, e.g. as in finite element or NURBS surface representations. Our goal is to use this method to solve numerically tensor-valued partial differential equations (PDEs) on surfaces. Previous methods use scalar potentials to numerically describe vector fields on surfaces, at the expense of requiring higher-order derivatives of the approximated fields and limited to simply connected surfaces, or represent tangential tensor fields as tensor fields in 3D subjected to constraints, thus increasing the essential number of degrees of freedom. In contrast, the LMP method uses an optimal number of degrees of freedom to represent a tensor, is general with regards to the topology of the surface, and does not increase the order of the PDEs governing the tensor fields. The main idea is to construct maps between the element parametrizations and a local Monge parametrization around each node. We test the LMP method by approximating in a least-squares sense different vector and tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply the LMP method to two physical models on surfaces, involving a tension-driven flow (vector-valued PDE) and nematic ordering (tensor-valued PDE), on different topologies. The LMP method thus solves the long-standing problem of the interpolation of tensors on general surfaces with an optimal number of degrees of freedom.
Generalizing the L-Kuramoto–Sivashinsky (L-KS) kernel from our earlier work, we give a novel explicit-kernel formulation useful for a large class of fourth order deterministic, stochastic, linear, ...and nonlinear PDEs in multispatial dimensions. These include pattern formation equations like the Swift–Hohenberg and many other prominent and new PDEs. We first establish existence, uniqueness, and sharp dimension-dependent spatio-temporal Hölder regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on {R+×Rd}d=13. The spatio-temporal Hölder exponents are exactly the same as the striking ones we proved for our recently introduced Brownian-time Brownian motion (BTBM) stochastic integral equation, associated with time-fractional PDEs. The challenge here is that, unlike the positive BTBM density, the L-KS kernel is the Gaussian average of a modified, highly oscillatory, and complex Schrödinger propagator. We use a combination of harmonic and delicate analysis to get the necessary estimates. Second, attaching order parameters ε1 to the L-KS spatial operator and ε2 to the noise term, we show that the dimension-dependent critical ratio ε2/ε1d/8 controls the limiting behavior of the L-KS SPDE, as ε1,ε2↘0; and we compare this behavior to that of the less regular second order heat SPDEs. Finally, we give a change-of-measure equivalence between the canonical L-KS SPDE and nonlinear L-KS SPDEs. In particular, we prove uniqueness in law for the Swift–Hohenberg and the law equivalence—and hence the same Hölder regularity—of the Swift–Hohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.
PDF
![A variable p[u] exponent re...](http://api.summon.serialssolutions.com/2.0.0/image/issn/XR4PS5YY4K/0022-247X/small "A variable p[u] exponent reaction-diffusion PDE for image denoising")
