PDEs Of Type Research Articles

Mild solutions of the stochastic MHD equations driven by fractional Brownian motions

The incompressible magnetohydrodynamic equations driven by additive fractional Brownian motions are considered. We firstly establish the local existence and uniqueness of the mild solution in Lp space on a smooth bounded domain in Rd(d=2,3). The proof is based on the semigroup theory, fixed point theorem and the results of stochastic PDEs of linear parabolic type. In the proof, the eigenvalue problem with perfectly conducting wall condition is considered to weaken the requirements of noise terms. Finally, the global existence of mild solutions is also established by energy estimate.

Necessary conditions for optimal boundary-control of couple linear parabolic partial differential equations

In this research, we will study the classical continuous optimal boundary- control problem(CCOPB-CPR) determine by couple linear PDEs of parabolic type (CCOPB-CCLPDEs) in details. the existence theorem for the uniqueness state vector solution (STVSO) of couple linear parabolic PDEs (CLPPDEs) for given (fixed) continuous classical boundary- control vector (CCB-CV) is considered and proved, the existence theorem of a continuous classical -boundary optimal control vector (CCOPB-CV) associated with the CLPPDEs is developed and proved, while the Frechet derivative (Féde) of the objective function is derived, the theorem for the existence of a unique solution of the adjoint vector equations (ADVEQ) congruous for the STVSO is considered. Lastly the necessary optimality conditions (NOPC) of the CCOPB-CPR is proved.

New exact solutions of nonlinear wave type PDEs with delay

We consider nonlinear wave type PDEs with delay of the form utt+H1(u)ut=[G(u)ux]x+H2(u)ux+F(u,w),where u=u(x,t) is the unknown function, w=u(x,t−τ), and τ is the delay time. The source function F(u,w) depends on one or several arbitrary functions of one argument. Using the modified method of functional constraints, we obtain a number of new exact solutions with generalized and functional separation of variables, as well as traveling-wave solutions. Most solutions are expressed in terms of elementary functions and contain free parameters. These solutions can be used to formulate test problems intended to evaluate the accuracy of numerical methods for solving nonlinear delay PDEs.

Optical soliton solutions to the (2+1)-dimensional Kundu–Mukherjee–Naskar equation

In this work, a new method has been applied for finding solutions to the Kundu–Mukherjee–Naskar (KMN) equation in (2[Formula: see text]+[Formula: see text]1) dimensions. These optical solutions were derived with the aid of a new method named the Exp-Function method. The method is used for solving the nonlinear governing equation, which contains high nonlinear term. This method is considered as an alternative technique for obtaining both analytical and approximate solutions for a different type of PDE with several applications in science and engineering. The efficiency of the proposed technique has been shown by a few test problems. By comparing the results obtained in this paper with other researchers it can be noticed that this method is a promising technique for finding solutions to other similar problems.

Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls

The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including mean field games and systemic risk. It also appears naturally in stochastic control problems with partial information and in time inconsistent problems. In this paper we propose a novel notion of viscosity solution for parabolic master equations, arising mainly from control problems, and establish its wellposedness. Our main innovation is to restrict the involved measures to a certain set of semimartingale measures which satisfy the desired compactness. As an important example, we study the HJB master equation associated with the control problems for McKean–Vlasov SDEs. Due to practical considerations, we consider closed-loop controls. It turns out that the regularity of the value function becomes much more involved in this framework than the counterpart in the standard control problems. Finally, we build the whole theory in the path dependent setting, which is often seen in applications. The main result in this part is an extension of Dupire’s (2009) functional Ito formula. This Ito formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions in the state dependent case. We provided an elementary proof for this well known result in the short note (2017), and the same arguments work in the path dependent setting here.

On approximating minimizers of convex functionals with a convexity constraint by singular Abreu equations without uniform convexity

AbstractWe revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.

A double mean field equation related to a curvature prescription problem

We study a double mean field–type PDE related to a prescribed curvature problem on compacts surfaces with boundary:(0.1){−Δu=2ρ(Keu∫ΣKeu−1|Σ|)in Σ,∂νu=2ρ′(heu2∫∂Σheu2−1|∂Σ|)on ∂Σ.Here ρ and ρ′ are real parameters, K,h are smooth positive functions on Σ and ∂Σ respectively and ν is the outward unit normal vector to ∂Σ.We provide a general blow–up analysis, then a Moser–Trudinger inequality, which gives energy–minimizing solutions for some range of parameters. Finally, we provide existence of min–max solutions for a wider range of parameters, which is dense in the plane if Σ is not simply connected.

Established and emerging therapeutic uses of PDE type 5 inhibitors in cardiovascular disease.

PDE type 5 inhibitors (PDE5Is), such as sildenafil, tadalafil and vardenafil, are a class of drugs used to prolong the physiological effects of NO/cGMP signalling in tissues through the inhibition of cGMP degradation. Although these agents were originally developed for the treatment of hypertension and angina, unanticipated side effects led to advances in the treatment of erectile dysfunction and, later, pulmonary arterial hypertension. In the last decade, accumulating evidence suggests that PDE5Is may confer a wider range of clinical benefits than was previously recognised. This has led to a broader interest in the cardiovascular therapeutic potential of PDE5Is, in conditions such as hypertension, myocardial infarction, stroke, peripheral arterial disease, chronic kidney disease and diabetes mellitus. Here, we review the pharmacological properties and established licensed uses of this class of drug, along with emerging therapeutic developments and possible future indications.

Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs

In this note, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? The aim of the paper is twofold. Firstly, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on recent results on nonclassical solutions to fully nonlinear equations. Secondly, to define an appropriate algebraic formalism which makes the analytic part of the construction of nonclassical solutions more transparent.

On hydrodynamic limits of Young diagrams

We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. ‘Static’ scaling limits of the shape functions, under these Gibbs measures, have been shown in the literature. The purpose of this article is to study corresponding, but less understood, ‘dynamical’ limits. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types of parabolic PDEs, depending on the energy structure.

Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators

Maximum Principles on unbounded domains play a crucial role in several problems related to linear second-order PDEs of elliptic and parabolic type. In the present notes, based on a joint work with prof. E. Lanconelli, we consider a class of sub-elliptic operators L in R^N and we establish some criteria for an unbounded open set to be a Maximum Principle set for L. We extend some classical results related to the Laplacian(proved by Deny, Hayman and Kennedy) and to the sub-Laplacians on homogeneous Carnot groups (proved by Bonfiglioli and Lanconelli).

Variational principle for a class of nonlocal boundary value problems for mixed type higher-order equations

For considered class of nonlocal boundary value problems for mixed type PDE of arbitrary high order an implicit symmetrizing operator [2,4] is built. Equivalence of the problem to the problem of minimization of appropriate quadratic functional is proved. Existence and uniqueness of the generalized solution in variation sense are obtained.

A mesh-free method for interface problems using the deep learning approach

In this paper, we propose a mesh-free method to solve interface problems using the deep learning approach. Two types of PDEs are considered. The first one is an elliptic PDE with a discontinuous and high-contrast coefficient. While the second one is a linear elasticity equation with discontinuous stress tensor. In both cases, we represent the solutions of the PDEs using the deep neural networks (DNNs) and formulate the PDEs into variational problems, which can be solved via the deep learning approach. To deal with inhomogeneous boundary conditions, we use a shallow neural network to approximate the boundary conditions. Instead of using an adaptive mesh refinement method or specially designed basis functions or numerical schemes to compute the PDE solutions, the proposed method has the advantages that it is easy to implement and is mesh-free. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method for interface problems.

On meromorphic solutions of first-order Briot-Bouquet type PDEs

Utilizing a theorem of Coman and Poletsky (see [4]) which characterizes the factorization of algebraically dependent meromorphic functions, and the parametrization of the polynomial functional equation they satisfy, along with results from Nevanlinna theory, we obtain results on the meromorphic solutions of the first order Briot-Bouquet type PDEs P(u,ux)=0.

Highly dispersive optical solitons using Kudryashov's method

This work is about highly dispersive optical solitons with Kerr law, quadratic-cubic law and non-local nonlinearities. The Kudryashov's method implemented in this paper to find the unique optical closed form of solutions. The method under consideration is consistent, effectual and can be used to establish new exact solitons of different types of PDEs arising in nonlinear optic.

Approximation of variational problems with a convexity constraint by PDEs of Abreu type

Motivated by some variational problems subject to a convexity constraint, we consider an approximation using the logarithm of the Hessian determinant as a barrier for the constraint. We show that the minimizer of this penalization can be approached by solving a second boundary value problem for Abreu's equation which is a well-posed nonlinear fourth-order elliptic problem. More interestingly, a similar approximation result holds for the initial constrained variational problem.

Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation

In linearized gravity, two linearized metrics are considered gauge-equivalent, , when they differ by the image of the Killing operator, . A universal (or complete) compatibility operator for K is a differential operator K1 such that and any other operator annihilating K must factor through K1. The components of K1 can be interpreted as a complete (or generating) set of local gauge-invariant observables in linearized gravity. By appealing to known results in the formal theory of overdetermined PDEs and basic notions from homological algebra, we solve the problem of constructing the Killing compatibility operator K1 on an arbitrary background geometry, as well as of extending it to a full compatibility complex Ki (), meaning that for each Ki the operator Ki+1 is its universal compatibility operator. Our solution is practical enough that we apply it explicitly in two examples, giving the first construction of full compatibility complexes for the Killing operator on these geometries. The first example consists of the cosmological FLRW spacetimes, in any dimension. The second consists of a generalization of the Schwarzschild–Tangherlini black hole spacetimes, also in any dimension. The generalization allows an arbitrary cosmological constant and the replacement of spherical symmetry by planar or pseudo-spherical symmetry.

An exact analytical model for fluid flow through finite rock matrix block with special saturation function

An exact analytical solution for one-dimensional fluid flow through rock matrix block is presented. The nonlinearity induced from flow functions makes the governing equations describing this mechanism difficult to be analytically solved. In this paper, an analytical solution to the infiltration problems considering non-linear relative permeability functions is presented for finite depth, despite its profound and fundamental importance. Elimination of the nonlinear terms in the equation, as a complex and tedious task, is done by applying several successive mathematical manipulations including: Hopf-Cole transformation to obtain a diffusive type PDE; an exponential type transformation to get a convective-diffusive type PDE with suitable boundary conditions; Laplace transformation; Integral transformation to homogenize the boundary condition in the Laplace domain; and finally Laplace inversion method to find the time domain solution. The obtained solution is used for developing a new matrix-fracture transfer function. The developed analytical equations are used for prediction of drying front in the case of evaporation from deep water table. The model results are in close agreement with the experimental data and numerical simulation of the process. Results of sensitivity analysis of different parameters on recovery rate and ultimate production based on design of experiment method are also discussed.

Stochastic nonlinear Fokker–Planck equations

The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker–Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with common noise. The uniqueness of solutions is obtained without any higher moment assumption on the solution by means of a duality argument to a backward stochastic PDE.

Numerical Simulation of Convection-Diffusion Phenomena by Four Inverse-Quadratic-RBF Domain-Meshfree Schemes

The convection-diffusion type of PDEs is numerically solved by four numerical methods in this work. These four comparatively young numerical approaches are categorized as ‘domain-meshfree’ as they require no internal meshing but rely only on the collocation process amongst nodes via. the inverse-quadratic radial basis function (IQ-RBF). They are; the well-known Kansa Collocation Method (KCM), the Hermite Collocation Method (HCM), the Radial Point Interpolation Method (RPIM), and the Dual Reciprocity Boundary Element Method (DRBEM). The work aims to demonstrate the use of IQ-RBF as well as to compare the practical use of the methods. Moreover, engineering senses of criteria judging the quality of the methods are considered. It is found in this work that while KCM is the simplest to construct and deploys, it is highly sensitive to the number of nodes and the IQ-RBF shape parameter. The unsymmetric and populated matrix problem are alleviated when HCM or DRBEM are in use yet more CPU-time and storage seem to be the price to pay, particularly HCM. However, it is actually RPIM that has appeared to be an optimal choice under all the criteria imposed.