PDEs Of Type Research Articles

Mathematical Analysis of a Seawater Intrusion Model Including Storativity

This paper is devoted to a seawater intrusion model in free aquifers, including rock and fluid compressibility effects. This original model is based on the superimposition of a phase-field model on sharp front tracking. It leads to the analysis of a two-dimensional strongly coupled system of PDEs of parabolic type describing the evolution of the depths of the two free surfaces. The main difficulty is dealing with the strong nonlinearity occurring in the time derivative due to the fluid storage coefficient. This point yields to the manipulation of very weak solutions---solutions of an alternative variational formulation focusing on the structural weakness of the problem.

A non-linear system of parabolic type PDEs for epidemic models

A non-linear system of parabolic type PDEs for epidemic models

Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ā + ∑j≥1yjψj for some parameter vector y = (yj)j ≥ 1 ∈ U = [−1,1]N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H01(D) . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under a uniform ellipticity assuption, for any 0 <p< 1, the ℓp summability of the (∥ψj∥L∞)j ≥ 1 implies the ℓp summability of the V-norms of the Taylor or Legendre coefficients. Such results ensure convergence rates n− s of polynomial approximations obtained by best n-term truncation of such series, with s = (1/p)−1 in L∞(U,V) or s = (1/p)−(1/2) in L2(U,V). In this paper we considerably improve these results by providing sufficient conditions of ℓp summability of the coefficient V-norm sequences expressed in terms of the pointwise summability properties of the (|ψj|)j ≥ 1. The approach in the present paper strongly differs from that of [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47], which is based on individual estimates of the coefficient norms obtained by the Cauchy formula applied to a holomorphic extension of the solution map. Here, we use weighted summability estimates, obtained by real-variable arguments. While the obtained results imply those of [7] as a particular case, they lead to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, these results imply that for all 0 <p< 2, the ℓp summability of the coefficient V-norm sequences follows from the weaker assumption that (∥ψj∥L∞)j ≥ 1 is ℓq summable for q = q(p) := 2p/(2−p) . We provide a simple analytic example showing that this result is in general optimal and illustrate our findings by numerical experiments. The analysis in the present paper applies to other types of linear PDEs with similar affine parametrization of the coefficients, and to more general Jacobi polynomial expansions.

Homotopy analysis Sumudu transform method for time—fractional third order dispersive partial differential equation

In this article, we apply the newly introduced numerical method which is a combination of Sumudu transforms and Homotopy analysis method for the solution of time fractional third order dispersive type PDE equations. It is also discussed generalized algorithm, absolute convergence and analytic result of the finite number of independent variables including time variable.

A Locally Gradient-Preserving Reinitialization for Level Set Functions

The level set method commonly requires a reinitialization of the level set function due to interface motion and deformation. We extend the traditional technique for reinitializing the level set function to a method that preserves the interface gradient. The gradient of the level set function represents the stretching of the interface, which is of critical importance in many physical applications. The proposed locally gradient-preserving reinitialization (LGPR) method involves the solution of three PDEs of Hamilton-Jacobi type in succession; first the signed distance function is found using a traditional reinitialization technique, then the interface gradient is extended into the domain by a transport equation, and finally the new level set function is achieved with the solution of a generalized reinitialization equation. We prove the well-posedness of the Hamilton-Jacobi equations, with possibly discontinuous Hamiltonians, and propose numerical schemes for their solutions. A subcell resolution technique is used in the numerical solution of the transport equation to extend data away from the interface directly with high accuracy. The reinitialization technique is computationally inexpensive if the PDEs are solved only in a small band surrounding the interface. As an important application, the LGPR method will enable the application of the local level set approach to the Eulerian Immersed boundary method.

Discretization of Liouville type nonautonomous equations preserving integrals

The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known Goursat list for which the integrals of minimal order are of the order less than or equal to two we presented a list of corresponding semi-discrete versions. The list contains new examples of non-autonomous Darboux integrable chains.

A BSDE with Delayed Generator Approach to Pricing under Counterparty Risk and Collateralization

We consider a nonlinear pricing problem that takes into account credit risk and funding issues. The aforementioned problem is formulated as a stochastic forward-backward system with delay, both in the forward and in the backward component, whose solution is characterized in terms of viscosity solution to a suitable type of path-dependent PDE.

Adaptive boundary observer for parabolic PDEs subject to domain and boundary parameter uncertainties

We are considering the problem of state observation for a class of infinite dimensional systems modeled by parabolic type PDEs. The model is subject to parametric uncertainty entering in both the domain equation and the boundary condition. An adaptive boundary observer, providing online estimates of the system state and parameters, is designed using finite- and infinite-dimensional backstepping-like transformations. The observer is exponentially convergent under an ad hoc persistent excitation condition.

On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy

Radial basis function-generated finite difference (RBF-FD) approximations generalize classical grid-based finite differences (FD) from lattice-based to scattered node layouts. This greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas. Many different types of radial functions have been considered in this RBF-FD context. In this study, we find that (i) polyharmonic splines (PHS) in conjunction with supplementary polynomials provide a very simple way to defeat stagnation (also known as saturation) error and (ii) give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter. In follow-up studies, we will focus on how to best use these hybrid RBF polynomial bases for FD approximations in the contexts of solving elliptic and hyperbolic type PDEs.

Arbitrary dimension convection–diffusion schemes for space–time discretizations

This note proposes embedding a time dependent PDE into a convection–diffusion type PDE (in one space dimension higher) with singularity, for which two discretization schemes, the classical streamline-diffusion and the EAFE (edge average finite element) one, are investigated in terms of stability and error analysis. The EAFE scheme, in particular, is extended to be arbitrary order which is of interest on its own. Numerical results, in combined space–time domain demonstrate the feasibility of the proposed approach.

Invariant partial differential equations with two-dimensional exotic centrally extended conformal Galilei symmetry

Conformal Galilei Algebras labeled by $d,\ell$ (where $d$ is the number of space dimensions and $\ell$ denotes a spin-${\ell}$ representation w.r.t. the $\mathfrak{sl}(2)$ subalgebra) admit two types of central extensions, the ordinary one (for any $d$ and half-integer $\ell$) and the exotic central extension which only exists for $d=2$ and ${\ell}\in\mathbb{N}$. For both types of central extensions invariant second-order PDEs with continuous spectrum were constructed in [1]. It was later proved in [2] that the ordinary central extensions also lead to oscillator-like PDEs with discrete spectrum. We close in this paper the existing gap, constructing \textcolor{black}{a new class of second-order invariant PDEs for the exotic centrally extended CGAs; they admit a discrete and bounded spectrum when applied to a lowest weight representation. These PDEs are markedly different with respect to their ordinary counterparts. The ${\ell}=1$ case (which is the prototype of this class of extensions, just like the $\ell=\frac{1}{2}$ Schr\"odinger algebra is the prototype of the ordinary centrally extended CGAs) is analyzed in detail.

Numerical solution of optimization problems for semi-linear elliptic equations with discontinuous coefficients and solutions

In this work we consider optimization problems for processes described by semi-linear partial differential equations of elliptic type with discontinuous coefficients and solutions (with imperfect contact matching conditions), with controls involved in the coefficients. Finite difference approximations of optimization problems are constructed. For the numerical implementation of finite optimization problems differentiability and Lipshitz-continuity of the grid functional of the approximating grid problems are proved. An iterative method for solving boundary value problems of contact for PDEs of elliptic type with discontinuous coefficients and solutions is developed and validated. The convergence of the iterative process is investigated. And the convergence rate of iterations (with calculated constants) is estimated.

Erectile Dysfunction Among Men Attending Surgical Outpatients Department in a Tertiary Hospital in South-Western Nigeria.

Background:Erectile dysfunction is becoming a public health issue with high incidences reported in community studies.Objective:To evaluate the characteristics and outcome of treatment in men with erectile dysfunction in a tertiary center in Ibadan southwestern Nigeria.Methods:Data of men with erectile dysfunction was retrieved between July 2004 and June 2014 and analyzed using SPSS version 16 statistical software.Results:Eighty-nine men with erectile dysfunction were managed which constituted 2% of all urological cases seen during the study period. Their median and mean ages were 39 years and 39.6 ± 1.2SD (range 19-76 years). The peak age incidence at 30-44 years was 41.6% and reduced with increasing age after 65 years to 4.5%. The etiologies were psychogenic in 55%, organic in 27%, idiopathic in 17% and 1% was familial. 67.5%, 31.5% and 3.4% were married, single and separated respectively. Seventy percent neither smoked cigarette nor drank alcohol, 21.3% drank alcohol and 9% took both alcohol and smoked cigarette. Seventy seven and half percent of men presented within 5 years of their symptom. The treatments offered were PDE type 5 inhibitors alone or in combination with psychotherapy or modification of medications. The outcome of these treatments ranged from 89% to 91% success rate.Conclusion:The number of men with erectile dysfunction managed in the tertiary hospital is very low though the outcome of treatment is within acceptable range. Increase public enlightenment may encourage increase hospital patronage and access to the available treatments for erectile dysfunction.

Derivation of a Sharp-Diffuse Interfaces Model for Seawater Intrusion in a Free Aquifer. Numerical Simulations

We derive a new model for seawater intrusion phenomena in free aquifers. It combines the efficiency of the sharp interface approach with the physical realism of the diffuse interface one. More precisely, a phase field is used as an intermediate variable for including the exchanges between the characteristic layers of the aquifer (salt water, fresh water, unsaturated zone). The three-dimensional problem is reduced to a two-dimensional model involving a strongly coupled system of PDEs of parabolic type describing the evolution of the depths of the two free surfaces. The model is discretized by finite elements in space and by a semi-implicit Euler method in time. Numerical simulations present some physical predictions from the model.

Optimal rate of convergence for stochastic Burgers-type equations

Recently, a solution theory for one-dimensional stochastic PDEs of Burgers type driven by space-time white noise was developed. In particular, it was shown that natural numerical approximations of these equations converge and that their convergence rate in the uniform topology is arbitrarily close to $\frac{1}{6}$. In the present article we improve this result in the case of additive noise by proving that the optimal rate of convergence is arbitrarily close to $\frac{1}{2}$.

Wave propagation in predator–prey systems

In this paper, we study a class of predator–prey systems of reaction–diffusion type. Specifically, we are interested in the dynamical behaviour for the solution with the initial distribution where the prey species is at the level of the carrying capacity, and the density of the predator species has compact support, or exponentially small tails near . Numerical evidence suggests that this will lead to the formation of a pair of diverging waves propagating outwards from the initial zone. Motivated by this phenomenon, we establish the existence of a family of travelling waves with the minimum speed. Unlike the previous studies, we do not use the shooting argument to show this. Instead, we apply an iteration process based on Berestycki et al 2005 (Arch. Rational Mech. Anal. 178 57–80) to construct a set of super/sub-solutions. Since the underlying system does not enjoy the comparison principle, such a set of super/sub-solutions is not based on travelling waves, and in fact the super/sub-solutions depend on each other. With the aid of the set of super/sub-solutions, we can construct the solution of the truncated problem on the finite interval, which, via the limiting argument, can in turn generate the wave solution. There are several advantages to this approach. First, it can remove the technical assumptions on the diffusivities of the species in the existing literature. Second, this approach is of PDE type, and hence it can shed some light on the spreading phenomenon indicated by numerical simulation. In fact, we can compute the spreading speed of the predator species for a class of biologically acceptable initial distributions. Third, this approach might be applied to the study of waves in non-cooperative systems (i.e. a system without a comparison principle).

BSDE, path-dependent PDE and nonlinear Feynman-Kac formula

In this paper, we introduce a type of path-dependent quasilinear (parabolic) partial differential equations in which the (continuous) paths on an interval [0,t] becomes the basic variables in the place of classical variables (t,x). This new type of PDE are formulated through a classical backward stochastic differential equation (BSDEs, for short) in which the terminal values and the generators are allowed to be general functions of Brownian paths. In this way we have established a new type of nonlinear Feynman-Kac formula for a general non-Markovian BSDE. Some main properties of regularities for this new PDE was obtained.

$G$-expectation weighted Sobolev spaces, backward SDE and path dependent PDE

We introduce a new notion of G-expectation-weighted Sobolev spaces, or in short, G-Sobolev spaces, and prove that a backward SDEs driven by G-Brownian motion are in fact path dependent PDEs in the corresponding Sobolev spaces under G-norms. For the linear case of G corresponding the classical Wiener probability space with Wiener measure P, we have established a 1-1 correspondence between BSDE and such new type of quasilinear PDE in the corresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1 correspondence between a fully nonlinear PDE in the corresponding G-Sobolev space and BSDE driven by G-Brownian. Consequently, the existence and uniqueness of such type of fully nonlinear path-dependence PDE in G-Sobolev space have been obtained via a recent results of BSDE driven by G-Brownian motion.

A Fokker-Planck type approximation of parabolic PDEs with oblique boundary data

We consider solutions of quasi-linear parabolic PDEs with zero oblique boundary data in a bounded domain. Our main result states that the solutions can be approximated by solutions of a Fokker-Planck type PDE in the whole space with a penalizing drift term which also converges to zero outside the original domain. The convergence is locally uniform, and optimal error estimates are obtained.

Object memory enhancement by combining sub-efficacious doses of specific phosphodiesterase inhibitors

The second messengers cGMP and cAMP have a vital role in synaptic plasticity and memory processes. As such, phosphodiesterases inhibitors (PDE-Is), which prevent the breakdown of these cyclic nucleotides, represent a potential treatment strategy in memory decline. Recently it has been demonstrated that cGMP and cAMP signaling act in sequence during memory consolidation, with early cGMP signaling requiring subsequent cAMP signaling. Here, we sought to confirm this relationship, and to evaluate its therapeutic implications. Combining sub-efficacious doses of the cGMP-specific PDE type 5 inhibitor vardenafil (0.1 mg/kg) and cAMP-specific PDE type 4 inhibitor rolipram (0.01 mg/kg) during the early and late memory consolidation phase, respectively, led to improved memory performance in a 24 h interval object recognition task. Similarly, such a sub-efficacious combination treatment enhanced the transition of early-phase long-term potentiation (LTP) to late-phase LTP in hippocampal slices. In addition, both object memory and LTP were improved after administration of two sub-efficacious doses of the dual substrate PDE type 2 inhibitor BAY60 7550 (0.3 mg/kg) at the early and late consolidation phase, respectively. Taken together, combinations of sub-efficacious doses of cAMP- and cGMP-specific PDE-Is have an additive effect on long-term synaptic plasticity and memory formation and might prove a superior alternative to single PDE-I treatment.