Homogeneous Dirichlet Research Articles

Analytical solutions of the nitrogen uptake model with Michaelis-Menten flux

In this paper, we describe the application of the generalized finite Hankel transform method to the convection diffusion problems of nitrogen in soil subject to Robin boundary conditions and the right-hand side of the left Robin boundary condition is the classic nonlinear Michaelis-Menten flux. If the Michaelis-Menten flux is taken as a function of time, the analytical solution of series form can be derived by the generalized finite Hankel transform. In the calculation, the Michaelis-Menten flux is evaluated by the numerical concentration at the root surface and the number of terms of the partial sum is determined after adequately approximating the numerical solution. Therefore, we improve and develop the analytical method built by Garg et al., and exemplify and verify it by the nitrogen uptake model for roots. The investigated model has a representatively and completely mathematical structure among most nutrient uptake models, and then the analytical method and the solutions developed here can also be used to nutrient uptake problems in soil subject to homogeneous or inhomogeneous Dirichlet or Neumann boundary conditions.

Attractors of 2D Navier–Stokes system of equations in a locally periodic porous medium

This article deals with two-dimensional Navier–Stokes system of equations with rapidly oscillating term in the equations and boundary conditions. Studying the problem in a perforated domain, the authors set homogeneous Dirichlet condition on the outer boundary and the Fourier (Robin) condition on the boundary of the cavities. Under such assumptions it is proved that the trajectory attractors of this system converge in some weak topology to trajectory attractors of the homogenized Navier–Stokes system of equations with an additional potential and nontrivial right hand side in the domain without pores. For this aim, the approaches from the works of A.V. Babin, V.V. Chepyzhov, J.-L. Lions, R. Temam, M.I. Vishik concerning trajectory attractors of evolution equations and homogenization methods appeared at the end of the XX-th century are used. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) system of equations and prove the existence of trajectory attractors for this system. Lastly, we formulate the main theorem and prove it through axillary lemmas.

Small perturbations in the type of boundary conditions for an elliptic operator

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this “background” situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a “small” subset ωε of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a “small” subset ωε of the Dirichlet boundary. The relevant quantity that measures the “smallness” of the subset ωε differs in the two cases: while it is the harmonic capacity of ωε in the former case, we introduce a notion of “Neumann capacity” to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general ωε, under the sole assumption that it is “small” in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset ωε is a vanishing surfacic ball.

On solutions for a generalized Navier-Stokes-Fourier system fulfilling the entropy equality.

We consider a flow of a non-Newtonian heat conducting incompressible fluid in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field and the Dirichlet boundary condition for the temperature. In three dimensions, for a power-law index greater or equal to , we show the existence of a solution fulfilling the entropy equality. The entropy equality can be formally deduced from the energy equality by renormalization. However, such a procedure can be justified by the DiPerna–Lions theory only for . The main novelty is that we do not renormalize the temperature equation, but rather construct a solution which fulfils the entropy equality.This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Upper bound on the number of determining modes for the 2D g-bénard problem

The "determining modes" concept introduced by Foias and Prodi in 1967 say that if two solutions agree asymptotically in their P projection, then they are asymptotical in their entirety. In this paper, we consider the 2D g-Bénard problem in domains satisfying the Poincaré inequality with homogeneous Dirichlet boundary conditions. We present an improved upper bound on the number of determining modes. Moreover, we slightly improve the estimate on the number of determining modes and obtain an upper bound of the order G. These estimates are in agreement with the heuristic estimates based on physical arguments, that have been conjectured by O.P. Manley and Y.M. Treve. The Gronwall lemma and Poincaré type inequality will play a central role in our computational technique as well as the proof of the main result of the paper. Studying the properties of solutions is important to determine the behavior of solutions over a long period of time. The obtained results particularly extend previous results for 2D g-Navier-Stokes equations and 2D Bénard problem.

The Stokes operator in two-dimensional bounded Lipschitz domains

We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain Ω subject to homogeneous Dirichlet boundary conditions. We prove Lp-resolvent estimates for p satisfying the condition |1/p−1/2|<1/4+ε for some ε>0. We further show that the Stokes operator admits the property of maximal regularity and that its H∞-calculus is bounded. This is then used to characterize domains of fractional powers of the Stokes operator. Finally, we give an application to the regularity theory of weak solutions to the Navier–Stokes equations in bounded planar Lipschitz domains.

An application of spectral localization to the critical SQG on a ball

We study the Cauchy problem for the quasi-geostrophic equations in a unit ball of the two-dimensional space with the homogeneous Dirichlet boundary condition. We show existence and uniqueness of the strong solution in the framework of Besov spaces. We also establish a spectral localization technique and commutator estimates.

An A Posteriori Error Estimator for a Non Homogeneous Dirichlet Problem Considering a Dual Mixed Formulation

In this paper, we describe an a posteriori error analysis for a conforming dual mixed scheme of the Poisson problem with non homogeneous Dirichlet boundary condition. As a result, we obtain an a posteriori error estimator, which is proven to be reliable and locally efficient with respect to the usual norm on H(div;Omega) x L^2(Omega). We remark that the analysis relies on the standard Ritz projection of the error, and take into account a kind of a quasi-Helmholtz decomposition of functions in H(div;Omega), which we have established in this work. Finally, we present one numerical example that validates the well behavior of our estimator, being able to identify the numerical singularities when they exist.

Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases

When one solves differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients an to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, u(±1) = 0, popular choices include the “Chebyshev difference basis” ςn(x) ≡ Tn+2(x)−Tn(x) with coefficients here denoted by bn and the “quadratic factor basis” ϱn(x) ≡ (1 − x2)Tn(x) with coefficients cn. If u(x) is weakly singular at the boundary, then the coefficients an decrease proportionally to \({\cal O}\left( {A\left( n \right)/{n^\kappa }} \right)\) for some positive constant κ, where A(n) is a logarithm or a constant. We prove that the Chebyshev difference coefficients bn decrease more slowly by a factor of 1/n while the quadratic factor coefficients cn decrease more slowly still as \({\cal O}\left( {A\left( n \right)/{n^{\kappa - 2}}} \right)\). The error for the unconstrained Chebyshev series, truncated at degree n = N, is \({\cal O}\left( {\left| {A\left( N \right)} \right|/{N^\kappa }} \right)\) in the interior, but is worse by one power of N in narrow boundary layers near each of the endpoints. Despite having nearly identical error norms in interpolation, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x. Meanwhile, for Chebyshev polynomials, the values of their derivatives at the endpoints are \({\cal O}\left( {{N^2}} \right)\), but only \({\cal O}\left( N \right)\) for the difference basis. Furthermore, we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases, solved by the least squares method. We also find an interesting fact that on the face of it, the aliasing error is regarded as a bad thing; actually, the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation. But the premise is under the same basis, and when involving different bases, it may not be established yet.

An Efficient Dissipation-Preserving Numerical Scheme to Solve a Caputo–Riesz Time-Space-Fractional Nonlinear Wave Equation

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results.

Positive solutions for a class of nonlocal problems with possibly singular nonlinearity

We study a class of elliptic problems with homogeneous Dirichlet boundary condition and a nonlinear reaction term f which is nonlocal depending on the \(L^{p}\)-norm of the unknown function. The nonlinearity f can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map, to produce multiplicity of classical positive solutions with ordered norms.

ON STRONGLY QUASILINEAR DEGENERATE ELLIPTIC SYSTEMS WITH WEAK MONOTONICITY AND NONLINEAR PHYSICAL DATA

This work is devoted to studying the quasilinear elliptic system $$\begin{aligned} -div ~ a(x,u,Du) + \vert u\vert ^{p-2} u +b(x,u,Du)=v(x)+f(x,u)+div ~g(x,u) \end{aligned}$$on a bounded open domain of \(\mathbb {R}^n\) with homogeneous Dirichlet boundary conditions. We show that there is a weak solution to this system under regularity, growth, and coercivity conditions for a, but only with very moderate monotonicity assumptions. We prove the existence result by using Galerkin’s approximation and the theory of Young measures.

Solving the heat equation with variable thermal conductivity

We consider the heat equation with spatially variable thermal conductivity and homogeneous Dirichlet boundary conditions. Using the Method of Fokas or Unified Transform Method, we derive solution representations as the limit of solutions of constant-coefficient interface problems where the number of subdomains and interfaces becomes unbounded. This produces an explicit representation of the solution, from which we can compute the solution and determine its properties. Using this solution expression, we can find the eigenvalues of the corresponding variable-coefficient eigenvalue problem as roots of a transcendental function. We can write the eigenfunctions explicitly in terms of the eigenvalues. The heat equation is the first example of more general variable-coefficient second-order initial–boundary value problems that can be solved using this approach.

Well-Posedness for Nonlinear Parabolic Stochastic Differential Equations with Nonlinear Robin Conditions

In this paper, we present the existence and uniqueness of strong probabilistic solutions for nonlinear parabolic Stochastic Partial Differential Equations (SPDEs) with nonlinear Robin boundary conditions in a domain with holes. On the boundary of the holes, a nonlinear Robin condition is imposed, while a homogeneous Dirichlet condition is prescribed on the exterior boundary. The coefficient matrix is assumed to be symmetric, while the nonlinear random forces are assumed to satisfy some types of regularities. We use Galerkin’s approximation method, probabilistic compactness results and some results from stochastic calculus.

Uniqueness for fractional nonsymmetric diffusion equations and an application to an inverse source problem

In this article, we discuss a solution to time‐fractional diffusion equation with the homogeneous Dirichlet boundary condition, where an elliptic operator is not necessarily symmetric. We prove that the solution is identically zero if its normal derivative with respect to the operator vanishes on an arbitrarily chosen subboundary of the spatial domain over a time interval. The proof is based on the Laplace transform and the spectral decomposition for a nonsymmetric elliptic operator. As a direct application, we prove the uniqueness result for an inverse problem on determining the spatial component in the source term by Neumann boundary data on subdoundary.

Linear regularized finite difference scheme for the quasilinear subdiffusion equation

Abstract A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with a time-fractional derivative and coefficients at the elliptic part that depend on the gradient of the solution is considered. Conditions on the coefficients ensure the monotonicity and Lipschitz property of the elliptic operator on the set of functions whose gradients in space variables are uniformly bounded. For this problem, a linear regularized mesh scheme is constructed and investigated. A sufficient condition is derived for the regularization parameter that ensures the so-called local correctness of the mesh scheme. On the basis of correctness and approximation estimates for model problems with time-fractional Caputo or Caputo–Fabrizio derivatives, accuracy estimates are given in terms of mesh and regularization parameters under the assumption of the existence of a smooth solution to the differential problem. The presented results of the numerical experiments confirm the obtained asymptotic accuracy estimates.

Weakly imposed Dirichlet boundary conditions for 2D and 3D Virtual Elements

In the framework of virtual element discretizations, we address the problem of imposing non homogeneous Dirichlet boundary conditions in a weak form, both on polygonal/polyhedral domains and on two/three dimensional domains with curved boundaries. We consider a Nitsche’s type method (Nitsche et al. 1970; Juntunen et al. 2009) and the stabilized formulation of the Lagrange multiplier method proposed by Barbosa and Hughes in Barbosa and Hughes (1991) . We prove that also for the virtual element method (VEM), provided the stabilization parameter is suitably chosen (large enough for Nitsche’s method and small enough for the Barbosa–Hughes Lagrange multiplier method), the resulting discrete problem is well posed, and yields convergence with optimal order on polygonal/polyhedral domains. On smooth two/three dimensional domains, we combine both methods with a projection approach similar to the one of Bramble et al. (1972). We prove that, given a polygonal/polyhedral approximation Ωh of the domain Ω, an optimal convergence rate can be achieved by using a suitable correction depending on high order derivatives of the discrete solution along outward directions (not necessarily orthogonal) at the boundary facets of Ωh. Numerical experiments validate the theory.

Mountain pass solutions to equations with subcritical Musielak-Orlicz-Sobolev growth

In this paper, we study the existence of solutions for equations driven by a new subcritical Musielak-Orlicz-Sobolev growth with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a non-trivial solution for them using the Mountain Pass Theorem.

Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

Using symmetrization techniques, we show that, for every N ≥ 2 N \geq 2 , any second eigenfunction of the fractional Laplacian in the N N -dimensional unit ball with homogeneous Dirichlet conditions is nonradial, and hence its nodal set is an equatorial section of the ball.

Monotonicity properties for the variational Dirichlet eigenvalues of the p-Laplace operator

Let D≥2 be an integer. For each open and bounded set Ω⊂RD and each integer k≥1 we denote by Rk(Ω) the largest number r>0 for which there exists k disjoint open balls in Ω of radius r. Next, for each open, bounded, convex set Ω⊂RD with smooth boundary and each real number p∈(1,∞) we denote by {λk(p;Ω)}k≥1 the sequence of eigenvalues of the p-Laplace operator subject to the homogeneous Dirichlet boundary conditions, given by the Ljusternik-Schnirelman theory. For each integer k≥1 we show that there exists Mk∈[(ke)−1,1] such that for any open, bounded, convex set Ω⊂RD with smooth boundary for which Rk(Ω) is less than or equal to Mk, the k-th eigenvalue of the p-Laplacian on Ω, λk(p;Ω), is an increasing function of p on (1,∞). Moreover, there exists Nk≥Mk such that for any real number s∈(Nk,∞)∖{1} there exists an open, bounded, convex set Ω⊂RD with smooth boundary which has Rk(Ω) equal to s such that λk(p;Ω) is not a monotone function of p∈(1,∞).