Neumann Value Research Articles

Exact Green's formula for the fractional Laplacian and perturbations

Let Ω be an open, smooth, bounded subset of $ \mathbb{R}^n $. In connection with the fractional Laplacian $(-\Delta )^a$ ($a>0$), and more generally for a $2a$-order classical pseudodifferential operator (ψdo) $P$ with even symbol, one can define the Dirichlet value $\gamma _0^{a-1}u$, resp. Neumann value $\gamma _1^{a-1}u$ of $u(x)$, as the trace, resp. normal derivative, of $u/d^{a-1}$ on $\partial \Omega $, where $d(x)$ is the distance from $x\in \Omega $ to $\partial \Omega $; they define well-posed boundary value problems for $P$.
 A Green's formula was shown in a preceding paper, containing a generally nonlocal term $(B\gamma _0^{a-1}u,\gamma _0^{a-1}v)_{\partial \Omega }$, where $B$ is a first-order ψdo on $\partial \Omega $. Presently, we determine $B$ from $L$ in the case $P=L^a$, where $L$ is a strongly elliptic second-order differential operator. A particular result is that $B=0$ when $L=-\Delta $, and that $B$ is multiplication by a function (is local) when $L$ equals $-\Delta $ plus a first-order term. In cases of more general $L$, $B$ can be nonlocal.

Admissible Boundary Values for the Gerdjikov-Ivanov Equation with Asymptotically Time-Periodic Boundary Data

We consider the Gerdjikov-Ivanov equation in the quarter plane with Dirichlet boundary data and Neumann value converging to single exponentials $\alpha {\rm e}^{{\rm i}\omega t}$ and $c{\rm e}^{{\rm i}\omega t}$ as $t\to\infty$, respectively. Under the assumption that the initial data decay as $x\to\infty$, we derive necessary conditions on the parameters $\alpha$, $\omega$, $c$ for the existence of a solution of the corresponding initial boundary value problem.

On the analytical solution of the two-phase Couette flow with wall transpiration

The two-phase Couette flow with transpiration through both walls is considered, where there is a constant blowing v0 at the lower wall and a corresponding suction at the upper wall. The interface between both fluids is initially flat and, hence, stays flat as it moves upward at the constant speed of the transpiration velocity v0. The corresponding initial value problem is subject to three dimensionless numbers consisting of the Reynolds number Re and the viscosity and density ratios, ϵ and γ. The solution is obtained by the unified transform method (Fokas method) in the form of an integral representation depending on initial and all boundary values including the Dirichlet and Neumann values at the interface. The unknown values at the moving interface are determined by a system of linear Volterra integral equations (VIEs). The VIEs are of the second kind with continuous and bounded kernels. Hence, the entire two-phase spatiotemporal 1 + 1 system has dimensionally reduced. The system of VIEs is solved via a standard marching method. For the numerical computation of the complex integral contours, a parameterized hyperbola is used. The influence of the dimensionless numbers Re, γ, and ϵ is studied exemplarily. The most notable effect results from ϵ that gives rise to a kink in the velocity at the moving interface. Both ratios, ϵ and γ, allow for very different flow regimes in each fluid phase such as nearly pure Couette flows and transpiration dominated flows with strongly curved velocity profiles. Those regimes are mainly determined by the effective Reynolds number in the respective phases.

A parallel unified transform solver based on domain decomposition for solving linear elliptic PDEs

A hybrid approach for the solution of linear elliptic PDEs, based on the unified transform method in conjunction with domain decomposition techniques, is introduced. Given a well-posed boundary value problem, the proposed methodology relies on the derivation of an approximate global relation, which is an equation that couples the finite Fourier transforms of all the boundary values. The computational domain is hierarchically decomposed into several nonoverlapping subdomains; for each of those subdomains, a unique approximate global relation is derived. Then, by introducing a modified Dirichlet-to-Neumann iterative algorithm, it is possible to compute the solution and its normal derivative at the resulting interfaces. By considering several hierarchical levels, higher spatial resolution can be achieved. There are three main advantages associated with the proposed approach. First, since the unified transform is a boundary-based technique, the interior of each subdomain does not need to be discretized; thus, no mesh generation is required. Additionally, the Dirichlet and Neumann values can be computed on the interfaces with high accuracy, using a collocation technique in the complex Fourier plane. Finally, the interface values at each hierarchical level can be computed in parallel by considering a quadtree decomposition in conjunction with the iterative Dirichlet-to-Neumann algorithm. The proposed methodology is analysed both regarding implementation details and computational complexity. Moreover, numerical results are presented, assessing the performance of the solver.

An iterative spatial-stepping numerical method for linear elliptic PDEs using the Unified Transform

A method for solving boundary value problems usually referred to as the unified transform or the Fokas method was introduced in the late nineties. A crucial role in this method is played by the so-called global relation, which characterizes the associated generalized Dirichlet-to-Neumann map, which can be used to determine the unknown boundary values in terms of the given boundary data. Then, the solution can be expressed via an integral formulated in the complex Fourier plane. This method can be considered as the spectral analogue of the classical Boundary Integral Method, which is formulated in the physical plane. Recently, it has been realized that the numerical implementation of the unified transform leads to a collocation-type method: this involves expanding the unknown boundary values in terms of appropriate basis functions and choosing a suitable set of complex collocation points. Herewith, an iterative spatial-stepping algorithm for computing the solution of a linear PDE in the interior of a convex polygon, is presented. The starting point of this algorithm is the analysis of the approximate global relation and its numerical solution. In more details, the interior of the polygon is partitioned into smaller concentric polygonal domains where the solution is computed recursively starting from the boundaries and proceeding towards the center. The solution of each inner polygon is computed by a spatial-stepping scheme using the Dirichlet and Neumann values of the respective outer polygon, which are computed via the approximate global relation.

An iterative method for the Cauchy problem for second-order elliptic equations

The problem of reconstructing the solution to a second-order elliptic equation in a doubly-connected domain from knowledge of the solution and its normal derivative on the outer part of the boundary of the solution domain, that is from Cauchy data, is considered. An iterative method is given to generate a stable numerical approximation to this inverse ill-posed problem. The procedure is physically feasible in that boundary data is updated with data of the same type in the iterations, meaning that Dirichlet values is updated with Dirichlet values from the previous step and Neumann values by Neumann data. Proof of convergence and stability are given by showing that the proposed method is an extension of the Landweber method for an operator equation reformulation of the Cauchy problem. Connection with the alternating method is discussed. Numerical examples are included confirming the feasibility of the suggested approach.

A class of unified transform techniques for solving linear elliptic PDEs in convex polygons

A class of numerical techniques is introduced for computing the solution and its normal derivative for two-dimensional, linear, elliptic partial differential equations (PDEs) in the interior of convex polygonal domains. The cornerstone of this approach is the Fokas approximate global relation, i.e. a discretized equation on the complex k-plane that couples the integral transforms of the given and the unknown boundary data, derived by analyzing the corresponding boundary value problem (BVP) with the Fokas unified transform method. The proposed techniques rely on the partitioning of a given polygonal computational domain into a number of concentric, polygonal subdomains. On each subdomain a global relation holds, and furthermore, the Dirichlet and Neumann values between consecutive polygons are related via prescribed algebraic relations. By assembling all the available equations into a linear system, it is possible to compute the solution as well as its normal derivative over the whole domain by solving a number of smaller, one-dimensional BVPs simultaneously. The advantage of the proposed methodology is the simultaneous approximation of both the solution and the normal derivative over the entire of a general polygonal domain with satisfactory accuracy, by solving one sparse linear system. In addition, our suggested approach provides a framework for designing different numerical schemes with prescribed order of accuracy, whereas it is possible to implement adaptive techniques in terms of the mesh size and the order of the expansion polynomials. Here, we present the details of the proposed methodology and provide numerical results for solving linear elliptic PDEs on convex polygons.

Linear Elliptic PDEs in a Cylindrical Domain with a Polygonal Cross‐Section

Integral representations for the solution of the Laplace, modified Helmholtz, and Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well‐posed boundary value problem (BVP) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs), usually referred to as the unified transform or the Fokas method, was introduced in the late 1990s. For linear elliptic PDEs in two dimensions, this method first, by employing two algebraic equations formulated in the Fourier plane, provides an elegant approach for determining the Dirichlet to Neumann map, i.e., for constructing the unknown boundary values in terms of the given boundary data. Second, this method constructs novel integral representations of the solution in terms of integrals formulated in the complex Fourier plane. In the present paper, we extend this novel approach to the case of the Laplace, modified Helmholtz, and Helmholtz equations, formulated in a three‐dimensional cylindrical domain with a polygonal cross‐section.

Admissible boundary values for the defocusing nonlinear Schrödinger equation with asymptotically time-periodic data

We consider solutions of the defocusing nonlinear Schrödinger equation in the quarter plane whose Dirichlet boundary data approach a single exponential αeiωt as t→∞. In order to determine the long time asymptotics of the solution, it is necessary to first characterize the asymptotic behavior of the Neumann value in terms of the given data. Assuming that the initial data decay as x→∞, we derive necessary conditions for the Neumann value to asymptote towards a single exponential of the form ceiωt. Since our approach yields expressions which relate α, ω, and c, the result can be viewed as a characterization of the large t behavior of the Dirichlet to Neumann map for single exponential profiles.

Boundedness and asymptotic behavior of solutions to a chemotaxis–haptotaxis model in high dimensions

We consider the Neumann value problem for the chemotaxis system { u t = ∇ ⋅ ( ∇ u − u ( α 1 + v ∇ v + ρ ∇ w ) ) + λ u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v − v − μ u v , x ∈ Ω , t > 0 , w t = γ u ( 1 − w ) , x ∈ Ω , t > 0 , in a bounded domain Ω ⊂ R n ( n ≥ 1 ) with smooth boundary, where α , ρ , λ , μ and γ are positive coefficients. It is shown that for any choice of reasonably regular initial data ( u 0 , v 0 , w 0 ) , there exists a constant λ ∗ depending on α , ρ , μ , γ , n , v 0 and w 0 such that for any λ > λ ∗ , the associated initial–boundary system possesses a global classical solution which is uniformly bounded. Moreover, building on this boundedness property, it is proved that as time tends to infinity, all the solution approaches the homogeneous steady state ( 1 , 0 , 1 ) in an appropriate sense.

The defocusing nonlinear Schrödinger equation with [formula omitted]-periodic data: New exact solutions

We consider solutions of the defocusing nonlinear Schrödinger (NLS) equation on the half-line whose Dirichlet and Neumann boundary values become periodic for sufficiently large t. We prove a theorem which, modulo certain assumptions, characterizes the pairs of periodic functions which can arise as Dirichlet and Neumann values for large t in this way. The theorem also provides a constructive way of determining explicit solutions with the given periodic boundary values. Hence our approach leads to a class of new exact solutions of the defocusing NLS equation on the half-line.

A numerical technique for linear elliptic partial differential equations in polygonal domains.

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

A Perturbative Approach for the Asymptotic Evaluation of the Neumann Value Corresponding to the Dirichlet Datum of a Single Periodic Exponential for the NLS

Boundary value problems for the nonlinear Schrödinger equation formulated on the half-line can be analyzed by the Fokas method. For the Dirichlet problem, the most difficult step of this method is the characterization of the unknown Neumann boundary value. For the case that the Dirichlet datum consists of a single periodic exponential, namely, a exp(iωt), a, ω real, it has been shown in [2–4] that if one assumes that the Neumann boundary value is given for large t by c exp(iωt), then c can be computed explicitly in terms of a and ω. Here, using the perturbative approach introduced in [16], it is shown that for typical initial conditions, it is indeed the case that at least up to third order in a perturbative expansion the Neumann boundary value is given by c exp(iωt) and the value of c is at least up to this order the value found in [2–4].

Perturbative and Exact Results on the Neumann Value for the Nonlinear Schrödinger on the Half-line

The most challenging problem in the implementation of the so-called unified transform to the analysis of the nonlinear Schrödinger equation on the half-line is the characterization of the unknown boundary value in terms of the given initial and boundary conditions. For the so-called linearizable boundary conditions this problem can be solved explicitly. Furthermore, for non-linearizable boundary conditions which decay for large t, this problem can be largely bypassed in the sense that the unified transform yields useful asymptotic information for the large t behavior of the solution. However, for the physically important case of periodic boundary conditions it is necessary to characterize the unknown boundary value. Here, we first present a perturbative scheme which can be used to compute explicitly the asymptotic form of the Neumann boundary value in terms of the given τ-periodic Dirichlet datum to any given order in a perturbation expansion. We then discuss briefly an extension of the pioneering results of Boutet de Monvel and co-authors which suggests that if the Dirichlet datum belongs to a large class of particular τ-periodic functions, which includes {a exp(iωt)|a > 0, ω ≥ a2}, then the large t behavior of the Neumann value is given by a τ-periodic function which can be computed explicitly.

Solving the geodesics on the ellipsoid as a boundary value problem

The geodesic between two given points on an ellipsoid is determined as a numerical solution of a boundary value problem. The secondorder ordinary differential equation of the geodesic is formulated by means of the Euler-Lagrange equation of the calculus of variations. Using Taylor’s theorem, the boundary value problem with Dirichlet conditions at the end points is replaced by an initial value problem with Dirichlet and Neumann conditions. The Neumann condition is determined iteratively by solving a system of four first-order differential equations with numerical integration. Once the correct Neumann value has been computed, the solution of the boundary value problem is also obtained. Using a special case of the Euler-Lagrange equation, the Clairaut equation is verified and the Clairaut constant is precisely determined. The azimuth at any point along the geodesic is computed by a simple formula. The geodesic distance between two points, as a definite integral, is computed by numerical integration. The numerical tests are validated by comparison to Vincenty’s inverse formulas.

A SOBOLEV SPACE APPROACH TO BOUNDARY VALUE PROBLEMS ON THE HALF-LINE

This paper addresses the existence of solutions u ∈ H1(ℝ+;ℝN) of ODE systems [Formula: see text], with boundary condition u1(0) = ξ, where u1is a (vector) component of u. Under general conditions, the problem corresponds to a functional equation involving a Fredholm operator with calculable index, which is proper on the closed bounded subsets of H1(ℝ+;ℝN). When the index is 0 and the solutions are bounded a priori, the existence follows from an available degree theory for such operators. Specific conditions are given that guarantee the existence of a priori bounds and second order equations with Dirichlet, Neumann or initial value conditions are discussed as applications.

Entropy Methods for Adaptive Utility Elicitation

This paper presents an optimal question-selection algorithm to elicit von Neumann and Morgenstern utility values for a set of ordered prospects of a decision situation. The approach uses information theory and entropy-coding principles to select the minimum expected number of questions needed for utility elicitation. At each stage of the questionnaire, we use the question that will provide the largest reduction in the entropy of the joint distribution of the utility values. The algorithm uses questions that require binary responses, which are easier to provide than numeric values, and uses an adaptive question-selection scheme where each new question depends on the previous response obtained from the decision maker. We present a geometric interpretation for utility elicitation and work through a full example to illustrate the approach.

Spatial critical points not moving along the heat flow II: The centrosymmetric case

We consider solutions of initial-boundary value problems for the heat equation on bounded domains in $\mathbb{R}^N,$ and their spatial critical points as in the previous paper [MS]. In Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, it is proved that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data being centrosymmetric with respect to the origin, then the domain must be centrosymmetric with respect to the origin. Furthermore, we consider spatial zero points instead of spatial critical points, and prove some similar symmetry theorems. Also, it is proved that these symmetry theorems hold for initial-boundary value problems for the wave equation.

The spatial critical points not moving along the heat flow

We consider solutions of the heat equation, in domains inRN, and their spatial critical points. In particular, we show that a solutionu has a spatial critical point not moving along the heat flow if and only ifu satisfies some balance law. Furthermore, in the case of Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, we prove that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data satisfying the balance law with respect to the origin, then the domain must be a ball centered at the origin.

Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces

We obtain here some inequalities for the eigenvalues of Dirichlet and Neumann value problems for general classes of operators (or system of operators) acting inL2(Ω) (orL2(Ω,Cm)),Ω⊂Rd,d⩾1.