Solution Of Dirichlet Research Articles

A plane-wave singularity subtraction technique for the classical Dirichlet and Neumann combined field integral equations

This paper presents expressions for the classical combined field integral equations for the solution of Dirichlet and Neumann exterior Helmholtz problems on the plane, in terms of smooth (continuously differentiable) integrands. These expressions are obtained by means of a singularity subtraction technique based on pointwise plane-wave expansions of the unknown density function. In particular, a novel regularization of the hypersingular operator is obtained, which, unlike regularizations based on Maue's integration-by-parts formula, does not give rise to involved Cauchy principal value integrals. Moreover, the expressions for the combined field integral operators and layer potentials presented in this contribution can be numerically evaluated at target points that are arbitrarily close to the boundary without severely compromising their accuracy. A variety of numerical examples in two spatial dimensions that consider three different Nyström discretizations for smooth domains and domains with corners—one of which is based on direct application of the trapezoidal rule—demonstrates the effectiveness of the proposed higher-order singularity subtraction approach.

Radial solutions to the Cauchy problem for □1+3U = 0 as limits of exterior solutions

We consider the strategy of realizing the solution of a Cauchy problem (CP) with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the three-dimensional (3d), linear wave equation [Formula: see text] with radial Cauchy data [Formula: see text], [Formula: see text]. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric CP. In treating the 3d wave equation, we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in [Formula: see text] whenever [Formula: see text]. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in [Formula: see text], we obtain [Formula: see text]-solutions via exterior Neumann solutions, and only [Formula: see text]-solutions via exterior Dirichlet solutions.

Existence and Multiplicity of Periodic Solutions for Dirichlet–Neumann Boundary Value Problem of a Variable Coefficient Wave Equation

Abstract In this paper, we consider the periodic solutions of a variable coefficient wave equation which models the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. Under Dirichlet–Neumann boundary conditions, we find some important properties for the variable coefficient wave operator. Then, based on these properties, we obtain the existence and multiplicity of periodic solutions by using the Leray–Schauder degree theory.

The plane waves method for axisymmetric Helmholtz problems

The plane waves method is employed for the solution of Dirichlet and Neumann boundary value problems for the homogeneous Helmholtz equation in two- and three-dimensional domains possessing radial symmetry. The appropriate selection of collocation points and unitary direction vectors in the method leads to circulant and block circulant coefficient matrices in two and three dimensions, respectively. We propose efficient matrix decomposition algorithms which make use of fast Fourier transforms for the solution of the systems resulting from such a discretization. In conjunction with the method of particular solutions, the method is extended to the solution of inhomogeneous axisymmetric Helmholtz problems. Several numerical examples are presented.

On the Existence and Multiplicity of Solutions for Dirichlet’s problem for Fractional Differential equations

Abstract In this paper, by using variational methods and critical point theorems, we prove the existence and multiplicity of solutions for boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. Our results extend the second order boundary value problem to the non integer case. Moreover, some conditions to determinate nonnegative solutions are presented and examples are given to illustrate our results.

THERMIC MINORANTS AND REDUCTIONS OF SUPERTEMPERATURES

Let $u$ be a supertemperature on an open set $E$, and let $v$ be a related temperature on an open subset $D$ of $E$. For example, $v$ could be the greatest thermic minorant of $u$ on $D$, if it exists. Putting $w=u$ on $E\setminus D$ and $w=v$ on $D$, we investigate whether $w$, or its lower semicontinuous smoothing, is a supertemperature on $E$. We also give a representation of the greatest thermic minorant on $E$, if it exists, in terms of PWB solutions on an expanding sequence of open subsets of $E$ with union $E$. In addition, in the case of a nonnegative supertemperature, we prove inequalities that relate reductions to Dirichlet solutions. We also prove that the value of any reduction at a given time depends only on earlier times.

Local and nonlocal boundary conditions forμ-transmission and fractional elliptic pseudodifferential operators

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega $. As shown recently by the author, the condition assures solvability of Dirichlet-type boundary problems for elliptic $P$ in full scales of Sobolev spaces with a singularity $d^{\mu -k}$, $d(x)=\operatorname{dist}(x,\partial\Omega)$. Examples include fractional Laplacians $(-\Delta)^a$ and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type or more general nonlocal. It is also shown how problems with data on $R^n\setminus \Omega $ reduce to problems supported on $\bar\Omega$, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces $F^s_{p,q}$ and $B^s_{p,q}$, including H\"older-Zygmund spaces $B^s_{\infty ,\infty}$. This leads to optimal H\"older estimates, e.g. for Dirichlet solutions of $(-\Delta)^au=f\in L_\infty (\Omega)$, $u\in d^aC^a(\bar\Omega)$ when $0<a<1$, $a\ne 1/2$ (in $d^aC^{a-\epsilon}(\bar\Omega)$ when $a=1/2$).

Analysis of a mixed space-time diffusion equation

An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank–Nicolson scheme. The Crank–Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.

О новом подходе в методе функций Грина при решении краевых задач Дирихле и Неймана для уравнения Лапласа

О новом подходе в методе функций Грина при решении краевых задач Дирихле и Неймана для уравнения Лапласа

Generalized Strategic Dual Image Method for Open Boundary Axisymmetrical Magnetic Field Problems

During the early 1990s, the strategic dual image (SDI) method for axisymmetric open boundary magnetic field problems was proposed. Although the method is powerful, the specific axis ratio of the ellipsoidal boundary has not been clarified, and therefore no further research has been done so far. Our aims are to clarify the relationship between the specific axis ratio of the ellipsoidal boundary and the harmonic solutions of Laplace equation, and to extend the strategic dual image method in general form. We have investigated the harmonic solutions of the Laplace equation in the oblate spheroidal coordinate to derive the explicit formula regarding the axis ratio and the averaging factor of Dirichlet and Neumann boundary value problems. Numerical analyses have also been carried out to verify the formula. We have determined the explicit formula regarding the axis ratio and averaging factor of Dirichlet and Neumann solutions, thus extending the SDI method in general form. Utilizing the derived formula, one can obtain the open boundary solutions of axisymmetrical magnetic field problems without alternating the existing software which are widely distributed through the Internet.

A spectrally accurate numerical implementation of the Fokas transform method for Helmholtz-type PDEs

A novel method for solving boundary value problems for linear partial differential equations in convex polygons was developed by A.S. Fokas in the late 1990s. A spectrally accurate numerical implementation for the case of the Laplace equation led the way for the present implementation for both the Helmholtz equation and the modified Helmholtz equation. We obtain again spectrally accurate numerical solutions for Dirichlet to Neumann maps, without any discretization of the domain interior. The eigenvalue problem is also examined using the present Legendre expansion method. A key feature of the current approach is that all integrals that arise in the map problem can be evaluated in closed form, reducing it to the solution of a small overdetermined linear system of equations.

A method for deriving hypergeometric and related identities from the H 2 Hardy norm of conformal maps

We explore a method which is implicit in a paper of Burkholder of identifying the H 2 Hardy norm of a conformal map with the explicit solution of Dirichlet's problem in the complex plane. Using the series form of the Hardy norm, we obtain an identity for the sum of a series obtained from the conformal map. We use this technique to evaluate several hypergeometric sums, as well as several sums that can be expressed as convolutions of the terms in a hypergeometric series. The most easily stated of the identities we obtain are Euler's famous Basel sum, as well as the sum We will be able to obtain the following hypergeometric reduction: A related identity is We will obtain two families of identities depending on a parameter, representative examples of which are and where C(k) is the kth Catalan number. We will also sum two series whose terms are defined by certain recurrence relations, and discuss an extension of the method to maps which are not conformal.

On the Applications of Cyclotomic Fields in Introductory Number Theory

In this essay, we study and comment on two number theoretical applications on prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime). We begin by giving a simplified proof of Kummer's case of Fermat's Last Theorem obtained by linking different versions of the proof in different textbooks. We finally modernize Dirichlet's solution to Pell's Equation.

Asymptotic solutions for turbulent mass transfer augmented by a first order chemical reaction

We present asymptotic solutions for turbulent mass transfer in a smooth conduit at high Schmidt number in the presence of a first-order chemical reaction in the fluid. Far-field solutions are derived for a case dominated by (1) mass transfer at the wall and (2) the first-order chemical reaction. An approximate solution is derived for the regime where both are important. The solutions are in good agreement with numerical solutions and with the literature. At high Damköhler numbers the system is governed by a reaction–diffusion equation and the observed increase in mass transfer coefficient is caused by thinning of the mass transfer boundary layer due to the fast chemical reaction in the fluid. We present closed-form far-field solutions for Dirichlet, Neumann and Robin boundary conditions and comment on grid resolution requirements to accurately resolve the mass transfer boundary layer. The solution strategy presented can be straightforwardly extended to non-linear wall- and bulk-reactions.

Functional analysis: introduction to further topics in analysis

This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors then use the Baire category theorem to illustrate several points, including the existence of Besicovitch sets. The second half of the book introduces readers to other central topics in analysis, such as probability theory and Brownian motion, which culminates in the solution of Dirichlet's problem. The concluding chapters explore several complex variables and oscillatory integrals in Fourier analysis, and illustrate applications to such diverse areas as nonlinear dispersion equations and the problem of counting lattice points. Throughout the book, the authors focus on key results in each area and stress the organic unity of the subject. * A comprehensive and authoritative text that treats some of the main topics of modern analysis * A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables * Key results in each area discussed in relation to other areas of mathematics * Highlights the organic unity of large areas of analysis traditionally split into subfields * Interesting exercises and problems illustrate ideas * Clear proofs provided

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any − 1 < m < 0 , positive functions f , g and u 0 ≥ 0 , we prove that under some mild conditions on f , g and u 0 as R → ∞ the solution u R of the Dirichlet problem u t = ( u m / m ) x x in ( − R , R ) × ( 0 , ∞ ) , u ( R , t ) = ( f ( t ) | m | R ) 1 / m , u ( − R , t ) = ( g ( t ) | m | R ) 1 / m for all t > 0 , u ( x , 0 ) = u 0 ( x ) in ( − R , R ) , converges uniformly on every compact subset of R × ( 0 , T ) to the solution of the equation u t = ( u m / m ) x x in R × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in R , which satisfies some mass loss formula on ( 0 , T ) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.

Boundary value problems on planar graphs and flat surfaces with integer cone singularities, II: The mixed Dirichlet–Neumann problem

In this paper we continue the study started in Hersonsky (in press) [16]. We consider a planar, bounded, m-connected region Ω, and let ∂ Ω be its boundary. Let T be a cellular decomposition of Ω ∪ ∂ Ω , where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair ( S , f ) where S is a special type of a (possibly immersed) genus ( m − 1 ) singular flat surface, tiled by rectangles and f is an energy preserving mapping from T ( 1 ) onto S. In Hersonsky (in press) [16] the solution of a Dirichlet problem defined on T ( 0 ) was utilized, in this paper we employ the solution of a mixed Dirichlet–Neumann problem.

A new approach to the solution of Dirichlet and Neumann boundary value problems for the Laplace equation

A new approach to the application of method of Green’s functions in the solution of Dirichlet and Neumann boundary value problems for the 2D Laplace equation.

Initial boundary value problems for integrable systems: towards the long time asymptotics

The long time behaviour of solutions of (Dirichlet) initial boundary value problems for the focusing nonlinear Schrödinger equation in the case of (asymptotically) periodic boundary conditions of special (one-frequency) structure is considered both theoretically and numerically. The results of numerical simulations are shown to confirm the theoretical description in the parameter ranges where the assumption about the one-frequency character of the Neumann boundary values is (theoretically) admissible, whereas in the other ranges they suggest a more complicated description of the behaviour of these values.

Topological sensitivity of energy cost functional for wave-based defect identification

This article is concerned with establishing the topological sensitivity (TS) against the nucleation of small trial inclusions of an energy-like cost function. The latter measures the discrepancy between two time-harmonic elastodynamic states (respectively defined, for cases where overdetermined boundary data is available for identification purposes, in terms of Dirichlet or Neumann boundary data for the same reference solid) as the strain energy of their difference. Such cost function constitutes a particular form of error in constitutive relation and may be used for e.g. defect identification. The TS is expressed in terms of four elastodynamic fields, namely the free and adjoint solutions for Dirichlet or Neumann data. A similar result is also given for the linear acoustic scalar case. A synthetic numerical example where the TS result is used for the qualitative identification of an inclusion is presented for a simple 2D acoustic configuration.