Neumann Kernel Research Articles

Green’s function for solving initial-boundary value problem of evolutionary partial differential equations

We propose a new method to solve the initial-boundary value problem for hyperbolic-dissipative partial differential equations (PDEs) based on the spirit of LY algorithm [T.-P. Liu and S.-H. Yu, Dirichlet–Neumann kernel for hyperbolic-dissipative system in half-space, Bull. Inst. Math. Acad. Sin. 7 (2012) 477–543]. The new method can handle more general domains than that of LYs’. We convert the evolutionary PDEs into the elliptic PDEs by the Laplace transformation. Using the Laplace transformation of the fundamental solutions of the evolutionary PDEs and the image method, we can construct Green’s functions for the corresponding elliptic PDEs. Finally, we obtain Green’s functions for the evolutionary PDEs by inverting the Laplace transformation. As a consequence, we establish Green’s functions for some basic PDEs such as the heat equation, the wave equation and the damped wave equation, in a half space and a quarter plane with various boundary conditions. On the other hand, the structure of hyperbolic-dissipative PDEs means its fundamental solution is non-symestric and hence the image method does not work. We utilize the idea of Laplace wave train introduced by Liu and Yu in [Navier–Stokes equations in gas dynamics: Greens function, singularity and well-posedness, Comm. Pure Appl. Math. 75(2) (2022) 223–348] to generalize the image method. Combining this with the notions of Rayleigh surface wave operators introduced in [S. J. Deng, W. K. Wang and S.-H. Yu, Green’s functions of wave equations in [Formula: see text], Arch. Ration. Mech. Anal. 216 (2015) 881–903], we are able to obtain the complete representations of Green’s functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions.

Computing the zeros of the Szegö kernel for doubly connected regions using conformal mapping

<abstract><p>An explicit formula for the zero of the Szegö kernel for an annulus region is well-known. There exists a transformation formula for the Szegö kernel from a doubly connected region onto an annulus. Based on conformal mapping, we derive an analytical formula for the zeros of the Szegö kernel for a general doubly connected region with smooth boundaries. Special cases are the explicit formulas for the zeros of the Szegö kernel for doubly connected regions bounded by circles, limacons, ellipses, and ovals of Cassini. For a general doubly connected region with smooth boundaries, the zero of the Szegö kernel must be computed numerically. This paper describes the application of conformal mapping via integral equation with the generalized Neumann kernel for computing the zeros of the Szegö kernel for smooth doubly connected regions. Some numerical examples and comparisons are also presented. It is shown that the conformal mapping approach also yields good accuracy for a narrow region or region with boundaries that are close to each other.</p></abstract>

Simulating local fields in carbon nanotube-reinforced composites for infinite strip with voids

A numerical simulation of the thermal properties is conducted for an isotropic and homogeneous infinite strip composite reinforced by carbon nanotubes (CNTs) and containing voids. The CNTs can be uniformly or randomly distributed but are non-overlapping. We model the CNTs as thin perfectly conducting elliptic inclusions and assume the voids to be of circular shape and act as barriers to heat flow. We also impose isothermal conditions on the external boundaries by assuming the lower infinite wall to be a heater under a given temperature, and the upper wall to be a cooler that can be held at a lower fixed temperature. The mathematical model, which takes the form of a mixed Dirichlet–Neumann problem, is solved by applying the boundary integral equation with the generalized Neumann kernel. We illustrate the performance of the proposed method through several numerical examples including the case of the presence of a large number of CNTs and voids.

Mikhlin’s Integral Equation and the Integral Equation with the Generalized Neumann Kernel on Simply Connected Domains

Mikhlin’s integral equation is a classical integral equation for solving boundary value problems for Laplace’s equation. The kernel of the integral equation is known as the Neumann kernel. Recently, an integral equation for solving the Riemann–Hilbert problem was derived. The kernel of the new integral equation is a generalization of the Neumann kernel, and hence, it is called the generalized Neumann kernel. The objective of this paper is to present a detailed comparison between these two integral equations with emphasis on their similarities and differences. This comparison is done through applying both equations to solve Laplace’s equation with Dirichlet boundary conditions in simply connected domains with smooth and piecewise smooth boundaries.

Numerical computation of preimage domains for spiral slit regions and simulation of flow around bodies.

In this paper, we propose the iterative numerical methods to calculate the conformal preimage domains for the specified logarithmic spiral slit regions and develop the applications of conformal mappings in the simulations of the flow around bodies. Firstly, we postulate that the boundaries of the preimage domains mapped onto logarithmic spiral slits are ellipses. The lengths of the long axes of ellipses and the coordinates of the centers are calculated using our iterative methods. Secondly, each type of the presented iterative method calculates numerical conformal mappings via solving the boundary integral equation with the generalized Neumann kernel. Finally, numerical examples show the convergence and availability of our iterative methods and display the simulations of the flow around the bodies as an application.

Topological groups without infinite precompact continuous homomorphic images

Let P be a property of topological groups. We say that a topological group G is MinAP moduloP if its image f[G] under each continuous homomorphism f:G→K from G to a compact group K has property P. When P is the property of being the trivial group, MinAP modulo P groups are precisely the minimally almost periodic groups of von Neumann and Wigner. We give a characterization of these new classes of groups for five properties P: finite, bounded, torsion, compact and connected. We show that these five classes are distinct and differ from the standard class of minimally almost periodic groups. We equip every Abelian group with a Hausdorff MinAP modulo finite group topology, thereby showing that, unlike the classical minimal almost periodicity, the property MinAP modulo finite imposes no restrictions whatsoever on the algebraic structure of the underlying group. At last but not least, we prove that the quotient group G/n(G) of every topological group G with respect to its von Neumann kernel n(G) is the categorical reflection of G in the class of maximally almost periodic (MAP) groups, and the natural quotient map plays the role of reflection homomorphism. As a consequence, this quotient group (G/n(G))+ equipped with its Bohr topology is topologically isomorphic to the image of G under the canonical homomorphism to its Bohr compactification.

Conformal Invariants in Simply Connected Domains

This paper studies the numerical computation of several conformal invariants of simply connected domains in the complex plane including, the hyperbolic distance, the reduced modulus, the harmonic measure, and the modulus of a quadrilateral. The used method is based on the boundary integral equation with the generalized Neumann kernel. Several numerical examples are presented. The performance and accuracy of the presented method is validated by considering several model problems with known analytic solutions.

Numerical computation of Mityuk's function and radius for circular/radial slit domains

We consider Mityuk's function and radius which have been proposed in [15] as generalizations of the reduced modulus and conformal radius to the cases of multiply connected domains. We present a numerical method to compute Mityuk's function and radius for canonical domains that consist of the unit disk with circular/radial slits. Our method is based on the boundary integral equation with the generalized Neumann kernel. Special attention is given to the validation of the theoretical results on the existence of critical points and the boundary behavior of Mityuk's radius.

Numerical Computing of Preimage Domains for Bounded Multiply Connected Slit Domains

In this paper, for a given bounded multiply connected slit domain $$\varOmega $$ , we present an iterative numerical method for computing a conformally equivalent multiply connected domain G bounded by smooth Jordan curves and the conformal mapping $$w=\varPhi (z)$$ from G onto $$\varOmega $$ . Each iteration of the proposed iterative method requires solving the boundary integral equation with the generalized Neumann kernel. We consider two cases of bounded slit domains, namely the unit disk with radial slit domain and an annulus with radial slit domain. Numerical examples are presented to illustrate that the proposed iterative method converges even for highly connected slit domains.

A fast numerical method for ideal fluid flow in domains with multiple stirrers

A collection of arbitrarily-shaped solid objects, each moving at a constant speed, can be used to mix or stir ideal fluid, and can give rise to interesting flow patterns. Assuming these systems of fluid stirrers are two-dimensional, the mathematical problem of resolving the flow field—given a particular distribution of any finite number of stirrers of specified shape and speed—can be formulated as a Riemann–Hilbert (R–H) problem. We show that this R–H problem can be solved numerically using a fast and accurate algorithm for any finite number of stirrers based around a boundary integral equation with the generalized Neumann kernel. Various systems of fluid stirrers are considered, and our numerical scheme is shown to handle highly multiply connected domains (i.e. systems of many fluid stirrers) with minimal computational expense.

Numerical Conformal Mapping onto the Parabolic, Elliptic and Hyperbolic Slit Domains

This paper presents a numerical method for computing the conformal mappings onto the parabolic slit domain, the elliptic slit domain and the hyperbolic slit domain. The method relies on a boundary integral equation with the generalized Neumann kernel. For a given multiply connected domain of connectivity $$m+1$$ , the proposed method requires $$O((m+1)n\log n)$$ operations where n is the number of nodes in the discretization of each boundary component of the given domain. Several numerical examples are presented to illustrate the performance of the proposed method.

Computing Robin Problem on Unbounded Simply Connected Domain via an Integral Equation with the Generalized Neumann Kernel

A Robin problem is a mixed problem with a linear combination of Dirichlet and Neumann D-N conditions. The aim of this paper are presents a new boundary integral equation BIE method for the solution of unbounded Robin boundary value problem BVP in the simply connected domain. The method show how to reformulate the Robin boundary value problem BVP as Riemann-Hilbert problem RHP which lead to the system of integral equation, and the related differential equations are also created that give rise to unique solutions. Numerical results on several tests regions by the Nyström method NM with the trapezoidal rule TR are presented to clarify the solution technique for the Robin problem when the boundaries are sufficiently smooth.

Fast and Accurate Computation of the Logarithmic Capacity of Compact Sets

We present a numerical method for computing the logarithmic capacity of compact subsets of $\mathbb{C}$, which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have any special symmetry. The method relies on the conformal map onto lemniscatic domains and, computationally, on the solution of a boundary integral equation with the Neumann kernel. Our numerical examples indicate that the method is fast and accurate. We apply it to give an estimate of the logarithmic capacity of the Cantor middle third set and generalizations of it.

Solving a class of Robin problems in simply connected regions via integral equations with a generalized Neumann kernel

Solving a class of Robin problems in simply connected regions via integral equations with a generalized Neumann kernel

Solving Robin problems in multiply connected regions via an integral equation with the generalized Neumann kernel

This paper presents a boundary integral equation method for finding the solution of Robin problems in bounded and unbounded multiply connected regions. The Robin problems are formulated as Riemann-Hilbert problems which lead to systems of integral equations and the related differential equations are also constructed that give rise to unique solutions, which are shown. Numerical results on several test regions are presented to illustrate that the approximate solution when using this method for the Robin problems when the boundaries are sufficiently smooth are accurate.

Numerical Computation of the Conformal Map onto Lemniscatic Domains

We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For $\ell$-times connected domains the method requires solving $\ell$ boundary integral equations with the Neumann kernel. This can be done in $O(\ell^2 n \log n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component of the multiply connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains of high connectivity.

Fast Computing of Conformal Mapping and Its Inverse of Bounded Multiply Connected Regions onto Second, Third and Fourth Categories of Koebe’s Canonical Slit Regions

This paper presents a boundary integral method with the adjoint generalized Neumann kernel for conformal mapping of a bounded multiply connected region onto a disk with spiral slits region $$\varOmega _1$$Ω1. This extends the methods that have recently been given for mappings onto annulus with spiral slits region $$\varOmega _2$$Ω2, spiral slits region $$\varOmega _3$$Ω3, and straight slits region $$\varOmega _4$$Ω4 but with different right-hand sides. This paper also presents a fast implementation of the boundary integral equation method for computing numerical conformal mapping of bounded multiply connected region onto all four regions $$\varOmega _1$$Ω1, $$\varOmega _2$$Ω2, $$\varOmega _3$$Ω3, and $$\varOmega _4$$Ω4 as well as their inverses. The integral equations are solved numerically using combination of Nystrom method, GMRES method, and fast multipole method (FMM). The complexity of this new algorithm is $$O((m + 1)n)$$O((m+1)n), where $$m+1$$m+1 is the multiplicity of the multiply connected region and n is the number of nodes on each boundary component. Previous algorithms require $$O((m+1)^3 n^3)$$O((m+1)3n3) operations. The algorithm is tested on several test regions with complex geometries and high connectivities. The numerical results illustrate the efficiency of the proposed method.

Representation of Green’s function of the Neumann problem for a multi-dimensional ball

Representation of the Green’s function of the classical Neumann problem for the Poisson equation in the unit ball of arbitrary dimension is given. In constructing this function, we use the representation of the fundamental solution of the Laplace equation in the form of a series. It is shown that the Green’s function can be represented in terms of elementary functions and its explicit form can be written out. An explicit form of the Neumann kernel at and .

Integral Equation for the Ahlfors Map on Multiply Connected Regions

This paper presents a new boundary integral equation with the adjoint Neumann kernel associated with where is the boundary correspondence function of Ahlfors map of a bounded multiply connected region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region. The integral equation is solved numerically for using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of we solve for the boundary correspondence function which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method.

A fast computational method for potential flows in multiply connected coastal domains

We present a fast and accurate numerical method for constructing incompressible, inviscid and irrotational flows in two-dimensional coastal domains, which are unbounded multiply connected domains above an infinitely long coastline boundary. In the numerical method, we utilize a numerical conformal mapping method based on a boundary integral equation with the generalized Neumann kernel in order to construct conformal mappings from coastal domains onto four of Koebe’s canonical domains. The numerical method is fast and accurate, since it just requires \(O((m+1)n\ln n)\) operations and it converges with \(O(e^{-cn})\) for coastal domains of connectivity \(m+1\), where \(n\) is the number of nodes in discretizing each smooth boundary component and \(c\) is a positive constant. With some examples, we also show that it is applicable to arbitrary coastal domains with high connectivity and complex geometry.