The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

In this article,a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed.The idea of using the P_(k)-harmonic polynomial space instead of the full polynomial space P_(k)is to use a much smaller number of basis functions to achieve the same accuracy when k≥2.The optimal rate of convergence is derived in both H^(1)and L^(2)norms.Numerical experiments have been conducted to verify the theoretical error estimates.In addition,numerical comparisons of using the P_(2)-harmonic polynomial space and using the standard P_(2)polynomial space are presented.

Harmonic Measures and Numerical Computation of Cauchy Problems for Laplace Equations

It is well known that the Cauchy problem for Laplace equations is an ill-posed problem in Hadamard’s sense.Small deviations in Cauchy data may lead to large errors in the solutions.It is observed that if a bound is imposed on the solution,there exists a conditional stability estimate.This gives a reasonable way to construct stable algorithms.However,it is impossible to have good results at all points in the domain.Although numerical methods for Cauchy problems for Laplace equations have been widely studied for quite a long time,there are still some unclear points,for example,how to evaluate the numerical solutions,which means whether they can approximate the Cauchy data well and keep the bound of the solution,and at which points the numerical results are reliable?In this paper,the authors will prove the conditional stability estimate which is quantitatively related to harmonic measures.The harmonic measure can be used as an indicate function to pointwisely evaluate the numerical result,which further enables us to find a reliable subdomain where the local convergence rate is higher than a certain order.

Properties of Solutions to Fractional Laplace Equation with Singular Term

The aim of the paper is to study the properties of positive classical solutions to the fractional Laplace equation with the singular term.Using the extension method,we prove the nonexistence and symmetric of solutions to the singular fractional equation.

Exponential Convergence Theory of the Multipole and Local Expansions for the 3-D Laplace Equation in Layered Media

In this paper,we establish the exponential convergence theory for the multipole and local expansions,shifting and translation operators for the Green's function of 3-dimensional Laplace equation in layered media.An immediate application of the theory is to ensure the exponential convergence of the FMM which has been shown by the numerical results reported in[27].As the Green's function in layered media consists of free space and reaction field components and the theory for the free space components is well known,this paper will focus on the analysis for the reaction components.We first prove that the density functions in the integral representations of the reaction components are analytic and bounded in the right half complex wave number plane.Then,by using the Cagniard-de Hoop transform and contour deformations,estimates for the remainder terms of the truncated expansions are given,and,as a result,the exponential convergence for the expansions and translation operators is proven.

Convergence Estimates for Some Regularization Methods to Solve a Cauchy Problem of the Laplace Equation

In this paper,we give a general proof on convergence estimates for some regularization methods to solve a Cauchy problem for the Laplace equation in a rectangular domain.The regularization methods we considered are:a non-local boundary value problem method,a boundary Tikhonov regularization method and a generalized method.Based on the conditional stability estimates,the convergence estimates for various regularization methods are easily obtained under the simple verifications of some conditions.Numerical results for one example show that the proposed numerical methods are effective and stable.

A Direct Method of Moving Planes to Fractional Power Sub Laplace Equations on the Heisenberg Group

We give the direct method of moving planes for solutions to the conformally invariant fractional power sub Laplace equation on the Heisenberg group.The method is based on four maximum principles derived here.Then symmetry and nonexistence of positive cylindrical solutions are proved.

The Christoffel problem by the fundamental solution of the Laplace equation

The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere S^(n).Necessary and sufficient conditions have been found by Firey(1967)and Berg(1969),by using the Green function of the Laplacian on the sphere.Expressing the Christoffel problem as the Laplace equation on the entire space R^(n+1),we observe that the second derivatives of the solution can be given by the fundamental solutions of the Laplace equation.Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem.We also study the Lp extension of the Christoffel problem and provide sufficient conditions for the problem,for the case p≥2.

Optimal Shape Factor and Fictitious Radius in the MQ-RBF:Solving Ill-Posed Laplacian Problems

To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).

Mathematical Modeling of Moored Ship Motion in Arbitrary Harbor utilizing the Porous Breakwater

The motion of the moored ship in the harbor is a classical hydrodynamics problem that still faces many challenges in naval operations,such as cargo transfer and ship pairings between a big transport ship and some small ships.A mathematical model is presented based on the Laplace equation utilizing the porous breakwater to investigate the moored ship motion in a partially absorbing/reflecting harbor.The motion of the moored ship is described with the hydrodynamic forces along the rotational motion(roll,pitch,and yaw)and translational motion(surge,sway,and heave).The efficiency of the numerical method is verified by comparing it with the analytical study of Yu and Chwang(1994)for the porous breakwater,and the moored ship motion is compared with the theoretical and experimental data obtained by Yoo(1998)and Takagi et al.(1993).Further,the current numerical scheme is implemented on the realistic Visakhapatnam Fishing port,India,in order to analyze the hydrodynamic forces on moored ship motion under resonance conditions.The model incorporates some essential strategies such as adding a porous breakwater and utilizing the wave absorber to reduce the port’s resonance.It has been observed that these tactics have a significant impact on the resonance inside the port for safe maritime navigation.Therefore,the current numerical model provides an efficient tool to reduce the resonance within the arbitrarily shaped ports for secure anchoring.

Electrical properties of m×n cylindrical network

We consider the problem of electrical properties of an m×n cylindrical network with two arbitrary boundaries,which contains multiple topological network models such as the regular cylindrical network,cobweb network,globe network,and so on.We deduce three new and concise analytical formulae of potential and equivalent resistance for the complex network of cylinders by using the RT-V method(a recursion-transform method based on node potentials).To illustrate the multiplicity of the results we give a series of special cases.Interestingly,the results obtained from the resistance formulas of cobweb network and globe network obtained are different from the results of previous studies,which indicates that our research work creates new research ideas and techniques.As a byproduct of the study,a new mathematical identity is discovered in the comparative study.

The Equal-Norm Multiple-Scale Trefftz Method for Solving the Nonlinear Sloshing Problem with Baffles

In this paper,the equal-norm multiple-scale Trefftz method combined with the implicit Lie-group scheme is applied to solve the two-dimensional nonlinear sloshing problem with baffles.When considering solving sloshing problems with baffles by using boundary integral methods,degenerate geometry and problems of numerical instability are inevitable.To avoid numerical instability,the multiple-scale characteristic lengths are introduced into T-complete basis functions to efficiently govern the high-order oscillation disturbance.Again,the numerical noise propagation at each time step is eliminated by the vector regularization method and the group-preserving scheme.A weighting factor of the group-preserving scheme is introduced into a linear system and then used in the initial and boundary value problems(IBVPs)at each time step.More importantly,the parameters of the algorithm,namely,the T-complete function,dissipation factor,and time step,can obtain a linear relationship.The boundary noise interference and energy conservation are successfully overcome,and the accuracy of the boundary value problem is also improved.Finally,benchmark cases are used to verify the correctness of the numerical algorithm.The numerical results show that this algorithm is efficient and stable for nonlinear two-dimensional sloshing problems with baffles.

A 3-D Numerical Model for the Calculation of Water Wave Transformation in Large Area

Based on the integral equation transformed from three dimensional Laplace equation and by the adoption of the division manner of sub-region boundary element method, the numerical computations of the velocity potential of each sub-region are given considering the continuity conditions of potential and normal derivatives at the interface of sub-regions. Therefore, computation of wave deformation in offshore flow field is realized. The present numerical model provides a good solution for the application of boundary element method to the calculation of wave deformation in large areas.

Numerical Solutions to the Robin Inverse Problem with Nonnegativity Constraints

We present iterative numerical methods for solving the inverse problem of recovering the nonnegative Robin coefficient from partial boundary measurement of the solution to the Laplace equation. Based on the boundary integral equation formulation of the problem, nonnegativity constraints in the form of a penalty term are incorporated conveniently into least-squares iteration schemes for solving the inverse problem. Numerical implementation and examples are presented to illustrate the effectiveness of this strategy in improving recovery results.

Data Recovering Problem Using a New KMF Algorithm for Annular Domain

This paper is interested at the Cauchy problem for Laplace’s equation, which is to recover both Dirichlet and Neumann conditions on the inaccessible part of the boundary (inner part) of an annular domain from the over specified conditions on the accessible one (outer part). This work is an extension of the proposed algorithm for a unit circle [1] to annular domain, where, we describe an alternating formulation of the KMF algorithm proposed by Kozlov, Mazya and Fomin, and its relationship with the standard formulation. The new KMF algorithm ameliorates the accuracy of the solution and reduces the number of iterations required to achieve convergence. In the last part, the discussion of the error estimation of solution is presented and some numerical tests, using the software Freefem are given to show the efficiency of the proposed method.

Communications in Theoretical Physics

The unified processing and research of multiple network models are implemented,and a new theoretical advance has been made,which sets up two new theorems on evaluating the exact electrical characteristics(potential and resistance)of the complex m×n resistor networks by the recursion-transform method with potential parameters,and applies to a variety of different types of lattice structure with arbitrary boundaries such as the nonregular m×n rectangular networks and the nonregular m×n cylindrical networks.Our research gives the analytical solutions of electrical characteristics of the complex networks(finite,semi-infinite and infinite),which has not been solved before.As applications of the theorems,a series of analytical solutions of potential and resistance of the complex resistor networks are discovered.

Singular Functions and Characterizations of Field Concentrations:a Survey

In the presence of closely located inclusions of the extreme material property,the physical fields,such as the electric field and the stress tensor,may be concentrated and arbitrarily large in the narrow region between two inclusions.Recently there has been significant progress on the quantitative characterization of the field concentration in the contexts of electrostatics(Laplace equation),linear elasticity(Lam´e system),and viscous flow(Stokes system).This paper is to review such progress in a coherent way.

Galerkin Formulations of the Method of Fundamental Solutions

In this paper,we introduce two Galerkin formulations of theMethod of Fundamental Solutions(MFS).In contrast to the collocation formulation of the MFS,the proposed Galerkin formulations involve the evaluation of integrals over the boundary of the domain under consideration.On the other hand,these formulations lead to some desirable properties of the stiffness matrix such as symmetry in certain cases.Several numerical examples are considered by these methods and their various features compared.

A Fast Accurate Boundary IntegralMethod for Potentials on Closely Packed Cells

Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties.However,accuracy deteriorates when the cell boundaries are close to each other.We present a boundary integralmethod in two dimensions which is specially designed tomaintain second order accuracy even if boundaries are arbitrarily close.Themethod uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy.For boundaries with many components we use the fast multipolemethod for efficient summation.We compute electric potentials on a domain with cells whose conductivity differs from that of the surrounding medium.We first solve an integral equation for a source term on the cell interfaces and then find values of the potential near the interfaces via integrals.Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region.A number of examples are presented.We demonstrate that increased refinement is not needed to maintain accuracy as interfaces become very close.

CALCULATION OF POWER LINE FIELDS BETWEEN CIRCULAR CONDUCTORS

It is important to calculate the electric field at the surface of high voltage direct current power transmission lines,since it is this field which governs the onset of corona discharge and the power loss arising therefrom.A method is presented here to calculate the electric field based on an implementation of the boundary element method for conductors of strictly circular cross section.Given the circular geometry it is possible to resolve all integrals involved analytically.A Galerkin approach is adopted,giving the solution in the spatial frequency domain.That allows a controlled truncation of the system matrix by choice of which frequency components to keep.It transpires that the low frequency components are the most important ones.Two test cases are used to quantify the accuracy of the solution with respect to truncation and distance from the surface.It is found that the accuracy increases with distance from the surface,but for all distances can be controlled by choosing an appropriate level of truncation.