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 MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION

In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is proposed to solve this problem. An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples illustrate the validity and effectiveness of this method.

HIGH-ORDER BOUNDARY CONDITIONS FOR THE PROBLEMS OF LAPLACE EQUATION IN INFINITE REGION AND THEIR APPLICATION

The high-order boundary conditions for the problems cf Laplace equation in infinite region have been developed. The improvement in accuracy for numerical solution is achieved by imposing the high-order boundary conditions on the exterior boundarv of a reduced finite region in which the numerical method is used. So both the computing efforts and the required storage in computer are reduced. The numerical examples show that the 1st-order boundary condition approaches to the exact boundary condition and it is clearly superior to the traditional boundary condition and the 2nd-order boundary condition.

UNIFORM MEYER SOLUTION TO THE THREE DIMENSIONAL CAUCHY PROBLEM FOR LAPLACE EQUATION

We consider the three dimensional Cauchy problem for the Laplace equation{uxx（x,y,z）＋uyy（x,y,z）＋uzz（x,y,z）=0,x∈R,y∈R,0〈z≤,u（x,y,0）=g（x,y）x∈R,y∈R,uz（x,y,0）=0,x∈R,y∈R,where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 〈 z 〈 1. The problem is ill-posed, the solution （if it exists） doesn＇t depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.

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.

Stable and Unstable Eigensolutions of Laplace's Tidal Equations for Zonal Wavenumber Zero

Laplace's tidal equations are of great importance in various fields of geophysics. Here, the special case of zonal symmetry (zonal wavenumber m = 0) is investigated, where degenerate sets of eigensolutions appear. New results are presented for the inclusion of dissipative processes and the case of unstable conditions. In both instances the (nonzero) eigenfrequencies are complex. In the latter case, additional stable (i.e. real) eigenfrequencies appear in the numerical results for the absolute value of the Lambparameter ε being larger than a critical value εc. Further, it is shown that any degeneracies are removed through the inclusion of dissipation. Moreover, asymptotic relations are derived employing the relation of Laplace's tidal equations for m = 0 to the spheroidal differential equation. The implications of these findings to numerical techniques are demonstrated and results of computations are presented.

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).

PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION

In this paper, we establish the partial Schauder estimates for the Kohn Laplace equation in the Heisenberg group based on the mean value theorem, the Taylor formula and a priori estimates for the derivatives of the Newton potential.

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.

Three-Dimensional Boundary Element Method Applied to Nonlinear Wave Transformation

For higher accuracy in simulating the transformation of three dimensional waves, in consideration of the advantages of constant panels and linear elements, a combined boundary elements is applied in this research. The method can be used to remove the transverse vibration due to the accumulation of computational errors. A combined boundary condition of sponge layer and Sommerfeld radiation condition is used to remove the reflected waves from the computing domain. By following the water particle on the water surface, the third order Stokes wave transform is simulated by the numerical wave flume technique. The computed results are in good agreement with theoretical ones.

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.

A New Pursuit Mode of Nonlinear Wave Surface

The numerical mode of nonlinear wave transformation based on both the Laplace equation for water field and the Bernoulli equation for water surface is a kind of time-domain boundary problem with initial conditions. And the basis for establishing the numerical mode of nonlinear wave in time domain is to trace the position of wave free surface and to calculate the instantaneous surface height and surface potential function. This paper firstly utilizes the ‘0-1' combined BEM to separate the boundary by means of discretization of Green's integral equation based on the Laplace equation, then separates the free surface of wave with FEM and derives the FEM equation of wave surface that satisfies the nonlinear boundary conditions. By jointly solving the above BEM and FEM equations, the wave potential and surface height could be obtained with iteration in time domain. Thus a new kind of nonlinear numerical mode is established for calculating wave transformation. The wave test in the numerical wave tank shows that the numerical simulation with this mode is of high accuracy.

An Improved Numerical Simulation Mode of Nonlinear Wave with Consideration of High Order Terms

SUN Da-peng BAO Wei-bin, WU Hao and LI Yu-cheng （ In this paper the 0-1 combined BEM is adopted to subdivide the computational domain boundary, and to discretize the Green＇s integral expression based on Laplace equation. The FEM is used to subdivide the wave surface and deduce the surface equation which satisfies the nonlinear boundary conditions on the surface. The equations with potential function and wave surface height as an unknown quantity by application of Taylor expansion approach can be solved by iteration within the time step. In m-time iteration within the computational process of time step （n-1）At to nat, the results of the previous iteration are taken as the initial value of the two-order unknown terms in the present iteration. Thus, an improved tracking mode of nonlinear wave surface is estabIished, and numerical results of wave tank test indicate that this mode is improved obviously and is more precise than the previous numerical model which ignored the two-order unknown terms of wave surface location and velocity potential function in comparison with the theoretical values.

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.