A harmonic polynomial method with a regularization strategy for the boundary value problems of Laplace’s equation

This paper concerns a boundary value problem of Laplace’s equation, which is solved by determining the unknown coefficients in the expansion of harmonic polynomials. A regularization method is proposed to tackle the resulting ill-posed linear system. The stability and convergence results are provided and a validating numerical experiment is presented.     

Solvability of a supercritical free surface flow problem under gravity-capillarity

In this paper, we consider a problem of a supercritical free surface flow over an obstacle lying on the bottom of a channel in 2D. The flow is irrotational, stationary and the fluid is ideal and incompressible. We take into account both the gravity and the effects of the superficial tension. The problem is nonlinear, it is formulated by the Laplace operator and the dynamic condition defined on the free surface of the fluid domain (Bernoulli equation). Using the perturbation stream function, we linearize the problem and we give a priori properties of the solution. These a priori properties allow us to construct a space where we can use the Lax–Milgram’s theorem to prove the existence and the uniqueness of the solution of the problem.     

On a partial integrodifferential equation of Seal’s type

In this paper we generalize a partial integrodifferential equation satisfied by the finite time ruin probability in the classical Poisson risk model. The generalization also includes the bivariate distribution function of the time of and the deficit at ruin. We solve the partial integrodifferential equation by Laplace transforms with the help of Lagrange’s implicit function theorem. The assumption of mixed Erlang claim sizes is then shown to result in tractable computational formulas for the finite time ruin probability as well as the bivariate distribution function of the time of and the deficit at ruin. A more general partial integrodifferential equation is then briefly considered.     

The homotopy analysis method for solving the time-fractional Fornberg–Whitham equation and comparison with Adomian’s decomposition method

In this paper, we applied relatively new analytical techniques, the homotopy analysis method (HAM) and the Adomian’s decomposition method (ADM) for solving time-fractional Fornberg–Whitham equation. The homotopy analysis method contains the auxiliary parameter, which provides us with a simple way to adjust and control the convergence region of solution series. The fractional derivatives are described in the Caputo sense. A comparison is made the between HAM and ADM results. The present methods performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented.     

Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions

In this paper, we consider the Cauchy problem of Laplace’s equation in the neighborhood of a circle. The method of fundamental solutions (MFS) combined with the discrete Tikhonov regularization is applied to obtain a regularized solution from noisy Cauchy data. Under the suitable choices of a regularization parameter and an a priori assumption to the Cauchy data, we obtain a convergence result for the regularized solution. Numerical experiments are presented to show the effectiveness of the proposed method.     

Drawdown in prolate spheroidal–spherical coordinates obtained via Green’s function and perturbation methods

When investigating aquifer behaviour it is important to note that there exists a close relationship between the geometrical properties of the aquifer and the behaviour of the solution. In this paper our concern is to solve the flow equation described by prolate spheroidal coordinates by means of perturbation and the Green’s function method, where the spheroid is considered to be a perturbation of a sphere. We transformed the spheroidal coordinates to spherical polar coordinates in the limit, as the shape factor tends to zero. The new groundwater flow equation is solved via an asymptotic parameter expansion and the Green’s function method. The approximate solution of the new equation is compared with experimental data from real world. To take into account the error committed while approximating, we estimate the error in the asymptotic expansion. The error functions obtained suggest that the error would be very small for the shape factor tending to zero if the first two terms of the expansion are taken as an approximation.     

Asymptotics of the solution of Laplace’s equation with third-type boundary conditions on the boundaries of two small holes

A boundary value problem for Laplace’s equation in a bounded domain with two small holes is considered. Third-type boundary conditions are set on the boundaries of the holes. A Neumann condition is specified on the outer boundary of the domain. A uniform asymptotic approximation of the solution is constructed and justified up to an arbitrary power of a small parameter.     

Analysis and Numerical Approximation of a Contact Problem Involving Nonlinear Hencky-Type Materials with Nonlocal Coulomb’s Friction Law

A static frictional contact problem between an elasto-plastic body and a rigid foundation is considered. The material’s behavior is described by the nonlinear elastic constitutive Hencky’s law. The contact is modeled with the Signorini condition and a version of Coulomb’s law in which the coefficient of friction depends on the slip. The existence of a weak solution is proved by using Schauder’s fixed-point theorem combined with arguments of abstract variational inequalities. Afterward, a successive iteration technique, based on the Ka?anov method, to solve the problem numerically is proposed, and its convergence is established. Then, to improve the conditioning of the iterative problem, an appropriate Augmented Lagrangian formulation is used and that will lead us to Uzawa block relaxation method in every iteration. Finally, numerical experiments of two-dimensional test problems are carried out to illustrate the performance of the proposed algorithm.     

Representation for the Green’s function of the Dirichlet problem for polyharmonic equations in a ball

We explicitly construct the Green’s function for the Dirichlet problem for polyharmonic equations in a ball in a space of arbitrary dimension. The formulas for the Green’s function are of interest in their own right. In particular, the explicit representations for a solution to the Dirichlet problem for the biharmonic equation are important in elasticity.     

Application of a 14-point averaging operator in the grid method

The Dirichlet problem for Laplace’s equation in a rectangular parallelepiped is solved by applying the grid method. A 14-point averaging operator is used to specify the grid equations on the entire grid introduced in the parallelepiped. Given boundary values that are continuous on the parallelepiped edges and have first derivatives satisfying the Lipschitz condition on each parallelepiped face, the resulting discrete solution of the Dirichlet problem converges uniformly and quadratically with respect to the mesh size. Assuming that the boundary values on the faces have fourth derivatives satisfying the Hölder condition and the second derivatives on the edges obey an additional compatibility condition implied by Laplace’s equation, the discrete solution has uniform and quartic convergence with respect to the mesh size. The convergence of the method is also analyzed in certain cases when the boundary values are of intermediate smoothness.     

Transient elastodynamic three-dimensional problems in cracked bodies

The general formulation of the transient elastodynamic second boundary value problem in an isotropic linear elastic body with a crack of arbitrary shape by combining the boundary integral equation method and the Laplace transform with respect to time is presented in this paper. Both finite and infinite elastic bodies are considered. A numerical solution of the transformed boundary integral equations is proposed.     

Some iterative methods for the largest positive definite solution to a class of nonlinear matrix equation

In this paper, we propose some inversion-free iteration methods for finding the largest positive definite solution of a class of nonlinear matrix equation. Then, we consider the properties of the solution for this nonlinear matrix equation. Also, we establish Newton’s iteration method for finding the largest positive definite solution and prove its quadratic convergence. Furthermore, we derive the semi-local convergence of the Newton’s iteration method. Finally, some numerical examples are presented to illustrate the effectiveness of the theoretical results and the behavior of the considered methods.     

Riemannian Newton-type methods for joint diagonalization on the Stiefel manifold with application to independent component analysis

The joint approximate diagonalization of non-commuting symmetric matrices is an important process in independent component analysis. This problem can be formulated as an optimization problem on the Stiefel manifold that can be solved using Riemannian optimization techniques. Among the available optimization techniques, this study utilizes the Riemannian Newton’s method for the joint diagonalization problem on the Stiefel manifold, which has quadratic convergence. In particular, the resultant Newton’s equation can be effectively solved by means of the Kronecker product and the vec and veck operators, which reduce the dimension of the equation to that of the Stiefel manifold. Numerical experiments are performed to show that the proposed method improves the accuracy of the approximate solution to this problem. The proposed method is also applied to independent component analysis for the image separation problem. The proposed Newton method further leads to a novel and fast Riemannian trust-region Newton method for the joint diagonalization problem.     

An 12 constrained best approximation problem

A problem of minimizing a quadratic function over the unit ball of l₂is considered,the motivation being a minimum norm problem for the heat equation controlled by the constrained initial condition. A constructive method for finding an ?-solution is developed and its convergence rate is estimated. The dependence of the solution, on the data is studied.     

Approximate analytic solutions to Radulet equations for the analysis of electromagnetic transients in single-phase power transmission lines

In this work, an analytic development for a transmission line with a corona effect for simulating an electromagnetic transient is presented. The asymptotic solution for the Radulet equations in which a nonlinear term is presented is obtained. The study is carried out for a single-phase transmission line. The electrical parameters for an overhead line are defined and several formulations for their calculation are presented. The frequency dependence of the electrical parameters is considered. In the first part, the linear problem solution is found; the Fourier and Laplace transforms are applied with respect to distance and time respectively. After finding the solution in the Fourier–Laplace domain, which is expressed in terms of a Green’s function integral, an approximate analytical solution is obtained in the distance–time domain by means of asymptotic methods. Finally, the nonlinear solution is found using as a first approach the linear solution. The results obtained show an attenuation in the voltage wave due to the corona effect.     

On a problem with nonperiodic frequent alternation of boundary conditions imposed on fast oscillating sets

A singularly perturbed eigenvalue problem for the Laplacian in a cylinder is considered. The problem is characterized by frequent nonperiodic alternation of boundary conditions imposed on narrow strips lying on the cylinder’s lateral surface. The width of the strips is an arbitrary function of a small parameter and can oscillate rapidly, with the nature of the oscillations being arbitrary. Sharp estimates are derived for the convergence rate of the eigenvalues and eigenfunctions in the problem.     

Dirichlet problem for the Stokes equation

The paper presents some coercive a priori estimates of the solution of the Dirichlet problem for the linear Stokes equation relating vorticity and the stream function of an axially symmetric flow of an incompressible fluid. This equation degenerates on the axis of symmetry. The method used to obtain the estimates is based on a differential substitution transforming the Stokes equation into the Laplace equation and on the subsequent transition from cylindrical to Cartesian coordinates in three-dimensional space.     

关于欧拉函数方程？（？（x））=2t的可解性

对任意的正整数 n,函数？（n）为著名的 Euler 函数,即在序列1,2,···, n 中与n 互质的整数的个数。本文利用初等方法研究了方程？（？（x））的可解性,并给出了该方程的全部正整数解。     

Derivation of transitional asymptotic expansions from integral representations

A method for deriving transitional asymptotic expansions from integral representations is described and applied to Anger function and modified Hankel function. The method consists in deriving asymptotic expansions of the function considered as well as its first derivativeat the transition point using conventional methods such as Laplace’s method or the method of steepest descents. Since both the functions considered satisfy a second order linear differential equation, it is possible to obtain asymptotic expansions of higher order derivatives of the functions from the first two expansions. Thus asymptotic expressions for all the derivatives at the transition point are known and a Taylor expansion of the function in the neighbourhood of the transition point can be written. The method is also applicable to the generalized exponential integral, Weber’s parabolic cylinder function and Poiseuille function.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.