Singular perturbation problems for time-reversible systems

In this paper singularly perturbed reversible vector fields defined in without normal hyperbolicity conditions are discussed. The main results give conditions for the existence of infinitely many periodic orbits and heteroclinic cycles converging to singular orbits with respect to the Hausdorff distance.

     

On averaging of nonlinear elliptic problems with inhomogeneous boundary conditions

A sequence of solutions of nonlinear elliptic problems is considered in the case where the Dirichlet conditions are given on the one part of the boundary and the Neumann conditions are given on the other part. The boundary-value problem is constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 269–276, February, 1995.     

Singular boundary method for 3D time-harmonic electromagnetic scattering problems

A frequency domain singular boundary method is presented for solving 3D time-harmonic electromagnetic scattering problem from perfect electric conductors. To avoid solving the coupled partial differential equations with fundamental solutions involving hypersingular terms, we decompose the governing equation into a system of independent Helmholtz equations with mutually coupled boundary conditions. Then the singular boundary method employs the fundamental solutions of the Helmholtz equations to approximate the scattered electric field variables. To desingularize the source singularity in the fundamental solutions, the origin intensity factors are introduced. In the novel formulation, only the origin intensity factors for fundamental solutions of 3D Helmholtz equations and its derivatives need to be considered which have been derived in the paper. Several numerical examples involving various perfectly conducting obstacles are carried out to demonstrate the validity and accuracy of the present method.     

一类四阶非线性奇摄动方程的边值问题

利用匹配渐近展开法,讨论了一类四阶非线性方程的具有两个边界层的奇摄动边值问题．引进伸长变量,根据边界条件与匹配原则,在一定的可解性条件下,给出了外部解和左右边界层附近的内层解,得到了该问题的二阶渐近解,并举例说明了这类非线性问题渐近解的存在性．     

A finite difference method for free boundary problems

Fornberg and Meyer-Spasche proposed some time ago a simple strategy to correct finite difference schemes in the presence of a free boundary that cuts across a Cartesian grid. We show here how this procedure can be combined with a minimax-based optimization procedure to rapidly solve a wide range of elliptic-type free boundary value problems.     

A stochastic approximation for fully nonlinear free boundary parabolic problems

We present a stochastic numerical method for solving fully nonlinear free boundary problems of parabolic type and provide a rate of convergence under reasonable conditions on the nonlinearity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 902–929, 2014     

On the boundary element method for some nonlinear boundary value problems

Summary Here we analyse the boundary element Galerkin method for two-dimensional nonlinear boundary value problems governed by the Laplacian in an interior (or exterior) domain and by highly nonlinear boundary conditions. The underlying boundary integral operator here can be decomposed into the sum of a monotoneous Hammerstein operator and a compact mapping. We show stability and convergence by using Leray-Schauder fixed-point arguments due to Petryshyn and Neas.Using properties of the linearised equations, we can also prove quasioptimal convergence of the spline Galerkin approximations.This work was carried out while the first author was visiting the University of Stuttgart     

A boundary-only treatment by singular boundary method for two-dimensional inhomogeneous problems

In this study, an effective singular boundary method (SBM) in conjunction with the recursive multiple reciprocity method (MRM) is developed and validated for inhomogeneous problems. It avoids the inner nodes or domain discretizations to evaluate the particular solution, and preserves the boundary-only property of the SBM. Rather than using only polyharmonic operators in the traditional MRM, a recursive MRM is proposed to annihilate source terms with different partial differential operators recursively. Nevertheless, high-order fundamental solutions are involved in the recursive MRM. The absence of the origin intensity factors of higher order fundamental solutions is a major bottleneck in applying the SBM. In order to remedy this difficulty, the origin intensity factors of higher order fundamental solutions are derived with simple formulas. Numerical examples are presented to illustrate the accuracy and efficiency of the proposed method.     

Newton's method for convex nonlinear elliptic boundary value problems

Summary Newton's method is applied to solving the boundary value problem for the equationLu=f(x,u) whereL is a linear second order uniformly elliptic operator andf(x,u) is a convex monotone increasing function ofu for each pointx in the domainD. The Newton iterates are shown to converge uniformly, quadratically and monotonically downward to the solution of the problem. The convergence is independent of the choice for the initial Newton iterate. Numerical results are presented for several problems of physical interest.     

A novel efficient method for nonlinear boundary value problems

We present two new two-level compact implicit variable mesh numerical methods of order two in time and two in space, and of order two in time and three in space for the solution of 1D unsteady quasi-linear biharmonic problem subject to suitable initial and boundary conditions. The simplicity of the proposed methods lies in their three-point discretization without requiring any fictitious points for incorporating the boundary conditions. The derived methods are shown to be unconditionally stable for a model linear problem for uniform mesh. We also discuss how our formulation is able to handle linear singular problem and ensure that the developed numerical methods retain their orders and accuracy everywhere in the solution region. The proposed difference methods successfully works for the highly nonlinear Kuramoto-Sivashinsky equation. Many physical problems are solved to demonstrate the accuracy and efficiency of the proposed methods. The numerical results reveal that the obtained solutions not only approximate the exact solutions very well but are also much better than those available in earlier research studies.