Application of generalized separation of variables to solving mixed problems with irregular boundary conditions

One of the methods for solving mixed problems is the classical separation of variables (the Fourier method). If the boundary conditions of the mixed problem are irregular, this method, generally speaking, is not applicable. In the present paper, a generalized separation of variables and a way of application of this method to solving some mixed problems with irregular boundary conditions are proposed. Analytical representation of the solution to this irregular mixed problem is obtained.     

The method of normal splines for linear DAEs on the number semi-axis

The method of normal spline-collocation (NSC), applicable to a wide class of ordinary linear singular differential and integral equations, is specified for the boundary value problems for differential-algebraic equations of second order on the number semi-axis. The method consists in minimization of a norm of the collocation systems' solutions in an appropriate Hilbert–Sobolev space. The NSC method does not use the notion of differentiation index and it is applicable to DAEs of any index as well as to equations not reducible to the normal form. The problems on the infinite interval can be solved in two ways. The first way is based on the use of the original space of functions defined on the semi-axis, and the second way is based on a singular transformation of the semi-axis into the unit segment. A new reproducing kernel, that provides the first way, is presented. An algorithm to create a non-uniform collocation grid is described.     

带跳随机波动率模型的高阶ADI分裂格式

针对带跳随机波动率模型满足的偏积分微分方程,提出一种新的高阶交替方向隐式（ADI）有限差分格式,该模型是一个具有混合导数和非常数系数的对流扩散型初边值问题.我们将不同的高阶空间离散与时间步ADI分裂格式相结合,得到了一种空间四阶精度、时间二阶精度的有效方法,并采用Fourier方法分析了高阶ADI格式的稳定性.最后,通过对欧式看跌期权定价模型进行数值实验证实了数值方法的高阶收敛性.     

Study of classical solution of a one-dimensional mixed problem for one class of fifth-order semilinear equations of the Korteweg-de Vries-Burgers type

As is well known, many problems of mathematical physics are reduced to one- and multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. The existence (local and global) and uniqueness of a classical solution to a one-dimensional mixed problem with homogeneous Riquier-type boundary conditions are analyzed for a class of fifth-order semilinear pseudoparabolic equations of the Korteweg-de Vries-Burgers type. For the classical solution of the mixed problem, a uniqueness theorem is proved using the Gronwall-Bellman inequality, a local existence theorem is proved by combining the generalized contraction mapping principle with the Schauder fixed point principle, and a global existence theorem is proved by applying the method of a priori estimates.     

Petrov-Galerkin Spectral Element Method for Mixed Inhomogeneous Boundary Value Problems on Polygons

The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.     

Parameter selection rules for path-following with singular embedding

Summary. Singular embedding methods require appropriately adjusted parameters to guarantee the contraction of locally quadratically convergent iterative methods. Firstly some general rule for the parameter selection is proposed and its rate of convergence is analyzed. Secondly, some modifications in the case of polynomial error and contraction bounds are studied. Finally, these results are applied to the embedding of an elliptic boundary value problem with discontinuous nonlinearities into a family of smooth problems. Here the regularization is done in such a way that the solutions of the resulting auxiliary problems require only one step of Newton's method. Received December 2, 1996 / Revised version reveived April 7, 1998     

On homogenization of solutions of the poisson equation in a perforated domain with different types of boundary conditions on different cavities

In this paper we consider boundary value problems in perforated domains with periodic structures and cavities of different scales, with the Neumann condition on some of them and mixed boundary conditions on others. We take a case when cavities with mixed boundary conditions have so called critical size (see [1]) and cavities with the Neumann conditions have the scale of the cell. In the same way other cases can be studied, when we have the Neumann and the Dirichlet boundary conditions or the Dirichlet condition and the mixed boundary condition on the boundary of cavities.There is a large literature where homogenization problems in perforated domains were studied [2];-[7];     

The boundary element method for stochastic potential problems

Stochastic Dirichlet and Neumann boundary value problems and stochastic mixed problems have been formulated. As a result the stochastic singular integral equations have been obtained. A way of solving these equations by means of discretization of a boundary using stochastic boundary elements has been presented, resulting in a set of random algebraic equations. It has been proved that for Dirichlet and Neumann problems probabilistic characteristics (i.e. moments: expected value and correlation function) fulfilled deterministic singular integral equations. A numerical method of evaluation of moments on a boundary and inside a domain has been presented.     

Application of orthogonal collocation on finite elements for solving non-linear boundary value problems

A simple, convenient and easy approach to solve non-linear boundary value problems (BVP) using orthogonal collocation on finite elements (OCFE) is presented. The algorithm is the conjunction of finite element method (FEM) and orthogonal collocation method (OCM). The stability of the numerical results is checked by a novel algorithm which not only justifies the stability of the results but also checks the convergence of the method. The method is applied to the non-symmetric boundary value problems having Dirichlet’s and mixed Robbin’s boundary conditions.     

Multiple solutions of mixed convection in a porous medium on semi-infinite interval using pseudo-spectral collocation method

One knows that calculation of all branches of solutions of nonlinear boundary value problems can be difficult even by numerical methods, especially when the boundary conditions occur at infinity. Regarding this matter, this paper considers a model of mixed convection in a porous medium with boundary conditions on semi-infinite interval which admits multiple (dual) solutions. Furthermore, pseudo-spectral collocation method is applied in erudite way to calculate both dual solutions analytically. Comparison to exact solutions reveals reliability and high accuracy of the procedure and convince to be used to obtain multiple solutions of these kind of nonlinear problems.     

Self‐adjoint singularly perturbed boundary value problems: an adaptive variational approach

This study is intended to provide a modified variational algorithm for the numerical solution of a class of self‐adjoint singularly perturbed boundary value problem, which is equally applicable to other classes of problems. The principle of the method lies in the introduction of a mixed piecewise domain decomposition and manipulating the variational iterative approach for tackling this class of problems. The uniform convergence of the technique to the exact solution is demonstrated. Numerical results, computational comparisons, suitable error measures and illustrations are provided to testify efficiently and demonstrate the convergence, efficiency and applicability of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

Enlarging the domain of convergence for multiple shooting by the homotopy method

Summary The homotopy method is a frequently used technique in overcoming the local convergence nature of multiple shooting. In this paper sufficient conditions are given that guarantee the homotopy process to be feasible. The results are applicable to a class of two-point boundary value problems. Finally, the numerical solution of two practical problems arising in physiology is described.     

Peripheral and ADI hopscotch methods for two dimensional parabolic and elliptic equations with a mixed term

The peripheral and ADI hopscotch methods are extended to solve problems in two space dimensions with a mixed derivative term. The method is compared numerically with existing hopscotch methods.     

An approximate factorization scheme for elliptic grid generation with control functions

An Alternating Direction Implicit (ADI), Approximate Factorization (AF) scheme is presented here for the solution of the two-dimensional elliptic partial differential equations, with control functions as source terms, used for grid generation. This scheme requires significantly less computational effort than a Successive Over Relaxation (SOR) scheme. The dependence of the choice of the acceleration parameter on the rate of convergence of the AF scheme has been studied. As an example, grids generated by this method are shown, along with a comparison of the convergence history for the present AF and SOR schemes. © 1994 John Wiley & Sons, Inc.     

Mixed boundary value problems and parametrices in the edge calculus

Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. We develop a new method for computing the interface conditions in terms of the index of boundary value problems in weighted spaces on infinite cones, combined with structures from the calculus of boundary value problems on a manifold with edges. This will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator. The approach itself is completely general.     

Flux intensity functions for the Laplacian at axisymmetric edges

We derive explicit representation formulas for the computation of flux intensity functions for mixed boundary value problems for the Poisson equation in axisymmetric domains with edges. We rely on the decomposition of the boundary value problems in three dimensions by means of partial Fourier analysis with respect to the rotational angle into boundary value problems in the two‐dimensional meridian domain of . Utilizing smooth cutoff functions, the solutions of the reduced problems are analyzed semi‐analytically near corners of the plane meridian domain, and the edge flux intensity functions are constructed via Fourier synthesis and convergence analysis. The formulas are also applicable in the case of crack fronts. The constructive nature of the formulas provides in a straightforward way an efficient strategy for the accurate computation of edge flux intensity functions in axisymmetric domains. A demonstration example that illustrates the application of the formulas is presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical solution of forward and backward problem for 2-D heat conduction equation

For a two-dimensional heat conduction problem, we consider its initial boundary value problem and the related inverse problem of determining the initial temperature distribution from transient temperature measurements. The conditional stability for this inverse problem and the error analysis for the Tikhonov regularization are presented. An implicit inversion method, which is based on the regularization technique and the successive over-relaxation (SOR) iteration process, is established. Due to the explicit difference scheme for a direct heat problem developed in this paper, the inversion process is very efficient, while the application of SOR technique makes our inversion convergent rapidly. Numerical results illustrating our method are also given.     

The Successive Over Relaxation Method (SOR) and Markov Chains

In the sixties SOR has been the working horse for the numerical solution of elliptic boundary problems; classical results for chosing the relaxation parameter have been derived by D. Young and R.S. Varga. In the last fifteen years SOR has been examined for the computation of the stationary distribution of Markov chains. In the paper there are pointed out similarities and differences compared with the application of SOR for elliptic boundary problems.     

An ADI Petrov–Galerkin method with quadrature for parabolic problems

We propose and analyze a fully discrete Laplace modified alternating direction implicit quadrature Petrov–Galerkin (ADI‐QPG) method for solving parabolic initial‐boundary value problems on rectangular domains. We prove optimal order convergence results for a restricted class of the associated elliptic operator and demonstrate accuracy of our scheme with numerical experiments for some parabolic problems with variable coefficients.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical Solution for the Helmholtz Equation with Mixed Boundary Condition

We consider the numerical solution for the Helmholtz equation in R~2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based on the Green formula,we express the solution in terms of the boundary data.The key to the numerical real- ization of this method is the computation of weakly singular integrals.Numerical performances show the validity and feasibility of our method.The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems.