A new method of fundamental solutions applied to nonhomogeneous elliptic problems

The classical method of fundamental solutions (MFS) has only been used to approximate the solution of homogeneous PDE problems. Coupled with other numerical schemes such as domain integration, dual reciprocity method (with polynomial or radial basis functions interpolation), the MFS can be extended to solve the nonhomogeneous problems. This paper presents an extension of the MFS for the direct approximation of Poisson and nonhomogeneous Helmholtz problems. This can be done by using the fundamental solutions of the associated eigenvalue equations as a basis to approximate the nonhomogeneous term. The particular solution of the PDE can then be evaluated. An advantage of this mesh-free method is that the resolution of both homogeneous and nonhomogeneous equations can be combined in a unified way and it can be used for multiscale problems. Numerical simulations are presented and show the quality of the approximations for several test examples. AMS subject classification 35J25, 65N38, 65R20, 74J20     

Three solutions for a class of higher dimensional singular problems

In the present paper we establish the existence of three positive weak solutions for a quasilinear elliptic problem involving a singular term of the type \({u^{-\gamma}}\). As far as we know this is the first contribution in the higher dimensional case for arbitrary \({\gamma > 0}\).     

The method of fundamental solutions for problems in potential flow

The method of fundamental solutions is a form of indirect boundary integral equation method. Its distinctive feature is adaptivity, gained through the use of an auxiliary boundary that is chosen automatically by a least squares procedure. The paper demonstrates the application of the method to problems in potential flow. A further advantage of the method is that the velocity field can be computed easily and accurately by a direct evaluation procedure.     

Multipole fast algorithm for the least‐squares approach of the method of fundamental solutions for three‐dimensional harmonic problems

In this article we describe an improvement in the speed of computation for the least‐squares method of fundamental solutions (MFS) by means of Greengard and Rokhlin's FMA. Iterative solution of the linear system of equations is performed for the equations given by the least‐squares formulation of the MFS. The results of applying the method to test problems from potential theory with a number of boundary points in the order of 80,000 show that the method can achieve fast solutions for the potential and its directional derivatives. The results show little loss of accuracy and a major reduction in the memory requirements compared to the direct solution method of the least squares problem with storage of the full MFS matrix. The method can be extended to the solution of overdetermined systems of equations arising from boundary integral methods with a large number of boundary integration points. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 828–845, 2003.     

Effect of rotation in lubrication problems: Existence of more fundamental solutions

A generalized Reynolds equation is derived in the present paper by taking into account the effect of rotation in lubrication problems. The existence of certain fundamental solutions is shown in this extended framework which is not allowed in the classical Reynolds theory. Results concerning the pressure and the load capacity of the resulting bearing system are obtained and interpreted in the respective cases when the film thickness is a linear or an exponential functions of the coordinate along the bearing length. One of the important results is that while the load capacity decreases with increasing values of α for an exponentially inclined slider in the classical Reynolds theory, it increases with increasing values of α in the present context.     

Exact solutions of functionally graded piezoelectric material sandwich cylinders by a modified Pagano method

The three-dimensional (3D) coupled analysis of simply-supported, functionally graded piezoelectric material (FGPM) circular hollow sandwich cylinders under electro-mechanical loads is presented. The material properties of each FGPM layer are regarded as heterogeneous through the thickness coordinate, and obey an exponent-law dependent on this. The Pagano method is modified to be feasible for the study of FGPM sandwich cylinders. The modifications are as follows: a displacement-based formulation is replaced by a mixed formulation; a set of the complex-valued solutions of the system equations is transferred to the corresponding set of real-valued solutions; a successive approximation method is adopted to approximately transform each FGPM layer into a multilayered piezoelectric one with an equal and small thickness for each layer in comparison with the mid-surface radius, and with the homogeneous material properties determined in an average thickness sense; and a transfer matrix method is developed, so that the general solutions of the system equations can be obtained layer-by-layer, which is significantly less time-consuming than the usual approach. A parametric study is undertaken of the influence of the aspect ratio, open- and closed-circuit surface conditions, and material-property gradient index on the assorted field variables induced in the FGPM sandwich cylinders.     

A new method for two dimensional hyperbolic problems in semi‐unbounded irregular domains

Anewmethod called the full rational differential quadrature method is presented to deal with two dimensional linear and nonlinear hyperbolic problems in semi‐unbounded irregular domains. The spacial and temporal discretizations are both implemented by the rational differential quadrature method (RDQM). The RDQM, proven to be A‐stable (Chen and Tanaka, Comput Mech 28 (2002) 331–338) in the temporal discretization, is much more efficient than the finite difference schemes widely used in earlier works. In addition, the irregular boundary conditions are treated by the direct expansion method(DEM). Numerical experiments show that the present method is of high efficiency and easy to implement. © 2011Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Numerical method for a class of parabolic problems in composite media

This paper deals with a numerical method, viz. a Rothe Galerkin finite element method, for a class of parabolic problems in composite media with jump conditions for the unknown at the interfaces of the subdomains, retaining the continuity of its conormal derivative. Crucial to our approach is a nonstandard variational formulation in a product Sobolev space setting. We begin with the discretization in time by the Rothe method, which, in essence, involves a backward finite difference scheme and which provides a constructive method for proving existence of a (unique) variational exact solution under weak conditions for the data. We are left with a recurrent system of elliptic problems at each subsequent time point which are solved numerically by a Galerkin finite element method. The resulting algorithm may be implemented by adapting standard codes for parabolic problems in one-component domains, when suitably taking into account the specific transmission conditions at the internal boundaries. The emphasis of the paper is on convergence results and error estimates for both the fully discrete and the semi-discrete approximation of the exact solution. The effectiveness of the approach is illustrated by means of a 1D and a 2D example, the analytical solution of which is known. © 1993 John Wiley & Sons, Inc.     

On a numerical solution of the Cauchy problem for the Laplace equation by the fundamental solutions method

We discuss a numerical method to solve a Cauchy problem for the Laplace equation in the two-dimensional annular domain. We consider the case that the Cauchy data is given on an arc. We develop an approximation method based of the fundamental solutions method using the least squares method with Tikhonov regularization. The effectiveness of our method is examined by a numerical experiment. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems

In this paper, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of the solutions of the variational inequality problem by using a new hybrid method. We obtain a new result for finding a solution of an equilibrium problem and the solutions of the variational inequality problem.     

A fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array

In this paper, we propose a fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array. Our problem is mathematically a boundary value problem of the Stokes equation with periodic boundary conditions, to which it is difficult to give a good approximation by the ordinary fundamental solution method. Our method gives an approximate solution by a linear combination of the periodic fundamental solutions. In addition, we can compute the drag forces on the obstacles by using the data obtained in our method. Numerical examples for the problems of flows past spheres show the effectiveness of our method.     

Weak linear bilevel programming problems: existence of solutions via a penalty method

We are concerned with a class of weak linear bilevel programs with nonunique lower level solutions. For such problems, we give via an exact penalty method an existence theorem of solutions. Then, we propose an algorithm.     

Ensemble feature selection for high dimensional data: a new method and a comparative study

The curse of dimensionality is based on the fact that high dimensional data is often difficult to work with. A large number of features can increase the noise of the data and thus the error of a learning algorithm. Feature selection is a solution for such problems where there is a need to reduce the data dimensionality. Different feature selection algorithms may yield feature subsets that can be considered local optima in the space of feature subsets. Ensemble feature selection combines independent feature subsets and might give a better approximation to the optimal subset of features. We propose an ensemble feature selection approach based on feature selectors’ reliability assessment. It aims at providing a unique and stable feature selection without ignoring the predictive accuracy aspect. A classification algorithm is used as an evaluator to assign a confidence to features selected by ensemble members based on their associated classification performance. We compare our proposed approach to several existing techniques and to individual feature selection algorithms. Results show that our approach often improves classification performance and feature selection stability for high dimensional data sets.     

Analysis of a new parareal algorithm based on waveform relaxation method for time-periodic problems

We present a new parallel algorithm for time-periodic problems by combining the waveform relaxation method and the parareal algorithm, which performs the parallelism both in sub-systems and in time. In the new algorithm, the waveform relaxation propagator is chosen as a new fine propagator instead of the classical fine propagator. And because of the characteristic of time-periodic problems, the new parareal waveform relaxation algorithm needs to solve a periodic coarse problem at the coarse level in each iteration. The new algorithm is proved to converge linearly at most. Then the theoretic parallel efficiency of the new algorithm is also considered. Numerical experiments confirm our analysis finally.     

Infinite domain potential problems by a new formulation of singular boundary method

This study proposes a new formulation of singular boundary method (SBM) and documents the first attempt to apply this new method to infinite domain potential problems. The essential issue in the SBM-based methods is to evaluate the origin intensity factor. This paper derives a new regularization technique to evaluate the origin intensity factor on the Neumann boundary condition without the need of sample solution and nodes as in the traditional SBM. We also modify the inverse interpolation technique in the traditional SBM to get rid of the perplexing sample nodes in the calculation of the origin intensity factor on the Dirichlet boundary condition. It is noted that this new SBM retains all merits of the traditional SBM being truly meshless, free of integration, mathematically simple, and easy-to-program without the requirement of a fictitious boundary as in the method of fundamental solutions (MFS). We examine the new SBM by the four benchmark infinite domain problems to verify its applicability, stability, and accuracy.     

A smoothing-type Newton method for second-order cone programming problems based on a new smooth function

A new smoothing function similar with the well known Fischer-Burmeister function is given. Based on this new function, a smoothing-type Newton method is proposed for solving second-order cone programming. At each iteration, the proposed algorithm solves only one system of linear equations and performs only one line search. This algorithm can start from an arbitrary point and it is Q-quadratically convergent under a mild assumption. Preliminary numerical results demonstrate the effectiveness of the method.     

Some new solutions of the problems of wave propagation in a two-layer fluid

We present new analytic solutions of the problem of wave propagation in a continuously stratified fluid in the Boussinesq approximation. We study the propagation of internal waves in an ideal fluid in systems of homogeneous-layer/continuously stratified layer and homogeneous-layer/continuously stratified half-space type. We obtain the dispersion equations and study several limiting cases. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, Vol. 27, 1997, pp. 132–137.