在局部区域上的奇摄动非线性方程Robin边值问题解的渐近性态

本文是讨论了一类在局部区域上的奇摄动非线性方程Ｒobin边值问题,利用泛函数分析及算子理论,得到了相应问题解的渐近性态。     

The effect of the slip condition on flows of an Oldroyd 6-constant fluid

The steady flows of a non-Newtonian fluid are considered when the slippage between the plate and the fluid is valid. The constitutive equations of the fluid are modeled by those for an Oldroyd 6-constant fluid. They give rise to non-linear boundary value problems. The analytical solutions are obtained using powerful, easy-to-use analytic technique for non-linear problems, the homotopy analysis method. It is shown that solutions exist for all values of non-Newtonian parameters. The solutions valid for no-slip condition for all values of non-Newtonian parameters can be derived as the special cases of the present analysis. Finally, graphs are plotted and critical assessment is made for the cases of slip and no-slip conditions.     

在部分区域上的奇摄动反应扩散方程初始边值问题解的渐近性态

讨论一类在部分区域上的奇摄动反应扩散方程初始边值问题,利用算子理论,得到了相应问题解的渐近性态。     

Non‐perturbative solution of three‐dimensional Navier–Stokes equations for the flow near an infinite rotating disk

In this paper, we present Homotopy perturbation method (HPM) and Padé technique, for finding non‐perturbative solution of three‐dimensional viscous flow near an infinite rotating disk. We compared our solution with the numerical solution (fourth‐order Runge–Kutta). The results show that the HPM–Padé technique is an appropriate method in solving the systems of nonlinear equations. The mathematical technique employed in this paper is significant in studying some other problems of engineering. Copyright © 2009 John Wiley & Sons, Ltd.     

A short note on the paper “Convergence of the TAGE iterative method for the system arisen from the cubic spline approximation for the solution of two-point BVPs with forcing function in integral form”, by Mohanty, Jain and Dhall

In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036-3047] and then correct it.     

粘性阻尼线性振动系统的复模态特征值问题的一种新的矩阵摄动分析解法

本文提出了对粘性阻尼线性振动系统的复模态二次广义特征值问题进行高效近似求解的一种新的矩阵摄动分析方法,即先将阻尼矩阵分解为比例阻尼部分和非比例阻尼部分之和,并求得系统的比例阻尼实模态特征解;然后以此为初始值,将阻尼矩阵的非比例部分作为对其比例部分的小量修改,利用摄动分析方法简捷地得到系统的复模态特征值问题的近似解.这一新方法适用于振系阻尼分布不十分偏离比例阻尼情况的问题,因此对大阻尼(非过阻尼)振动系统也有效.这是它优于以前提出的基于无阻尼实模态特征解的类似摄动分析方法的重要特点.文中建立了复模态特征值和特征向量的二阶摄动解式,并通过算例证实了其有效性.此外还讨论了利用比例阻尼假定估计阻尼系统固有振动的复特征值的可行性.     

Study on connections of the MFS, Trefftz method, indirect BIEM and invariant MFS in the three-dimensional Laplace problems containing spherical boundaries

Following the success of a study on the method of fundamental solutions using an image concept [13], we extend to solve the three-dimensional Laplace problems containing spherical boundaries by using the three approaches. The case of eccentric sphere for the Laplace problem is considered. The optimal locations for the source distribution to include the foci in the MFS are also examined by using the image concept in the 3D problems. Whether a free constant is required or not in the MFS is also studied. The error distribution is discussed after comparing with the analytical solution derived by using the bispherical coordinates. Besides, the relationship between the Trefftz bases and the singularity in the MFS for the three-dimensional Laplace problems is also addressed. It is found that one source of the MFS contains several interior and exterior Trefftz sets through a degenerate kernel. On the contrary, one single Trefftz base can be superimposed by some lumped sources in the MFS through an indirect BIEM. Based on this finding, the relationship between the fictitious boundary densities of the indirect BIEM and the singularity strength in the MFS can be constructed due to the fact that the MFS is a lumped version of an indirect BIEM.     

A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation

In this paper, we consider an inverse source problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine a space-dependent source term in the time-fractional diffusion equation from a noisy final data. Based on a series expression of the solution, we can transform the original inverse problem into a first kind integral equation. The uniqueness and a conditional stability for the space-dependent source term can be obtained. Further, we propose a modified quasi-boundary value regularization method to deal with the inverse source problem and obtain two kinds of convergence rates by using an a priori and an a posteriori regularization parameter choice rule, respectively. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.     

Bernstein Ritz‐Galerkin method for solving an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

In this article, the Ritz‐Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then Ritz‐Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

Wavelet method is a recently developed tool in applied mathematics. Investigation of various wavelet methods, for its capability of analyzing various dynamic phenomena through waves gained more and more attention in engineering research. Starting from ‘offering good solution to differential equations’ to capturing the nonlinearity in the data distribution, wavelets are used as appropriate tools at various places to provide good mathematical model for scientific phenomena, which are usually modeled through linear or nonlinear differential equations. Review shows that the wavelet method is efficient and powerful in solving wide class of linear and nonlinear reaction–diffusion equations. This review intends to provide the great utility of wavelets to science and engineering problems which owes its origin to 1919. Also, future scope and directions involved in developing wavelet algorithm for solving reaction–diffusion equations are addressed.     

The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier-Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε¹ as the viscosity ε tends to zero.     

A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation

This paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations. An algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed. Several illustrative examples, with linear and nonlinear fractional differential equations, are given to demonstrate the effectiveness of the method.     

Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time

We consider the inverse problem of finding the temperature distribution and the heat source whenever the temperatures at the initial time and the final time are given. The problem considered is one dimensional and the unknown heat source is supposed to be space dependent only. The existence and uniqueness results are proved.     

The fractional Tikhonov regularization method for simultaneous inversion of the source term and initial value in a space-fractional Allen-Cahn equation

In this paper, we consider the inverse problem for identifying the source term and initial value simultaneously in a space-fractional Allen-Cahn equation. This problem is ill-posed, i.e., the solution of this problem does not depend continuously on the data. The fractional Tikhonov method is used to solve this problem. Under the a priori and the a posteriori regularization parameter choice rules, the error estimates between the regularization solutions and the exact solutions are obtained, respectively. Different numerical examples are presented to illustrate the validity and effectiveness of our method.     

Harmonic and Refined Extraction Methods for the Singular Value Problem,with Applications in Least Squares Problems

For the accurate approximation of the minimal singular triple (singular value and left and right singular vector) of a large sparse matrix, we may use two separate search spaces, one for the left, and one for the right singular vector. In Lanczos bidiagonalization, for example, such search spaces are constructed. In SIAM J. Sci. Comput., 23(2) (2002), pp. 606–628, the author proposes a Jacobi–Davidson type method for the singular value problem, where solutions to certain correction equations are used to expand the search spaces. As noted in the mentioned paper, the standard Galerkin subspace extraction works well for the computation of large singular triples, but may lead to unsatisfactory approximations to small and interior triples. To overcome this problem for the smallest triples, we propose three harmonic and a refined approach. All methods are derived in a number of different ways. Some of these methods can also be applied when we are interested in the largest or interior singular triples. Theoretical results as well as numerical experiments indicate that the results of the alternative extraction processes are often better than the standard approach. We show that when Lanczos bidiagonalization is used for the subspace expansion, the standard, harmonic, and refined extraction methods are all essentially equivalent. This gives more insight in the success of Lanczos bidiagonalization to find the smallest singular triples. Finally, we show that the extraction processes for the smallest singular values may give an approximation to a least squares problem at low additional costs. The truncated SVD is also discussed in this context. AMS subject classification (2000) 65F15, 65F50, (65F35, 93E24).Submitted December 2002. Accepted October 2004. Communicated by Haesun Park.M. E. Hochstenbach: The research of this author was supported in part by NSF grant DMS-0405387. Part of this work was done when the author was at Utrecht University.     

Stochastic Models for the Amount of Overflow in a Finite Dam with Random Inputs,Random Outputs,and Exponential Release Policy

In this article, we discuss finite dam models to study the expected amount of overflow in a given time. The inputs into the dam are taken as random and there are two types of outputs—one is random and the other is deterministic which is proportional to the content of the dam. The master equation for the expected amount of overflow is an one dimensional equation with separable kernel. For this class of master equation, the integral equation for the expected amount of overflow has been transformed exactly into ordinary differential equation with variable coefficients. The imbedding method is used to study the expected amount of overflow in a given time without emptiness in this period. We also consider the model for the expected amount of overflow in a given time with any number of emptiness of the dam in this period. The results are derived in the form of a third order differential Equation for the Laplace transformation function for the expected overflow. The closed form analytical solutions are obtained in terms of beta functions and degenerate hyper-geometric functions of two variables.     

Monotone method for singular BVP in the presence of upper and lower solutions

In this paper we examine existence of monotone approximations of solutions of singular boundary value problem -(p(x)y^′(x))^′=q(x)f(x,y,py^′) for 0<x?b and lim_x→0⁺p(x)y^′(x)=0,α₁y(b)+β₁p(b)y^′(b)=γ₁. Under quite general conditions on f(x,y,py^′) we show that solution of the singular two point boundary value problem is unique. Here is allowed to have integrable singularity at x=0 and we do not assume .