A new application of the homotopy analysis method: Solving the Sturm–Liouville problems

In this paper, the homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourth-order Sturm–Liouville problems. These eigenvalues are calculated by starting the HAM algorithm with one initial guess. In this paper, it can be observed that the auxiliary parameter ℏ, which controls the convergence of the HAM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more important qualitative difference in analysis between HAM and other methods.     

Locally Optimal Block Preconditioned Conjugate Gradient Method for Hierarchical Matrices

We present a method of almost linear complexity to approximate some (inner) eigenvalues of symmetric self-adjoint integral or differential operators. Using ℋ-arithmetic the discretisation of the operator leads to a large hierarchical (ℋ-) matrix M. We assume that M is symmetric, positive definite. Then we compute the smallest eigenvalues by the locally optimal block preconditioned conjugate gradient method (LOBPCG), which has been extensively investigated by Knyazev and Neymeyr. Hierarchical matrices were introduced by W. Hackbusch in 1998. They are data-sparse and require only O(nlog₂ n) storage. There is an approximative inverse, besides other matrix operations, within the set of ℋ-matrices, which can be computed in linear-polylogarithmic complexity. We will use the approximative inverse as preconditioner in the LOBPCG method. Further we combine the LOBPCG method with the folded spectrum method to compute inner eigenvalues of M. This is equivalent to the application of LOBPCG to the matrix M_μ = (M − μI)² . The matrix M_μ is symmetric, positive definite, too. Numerical experiments illustrate the behavior of the suggested approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the L ¹-estimate between the entropy solution and the geometric optics expansion function is bounded by O(?²), independent of the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.     

Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree

In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic.     

Continuity in weak topology: First order linear systems of ODE

In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L^p topologies, but also with respect to the weak topologies of the Lp spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way.     

ON MULTI-DIMENSIONAL SONIC-SUBSONIC FLOW

In this paper, a compensated compactness framework is established for sonicsubsonic approximate solutions to the n-dimensional (n ≥ 2) Euler equations for steady irrotational flow that may contain stagnation points. This compactness framework holds provided that the approximate solutions are uniformly bounded and satisfy H 1 loc (Ω) compactness conditions. As illustration, we show the existence of sonic-subsonic weak solution to n-dimensional (n ≥ 2) Euler equations for steady irrotational flow past obstacles or through an infinitely long nozzle. This is the first result concerning the sonic-subsonic limit for n-dimension (n ≥ 3).     

A divide and conquer method for unitary and orthogonal eigenproblems

Summary LetH ^{n xn} be a unitary upper Hessenberg matrix whose eigenvalues, and possibly also eigenvectors, are to be determined. We describe how this eigenproblem can be solved by a divide and conquer method, in which the matrixH is split into two smaller unitary upper Hessenberg matricesH ₁ andH ₂ by a rank-one modification ofH. The eigenproblems forH ₁ andH ₂ can be solved independently, and the solutions of these smaller eigenproblems define a rational function, whose zeros on the unit circle are the eigenvalues ofH. The eigenvector ofH can be determined from the eigenvalues ofH and the eigenvectors ofH ₁ andH ₂. The outlined splitting of unitary upper Hessenberg matrices into smaller such matrices is carried out recursively. This gives rise to a divide and conquer method that is suitable for implementation on a parallel computer.WhenH ^{n xn} is orthogonal, the divide and conquer scheme simplifies and is described separately. Our interest in the orthogonal eigenproblem stems from applications in signal processing. Numerical examples for the orthogonal eigenproblem conclude the paper.Research supported in part by the NSF under Grant DMS-8704196 and by funds administered by the Naval Postgraduate School Research Council     

Waiting-Time Behavior in a Nonlinear Diffusion Equation

We study the nonlinear diffusion equation u_t*=(uⁿu_x)_x, which occurs in the study of a number of problems. Using singular-perturbation techniques, we construct approximate solutions of this equation in the limit of small n. These approximate solutions reveal simply the consequences of this variable diffusion coefficient, such as the finite propagation speed of interfaces and waiting-time behavior (when interfaces wait a finite time before beginning to move), and allow us to extend previous results for this equation.     

Separation of Eigenvalues of the Wave Equation for the Unit Ball in RN

It is shown that π is the infinium gap between the consecutive square roots of the eigenvalues of the wave equation in a hypespherical domain for both the Neumann (free) and the full range of mixed (elastic) homogeneous boundary conditions. Previous literature contains the same information apparently only for the Dirichlet (fixed) boundary condition. These square roots of the eigenvalues are the zeros of solutions of a differential equation in Bessel functions (first kind) and their first derivatives. The infinium gap is uniform for Bessel functions of orders x ≥ ½ (as well as for x = 0). The intervals between the roots are actually monotone decreasing in length. These results are obtained by interlacing zeros of Bessel and associated functions and comparing their relative displacements with oscillation theory. If W_l denotes the lth positive root for some fixed order x, the minimum gap property assures that {exp(±iw_lt|l = 1, 2,...} form a Riesz basis in L²(0, τ) for τ > 2. This has application to the problem of controlling solutions of the wave equation by controlling the boundary values.     

Homotopy Analysis Method for Solving(2+1)-dimensional Navier-Stokes Equations with Perturbation Terms

In this paper Homotopy Analysis Method (HAM) is implemented for obtaining approximate solutions of (2+1)-dimensional Navier-Stokes equations with perturbation terms. The initial approximations are obtained using linear systems of the Navier-Stokes equations;by the iterations formula of HAM,the first approxima-tion solutions and the second approximation solutions are successively obtained and Homotopy Perturbation Method(HPM)is also used to solve these equations;finally, approximate solutions by HAM of (2+1)-dimensional Navier-Stokes equations with-out perturbation terms and with perturbation terms are compared. Because of the freedom of choice the auxiliary parameter of HAM,the results demonstrate that the rapid convergence and the high accuracy of the HAM in solving Navier-Stokes equa-tions;due to the effects of perturbation terms,the 3rd-order approximation solutions by HAM and HPM have great fluctuation.     

Greek ladders via linear algebra

Theon's ladder is an ancient method for easily approximating nth roots of a real number k. Previous work in this area has focused on modifying Theon's ladder to approximate roots of quadratic polynomials. We extend this work using techniques from linear algebra. We will show that a ladder associated to the quadratic polynomial ax ² + bx + c can be adjusted to approximate either root. Other situations such as quadratics with no real roots and corresponding matrices with complex eigenvalues are also addressed.     

Best Lp approximate solutions of nonlinear integrodifferential equations

In this paper best L_p approximate solutions are shown to exist for a wide class of integrodifferential equations. Using approximation theory techniques, a local existence theorem for solutions is established, and the convergence of the best approximate solutions to a solution is shown.     

A note on harmonic Ritz values and their reciprocals

This note summarizes an investigation of harmonic Ritz values to approximate the interior eigenvalues of a real symmetric matrix A while avoiding the explicit use of the inverse A^?1. We consider a bounded functional ψ that yields the reciprocals of the harmonic Ritz values of a symmetric matrix A. The crucial observation is that with an appropriate residual s, many results from Rayleigh quotient and Rayleigh–Ritz theory naturally extend. The same is true for the generalization to matrix pencils (A, B) when B is symmetric positive definite. These observations have an application in the computation of eigenvalues in the interior of the spectrum of a large sparse matrix. The minimum and maximum of ψ correspond to the eigenpairs just to the left and right of zero (or a chosen shift). As a spectral transformation, this distinguishes ψ from the original harmonic approach where an interior eigenvalue remains at the interior of the transformed spectrum. As a consequence, ψ is a very attractive vehicle for a matrix‐free, optimization‐based eigensolver. Instead of computing the smallest/largest eigenvalues by minimizing/maximizing the Rayleigh quotient, one can compute interior eigenvalues as the minimum/maximum of ψ. Copyright © 2009 John Wiley & Sons, Ltd.     

Analytical approximation for laminar film condensation of saturated stream on an isothermal vertical plate

In this paper, the laminar film condensation of saturated stream on an isothermal vertical plate is studied. The boundary layer equations of momentum and thermal energy are reduced to two ordinary differential equations by means of a set of similarity transformations. The problem is then solved analytically using the homotopy analysis method (HAM). The dual solutions are obtained for a range of values of the parameter η_δ. However, it should be noted that the second branch solution of the considered problem has only mathematical meanings. The present work shows the validity and the great potentiality of the proposed technique for the nonlinear problems with multiple solutions.     

Eigenvalues of a symmetric tridiagonal matrix: A divide-and-conquer approach

Summary The Symmetric Tridiagonal Eigenproblem has been the topic of some recent work. Many methods have been advanced for the computation of the eigenvalues of such a matrix. In this paper, we present a divide-and-conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial. The problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem. We give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation. Furthermore, this set of recursions can be implemented on a regulartree structure. If the concurrency exhibited by this algorithm is exploited, it can be shown that thetime for computation of all the eigenvalues becomesO(nlogn) instead ofO(n ²) as is the case for the approach where the order is increased by only one at every step. We address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation.     

Sums of reciprocal Stekloff eigenvalues

Let 0 = λ₁ < λ₂ ≤ λ₃ ≤ … be the Stekloff eigenvalues of a plane domain. The paper is concerned with formulas for ∑^∞₂λ^(–2)_j in simply and doubly connected domains. In the simply connected case it is proven that the disk minimized this sum. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A New Algorithm for the Symmetric Tridiagonal Eigenvalue Problem

We apply a novel approach to approximate within ϵ to all the eigenvalues of an n × n symmetric tridiagonal matrix A using at most n²([3 log₂(625n⁶)] + (83n − 34)[log₂ (log₂((λ₁ − λ_n)/(2ϵ))/log₂(25_n))]) arithmetic operations where λ₁ and λ_n denote the extremal eigenvalues of A. The algorithm can be modified to compute any fixed numbers of the largest and the smallest eigenvalues of A and may also be applied to the band symmetric matrices without their reduction to the tridiagonal form.     

The preconditioned inverse iteration for hierarchical matrices

The preconditioned inverse iteration is an efficient method to compute the smallest eigenpair of a symmetric positive definite matrix M. Here we use this method to find the smallest eigenvalues of a hierarchical matrix. The storage complexity of the data‐sparse ‐matrices is almost linear. We use ‐arithmetic to precondition with an approximate inverse of M or an approximate Cholesky decomposition of M. In general, ‐arithmetic is of linear‐polylogarithmic complexity, so the computation of one eigenvalue is cheap. We extend the ideas to the computation of inner eigenvalues by computing an invariant subspace S of (M ? μI)² by subspace preconditioned inverse iteration. The eigenvalues of the generalized matrix Rayleigh quotient μ_M(S) are the desired inner eigenvalues of M. The idea of using (M ? μI)² instead of M is known as the folded spectrum method. As we rely on the positive definiteness of the shifted matrix, we cannot simply apply shifted inverse iteration therefor. Numerical results substantiate the convergence properties and show that the computation of the eigenvalues is superior to existing algorithms for non‐sparse matrices.Copyright © 2012 John Wiley & Sons, Ltd.     

APPROXIMATION OF A KIND OF NEVAI-DURRMEYER OPERATORS IN Lω^p SPACES

The present paper introduces a kind of Nevai-Durrmeyer operators which can be used to approximate functions in Lω^p, spaces with the weight ω（x）=1/√（1-x^2） and the approximate rate is also estimated.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.