The Saffman-Taylor problem for an extremely shear-thinning fluid

We consider a steady flow driven by pushing a finger of gasinto a highly shear-thinning power-law fluid, with exponentn, in a Hele-Shaw channel. We formulate the problem in termsof the streamfunction , which satisfies the p-Laplacian equation (with ), and investigate travelling wave solutions in the large-n (extreme shear-thinning) limit.We take a Legendre transform of the free-boundary problem for, which reduces it to a linear problem on a fixed domain. The solution to this problem is foundby using matched asymptotic expansions and the resulting shapeof the finger deduced (being, to leading order, a semi-infinitestrip). The nonlinear problem for the streamfunction is alsotreated using matched asymptotic expansion in the physical plane.The finger-width selection problem is briefly discussed in termsof our results.     

On a Functional Differential Equation

This paper considers some analytical and numerical aspects ofthe problem defined by an equation or systems of equations ofthe type (d/dt)y(t) = ay(t)+by(t), with a given initial conditiony(0) = 1. Series, integral representations and asymptotic expansions fory are obtained and discussed for various ranges of the parametersa, b and (> 0), and for all positive values of the argumentt. A perturbation solution is constructed for |1–| <<1, and confirmed by direct computation. For > 1 the solutionis not unique, and an analysis is included of the eigensolutionsfor which y(0) = 0. Two numerical methods are analysed and illustrated. The first,using finite differences, is applicable for < 1, and twotechniques are demonstrated for accelerating the convergenceof the finite-difference solution towards the true solution.The second, an adaptation of the Lanczos method, is applicablefor any > 0, though an error analysis is available onlyfor < 1. Numerical evidence suggests that for > 1 themethod still gives good approximations to some solution of theproblem.     

Asymptotic Solutions of a Model Diffusion-Reaction Equation

In this paper we study the large-time solution of the nonlineardiffusion reaction equation , m>1, p>0 subject to u(0, t)= and initial data with finitesupport. If we regard the steady state as the leading term inan asymptotic expansion of the solution as t then we show thatthis expansion is non-uniform in x. The nature of the non-uniformity,located at the moving interface, is shown to depend cruciallyon the location in (p, m) parameter space. For p<1<m weconstruct a uniformly valid solution using strained coordinates.In the remaining regions uniform zeroth-order composite solutionsare constructed via matched expansions.     

Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods

This paper derives a general procedure to produce an asymptotic expansion for eigenvalues of the Stokes problem by mixed finite elements. By means of integral expansion technique, the asymptotic error expansions for the approximations of the Stokes eigenvalue problem by Bernadi–Raugel element and _Q2-_P1$Q_2-P_1$ element are given. Based on such expansions, the extrapolation technique is applied to improve the accuracy of the approximations.     

On the Convergence of Sequences of Rational Approximations to Analytic Functions of a Certain Class

Complex analytic methods based on the theory of Walsh (1935)and on properties of orthonormal polynomials with general weight-functionon [–1, 1] are applied to the construction of variousrational approximations on the interval [–k, k] to a functiong(x) defined by or by where Remainder estimates are obtained, and from these, in the caseswhere g(x) is real on [–k, k], an asymptotic formula isobtained for the maximum error of the best rational approximationin the sense of the uniform norm. It is also shown that therate of convergence of the sequence of best approximations ofdegree n is twice the minimum rate predicted by Walsh's theory,in the sense that the degree of the approximation required fora given precision is approximately only half as great. Graphs are shown which illustrate that, for a simple example,the remainder estimates on which this asymptotic formula isbased are remarkably accurate even for approximations of lowdegree.     

Stirling Number Asymptotics from Recursion Equations Using the Ray Method

A technique for computing asymptotic expansions of combinatorial quantities from their recursion relations is presented. It is applied to the Stirling numbers of the first and second kinds, s(n, k) and S(n, k), for n ? 1 and three ranges of k: (i) k = O(1), (ii) n ? k = O(1), (iii) k ? 1, 0 < k / n < 1, The technique uses asymptotic methods of applied mathematics such as the ray method and the method of matched asymptotic expansions.     

Arithmetic properties of Apery numbers

Let (A_n)_n1 be the sequence of Apéry numbers with a generalterm given by . In thispaper, we prove that both the inequalities (A_n) > c₀ loglog log n and P(A_n) > c₀ (log n log log n)^1/2 hold fora set of positive integers n of asymptotic density 1. Here,(m) is the number of distinct prime factors of m, P(m) is thelargest prime factor of m and c₀ > 0 is an absolute constant.The method applies to more general sequences satisfying botha linear recurrence of order 2 with polynomial coefficientsand certain Lucas-type congruences.     

A Method for Incorporating Transcendentally Small Terms into the Method of Matched Asymptotic Expansions

The essential ideas behind a method for incorporating exponentially small terms into the method of matched asymptotic expansions are demonstrated using an Ackerberg–O'Malley resonance problem and a spurious solutions problem of Carrier and Pearson. One begins with the application of the standard method of matched asymptotic expansions to obtain at least the leading terms in outer and inner (Poincaré-type) expansions; some, although not all, matching can be carried out at this stage. This is followed by the introduction of supplementary expansions whose gauge functions are transcendentally small compared to those in the standard expansions. Analysis of terms in these expansions allows the matching to be completed. Furthermore, the method allows for the inclusion of globally valid transcendentally small contributions to the asymptotic solution; it is well known that such terms may be numerically significant.     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

Composite Approximations to the Solutions of the Orr-Sommerfeld Equation

The method of matched asymptotic expansions is used to derive composite approximations to the solutions of the Orr-Sommerfeld equation which satisfy Olver's completeness requirement. It is shown that the inner expansions can be obtained to all orders in terms of a certain class of generalized Airy functions, and these expansions are then used to derive approximations to the connection formulae. Because of the linearity of the problem it is possible and convenient to fix the normalization of the inner and outer expansions separately and then to relate them through the central matching coefficients. The Stokes multipliers can then be expressed in terms of the central matching coefficients and the coefficients which appear in the connection formulae. Once the inner and outer expansions have been matched they can be combined, if desired, to form composite approximations of either the additive or multiplicative type. For example, the ‘modified’ viscous solutions of Tollmien emerge in a natural way as first-order composite approximations obtained by multiplicative composition; similarly, the form of the ‘viscous correction’ to the singular inviscid solutions which I conjectured some years ago emerges as a first-order additive composite approximation. Because of the completeness requirement, however, these composite approximations are valid only in certain wedge shaped domains; approximations which are valid in the complementary sectors can then be obtained by the use of the connections formulae. The theory thus provides a relatively simple and explicit method of obtaining higher approximations, and its structure permits a direct comparison of the present results not only with the older heuristic theories but also with the comparison equation method.     

On the Convergence Rate of Generalized Fourier Expansions

We consider the problem of estimating the generalized Fouriercoefficients b_nin an expansion of the type . We give a simple method of obtaining a priori estimates of b_nandpresent a detailed analysis of the convergence obtained withseveral frequently employed orthogonal expansion sets. The resultsdepend only on the general analytic structure of the functionf(x), and are relevant to a recent discussion of the convergenceof variational calculations (Delves & Mead, 1971). A simpleextension of the results also gives estimates of the pointwiseconvergence of the expansions.     

The problem of differentiating an asymptotic expansion in real powers. Part II: Factorizational theory

In Part II of our work we approach the problem discussed in Part I from the new viewpoint of canonical factorizations of a certain nth order differential operator L. The main results include:

1. characterizations of the set of relations $$ f^{(k)} (x) = P^{(k)} (x) + o^{(k)} (x^{\alpha _n - k} ),x \to + \infty ,0 \leqslant k \leqslant n - 1, $$ where $$ P(x) = a_1 x^{\alpha _1 } + \cdots + a_n x^{\alpha _n } and \alpha _1 > \alpha _2 > \cdots > \alpha _n , $$ by means of suitable integral conditions
2. formal differentiation of a real-power asymptotic expansion under a Tauberian condition involving the order of growth of L
3. remarkable properties of asymptotic expansions of generalized convex functions.

     

On Some Steady-state Moving Boundary Problems in the Linear Theory of Viscoelasticity

Three steady-state moving boundary problems in the linear theoryof viscoelasticity are considered when the inertia terms aretaken into account. Attention is directed to the dimensionlessparameter = V/L where V is the speed of the boundary, is therelaxation time of the medium, and L is a length associatedwith the geometry of the problem. When is small the problemsare recognized as having a singular perturbation character andthe method of matched asymptotic expansions is used. The problemsare (i) a semi-infinite crack moving steadily in a clamped viscoelasticstrip, (ii) a finite length crack moving in an infinite viscoelasticmedium, and (iii) a steady rolling cylinder on a visco-elastichalf-space.     

Local Similarity Solutions for the Initial-value Problem in Non-linear Diffusion

The paper considers the initial-value problem for the equation with ß>0. For initial data with finite supportthe interfaces where v is zero may move immediately or afterthe lapse of a finite time. The structure of the solution, inthe cases where this is governed by local conditions on theinitial data, is described by local similarity solutions whichin the appropriate limit can be matched to the global solution.     

Merging asymptotic expansions for cooperative gamblers in generalized St. Petersburg games

Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any one of n cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic distributions for the total winnings of the n players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations even for very small n.     

On a Problem of Brocard

It is proved that, if P is a polynomial with integer coefficients,having degree 2, and 1 > > 0, then n(n – 1) ...(n – k + 1) = P(m) has only finitely many natural solutions(m,n,k), n k > n, provided that the abc conjecture is assumedto hold under Szpiro's formulation. 2000 Mathematics SubjectClassification 11D75, 11J25, 11N13.     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

Two Examples Concerning Martingales in Banach Spaces

The analytic concepts of martingale type p and cotype q of aBanach space have an intimate relation with the geometric conceptsof p-concavity and q-convexity of the space under consideration,as shown by pisier. In particular, for a banach space X, havingmartingale type p for some p > 1 implies that X has martingalecotype q for some q < . The generalisation of these concepts to linear operators wasstudied by the author, and it turns out that the duality aboveonly holds in a weaker form. An example is constructed showingthat this duality result is best possible. So-called random martingale unconditionality estimates, introducedby Garling as a decoupling of the unconditional martingale differences(UMD) inequality, are also examined. It is shown that the random martingale unconditionality constantof for martingales of length n asymptotically behaves like n. This improves previous estimatesby Geiss, who needed martingales of length 2ⁿ to show this asymptotic.At the same time the order in the paper is the best that canbe expected.     

Eliminating Indeterminacy in Singularly Perturbed Boundary Value Problems with Translation Invariant Potentials

Solutions exhibiting an internal layer structure are constructed for a class of nonlinear singularly perturbed boundary value problems with translation invariant potentials. For these problems, a routine application of the method of matched asymptotic expansions fails to determine the locations of the internal layer positions. To overcome this difficulty, we present an analytical method that is motivated by the work of Kath, Knessl and Matkowsky [4]. To construct a solution having n internal layers, we first linearize the boundary value problem about the composite expansion provided by the method of matched asymptotic expansions. The eigenvalue problem associated with the homogeneous form of this linearization is shown to have n exponentially small eigenvalues. The condition that the solution to the linearized problem has no component in the subspace spanned by the eigenfunctions corresponding to these exponentially small eigenvalues determines the internal layer positions. These “near” solvability conditions yield algebraic equations for the internal layer positions, which are analyzed for various classes of nonlinearities.     

Properties of the Approximations of a Matrix which Lower its Rank

Let C be a real m ? n matrix. This paper considers the generalproperties of the best approximations E to C in an arbitrarynorm by matrices of rank r. The case in which rank C = n, r= n – 1, and ||?|| is a separable norm is dealt with separately.For this particular case, the form of every best approximationE and an estimate of the error ||C – E|| are given. Thepaper also investigates when, for this particular case, thebest approximation E is unique.