R. S. Anderssen

On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems

The use of algebraic eigenvalues to approximate the eigenvalues of Sturm-Liouville operators is known to be satisfactory only when approximations to the fundamental and the first few harmonics are required. In this paper, we show how the asymptotic error associated with related but simpler Sturm-Liouville operators can be used to correct certain classes of algebraic eigenvalues to yield uniformly valid approximations.ZusammenfassungDie Benutzung algebraischer Eigenwerte zur näherungsweisen Berechnung der Eigenwerte von Sturm-Liouville-Operatoren ist bekanntlich nur für die Grundschwingung und einige weitere Harmonische zufriedenstellend. In dieser Arbeit zeigen wir, wie man den asymptotischen Fehler, der bei verwandten aber einfachen Sturm-Liouville-Operatoren auftritt, dazu benutzen kann, um gewisse Klassen algebraischer Eigenwerte so zu korrigieren, daß die gleichmäßig gute Approximationen liefern.

Sampling localization in determining the relaxation spectrum

Abstract It is widely believed that the storage and loss moduli over the frequency range ω min ω ω max yield information about the relaxation spectrum over the range of relaxation times ( ω max ) −1 τ ω min ) −1 ; i.e. the reciprocal frequency range. This rule-of-thumb is often used to discard estimates of the relaxation spectrum provided by indirect methods outside the reciprocal frequency range of the data. We will show, however, that this rule is imprecise, and that the relaxation spectrum is determined on a shorter interval of relaxation times than the reciprocal frequency range. This is formalized in the paper as sampling localization.

For numerical differentiation, dimensionality can be a blessing!

Finite difference methods, such as the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial differential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly amplified in the differentiation process, especially if small step-sizes are chosen and higher derivatives are required. A number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data. The size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h. In this paper, it is initially shown how first (and higher) order single-variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one-dimensional data. The result is then extended to the multivariate differentiation of higher dimensional data. The nature of the trade-off between convergence and stability is explicitly characterized, and the complexity of various implementations is examined.

A product integration method for a class of singular first kind Volterra equations

SummaryConvergence of a midpoint product integration method for singular first kind Volterra equations with kernels of the formk(t, s)(t−s)−α, 0<α<1, wherek(t, s) is continuous, is examined. It is shown that convergence of order one holds if the solution of the Volterra equation has a Lipschitz continuous first derivative andk(t, s) is suitably smooth. In addition, convergence is shown to hold when the solution has only Lipschitz continuity and the same conditions onk(t, s) apply. An existence theorem of Kowalewski is used to relate these conditions on the solution to conditions on the data andk(t, s).

On the correction of finite difference eigenvalue approximations for sturm-liouville problems with general boundary conditions

When finite difference and finite element methods are used to approximate continuous (differential) eigenvalue problems, the resulting algebraic eigenvalues only yield accurate estimates for the fundamental and first few harmonics. One way around this difficulty would be to estimate the error between the differential and algebraic eigenvalues by some independent procedure and then use it to correct the algebraic eigenvalues. Such an estimate has been derived by Paine, de Hoog and Anderssen for the Liouville normal form with Dirichlet boundary conditions. In this paper, we extend their result to the Liouville normal form with general boundary conditions.

Theoretical derivation of molecular weight scaling for rheological parameters

Abstract Recently, a method has been established to determine moments (functionals) of the molecular weight distribution (MWD) of a given polymer directly from measurements of the linear viscoelastic relaxation modulus of that polymer. In part, the need to compute such quantities (functionals) is motivated by the experimentally observed scaling of rheological properties of polymers with respect to moments of their MWD. Although various authors have advanced different ad hoc arguments to derive various molecular weight scaling results for a variety of rheological parameters, such as the zero-shear viscosity, no formal procedure for deriving molecular weight scaling for rheological parameters has been proposed. In this paper, a natural parametric generalization of the reptation based mixing rules is introduced which includes single and double reptation as special cases. For this generalization, it is shown, by invoking the mean value theorem for integrals, how to formalize the derivation of molecular weight scaling for rheological parameters. In particular, from the point of view of choosing practical mixing rules, this paper establishes that when the relaxation function is characterized by a single time constant, the molecular weight scaling is independent of the standard linear mixing rules.

Finite difference methods for the numerical differentiation of non-exact data

In this paper, we derive results about the numerical performance of multi-point (moving average) finite difference formulas for the differentiation of non-exact data. In particular, we show that multi-point differentiators can be constructed which are asymptotically unbiased and have a bounded amplification factor as the steplength decreases and the number of points increases.ZusammenfassungIn dieser Arbeit werden Ergebnisse über die numerische Güte von Mehrpunktdifferenzenformeln für die Differentation empirischer Funktionen hergeleitet. Insbesondere wird gezeigt, daß Mehrpunktdifferenzenoperatoren konstruiert werden können, die asymptotisch verzerrungsfrei sind und einen für abnehmende Schrittweite und zunehmende Punkteanzahl beschränkten Amplifikationsfaktor besitzen.

Sampling localization and duality algorithms in practice

Abstract In a recent paper, Anderssen and Davies [Simple moving-average formulae for direct recovery of the relaxation spectrum, Mathematics Research Report MRR 016-98, Centre for Mathematics and its Applications, The Australian National University] have derived moving-average formulae which can be applied to oscillatory shear data to recover estimates of the relaxation spectrum of the viscoelastic material tested. These moving-average formulae represent an improvement over commercial packages currently available, for two reasons. First, they take the limits imposed by sampling localization in determining the relaxation spectrum fully into account. Secondly, to within finite resolution, these formulae yield accurate relaxation spectra in a fraction of a second on a PC. Anderssen and Davies have also indicated that their formulae are best employed within an iterative algorithm which exploits the natural duality between storage and loss moduli. The purpose of this paper is to pursue this natural duality further, and present a class of fast algorithm accessible to the experimentalist. Their performance when applied to noisy data is described. Their success is attributed to the implicit duality constraints imposed through sampling localization and the Kramers–Kronig relations, and to the nature of the regularization imposed.

Rheological implications of completely monotone fading memory

In the constitutive equation modeling of a (linear) viscoelastic material, the “fading memory” of the relaxation modulus G(t) is a fundamental concept that dates back to Boltzmann [Ann. Phys. Chem. 7, 624 (1876)]. There have been various proposals that range from the experimental and pragmatic to the theoretical about how fading memory should be defined. However, if, as is common in the rheological literature, one assumes that G(t) has the following relaxation spectrum representation: G(t)=∫0∞ exp(−t/τ)[H(τ)/τ]dτ, t > 0, then it follows automatically that G(t) is a completely monotone function. Such functions have quite deep mathematical properties, that, in a rheological context, spawn interesting and novel implications. For example, because the set of completely monotone functions is closed under positive linear combinations and products, it follows that the dynamics of a linear viscoelastic material, under appropriate stress–strain stimuli, will involve a simultaneous mixture of different molecular int...

COMPLETELY MONOTONE FADING MEMORY RELAXATION MODULII

J—oo which defines how the stress a(t) at time t depends on the earlier history of the shear rate j(r) via the relaxation modulus (kernel) G(t). Physical reality is achieved by requiring that the form of the relaxation modulus G(t) gives the Boltzmann equation fading memory, so that changes in the distant past have less effect now than the same changes in the more recent past. A popular choice, though others have previously been proposed and investigated, is the assumption that G(t) be a completely monotone function. This assumption has much deeper ramifications than have been identified, discussed or exploited in the rheological literature. The purpose of this paper is to review the key mathematical properties of completely monotone functions, and to illustrate how these properties impact on the theory and application of linear viscoelasticity and polymer dynamics. A more general representation of a completely monotone function, known in the mathematical literature, but not the rheological, is formulated and discussed. This representation is used to derive new rheological relationships. In particular, explicit inversion formulas are derived for the relationships that are obtained when the relaxation spectrum model and a mixing rule are linked through a common relaxation modulus.