Uniform Asymptotic Expansion Research Articles

On the method of modified equations. VI: Asymptotic analysis of and asymptotic successive-corrections techniques for two-point, boundary-value problems in ODE's

The modified equation technique is extended to two-point, boundary-value problems, and a second-order accurate, implicit, centered, finite difference scheme for non-homogeneous, second-order, ordinary differential equations with linear boundary conditions is analyzed. The first, second and third modified equations, or equivalent, second equivalent and (simply) modified equations, respectively, for this scheme and its boundary conditions are presented. It is shown that the three kinds of modified equations are asymptotically equivalent when the equivalent equation is used for the boundary conditions, since an asymptotic analysis of these equations with the grid size as small parameter yields exactly the same results. For a linear problem, multiple scales and summed-up asymptotic techniques are used and the resulting uniform asymptotic expansions are shown to be equivalent to the solution of the original finite difference scheme. Asymptotic successive-corrections techniques are also applied to the three kinds of modified equations to obtain higher-order schemes. Higher-order boundary conditions are easily treated in the asymptotic successive-corrections technique, although these boundary conditions must be obtained by using the equivalent equation in order to obtain a correct estimate of the global error near the domain boundaries. The methods introduced in this paper are applied to homogeneous and non-homogeneous, second-order, linear and non-linear, ordinary differential equations, and yield very accurate results.

On the structure of the solution of singularly perturbed initial boundary value problems with an unbounded spectrum of the limit operator

Singularly perturbed initial boundary value problems are studied for some classes of linear systems of ordinary differential equations on the semiaxis with an unbounded spectrum of the limit operator. We give a new version of the proof of the existence of a unique and bounded (as e→+0) solution for which with the help of the splitting method we construct a uniform asymptotic expansion on the entire semiaxis and describe all singularities (reflecting the structure of the corresponding boundary layers) in closed analytic form, including the critical case in which the points of the spectrum of the limit operator can touch the imaginary axis; this supplements previous results.

On the radiation of ultrasound into an isotropic elastic half-space via wavefront expansions of the impulse response

The problem of propagation of pulses in the radiating near zone of a large circular normal transducer directly coupled to a homogeneous and isotropic elastic half-space is re-visited. It is shown that for certain observation angles the impulse response approach is computationally inefficient. A new method based on the so-called wavefront expansions of the impulse response is developed instead. The expansions are obtained by the analytical harmonic synthesis of the high-frequency asymptotics of the transducer field. Unlike these asymptotics the wavefront expansions are expressed in terms of elementary functions only. The direct P, edge P and S waves as well as the transition regions (penumbra and axial region) are described. The uniform asymptotic expansions applicable throughout the radiating near zone are derived as well. The code based on the time convolution of the pressure input function with the wavefront expansions is compared to a direct numerical code. It is thousands of times faster but practically just as accurate except that the phenomena related to the head waves are not described. Formulas pertaining to the far field are also offered.

Leakage fields from planar semi-infinite transmission lines

The leakage fields radiated by planar semi-infinite transmission-line currents are studied for two canonical structures: the air-gap stripline and the planar strip of current in free space. The first structure represents a general class of printed-circuit structures where leakage occurs into a substrate mode. The second structure is representative of structures for which leakage occurs primarily into space. It is found that the transverse behavior of the leakage fields for a semi-infinite line differs in significant ways from that expected from an infinite line, a result that is at first surprising, but is understandable on physical grounds, and which is explained by the analysis presented here. The exact leakage fields are compared with two asymptotic approximations: a stationary-phase [geometrical optics (GO)] evaluation, and a uniform asymptotic expansion (UAE). The UAE solution is the more accurate one, and accounts for diffraction-like effects that arise due to the semi-infinite nature of the line. One conclusion is that the observation distance must be very large for the true leakage fields to closely resemble the GO fields, although a quasi-GO behavior may still be evident at shorter distances, depending on the value of the leakage angle. Another conclusion is that the transverse behavior of the GO leakage field of the printed-circuit structure is an exponentially increasing field within the leakage region, as expected from a simple ray-optics picture, while it is not for the strip in free space.

Asymptotic expansions of integrals: The term-by-term integration method

The classical term-by-term integration technique used for obtaining asymptotic expansions of integrals requires the integrand to have an uniform asymptotic expansion in the integration variable. A modification of this method is presented in which the uniformity requirement is substituted by a much weaker condition. As we show in some examples, the relaxation of the uniformity condition provides the term-by-term integration technique a large range of applicability. As a consequence of this generality, Watson's lemma and the integration by parts technique applied to Laplace's and a special family of Fourier's transforms become corollaries of the term-by-term integration method.

Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces

This paper consists of two parts. In the first part, the solution of the Helmholtz equation under forward-scattering or propagation conditions is sought as a uniform asymptotic perturbation expansion using the method of multiple scales. It is then shown that the parabolic wave equation (PWE) solution is the zeroth-order term in this expansion. In the second part, the electric-field integral equation and the magnetic-field integral equation, derived under the PWE approximation, are solved for surface currents induced on a sinusoidal surface. The scattered fields produced by these currents are then calculated using the appropriate radiation integrals. Results are compared to those obtained using the method of ordered multiple interactions developed by Kapp and Brown.

On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function

We examine the asymptotic nature as k → ∞ of the coefficientsck(�) appearing in the uniform asymptotic expansion of the incomplete gamma function ( a,z) whereis a variable that depends on the ratio z/a. It is shown that this expansion diverges like the familiar factorial divided by a power dependence multiplied by a function fk(�). For values ofnear the real axis, fk(�) is a slowly varying function, but in the left half-plane, there are two lobes situated symmetrically about the negative realaxis in which fk(�) becomes large. The asymptotic expansion of fk(�) as k → ∞ is found to reveal a resurgence-type structure in which the high-order coefficients are related tothe low-order coeffi- cients. Numerical examples are given to illustrate the growth of the coefficients ck(�).

On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function

On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function

Efficient Evaluation of Sommerfeld Integrals for Tm Wave Scattering By Buried Objects

Unlike the scattering problem in homogeneous space, intensive computations of Sommerfeld integrals are involved in the EM scattering of buried scatterers. There are mainly three bottlenecks of the CPU time for such problem: matrix filling, matrix inversion and calculation of the scattered fields. For moderately sized problems, extensive numerical experience shows that the CPU time used in the first and third items (concerned with the Sommerfeld integrals) is much more than that used in the matrix inversion. Therefore an efficient method for solving such buried object problems requires the fast evaluation of such Sommerfeld integrals. In this paper, several efficient methods are presented to evaluate the integrals that appear in the TM wave scattering by two-dimensional buried dielectric and conducting cylinders. In the numerical integration method, the original integrating path is deformed to the steepest-descent paths to expedite the numerical integration and yield a more stable computation result. In the method of steepest descent, the leading and higher-order approximation of the saddle point, and the contribution of the branch point are formulated. However these formulations are invalid near the so called "critical angles". In the uniform asymptotic expansion method, we improve the approximations near such critical angles to yield satisfactory results. Finally, numerous numerical examples are given by using the fast evaluation methods. Numerical results show that the leading-order approximation and the uniform asymptotic solution can expedite the buried object scattering problem, and the CPU time will be reduced several thousands times.

Uniform asymptotic analysis for waves in an incompressible elastic rod I. Disturbances superimposed on an initially stress-free state

This paper studies the propagation of disturbances in an initially stress-free elastic circular rod. Starting from the Mindlin-Hermann equations, we derive a fourth-order partial differential equation as the governing equation for small axial-radial deformations in a rod composed of an incompressible material. Then, we consider an initial-value problem with an initial singularity in the shear strain. Using the technique of Fourier transforms, we manage to express the physical quantities in terms of integrals. The classical method of stationary phase is first used to obtain some important information. However, for material points in a neighbourhood behind the shear-wave front, the phase function of the integrals has a stationary point which approaches positive infinity. Consequently, the classical method of stationary phase fails completely. Here, instead, we use a new method developed by us (Dai & Wong 1994 Wave Motion 19, 293-308) to handle this case. An asymptotic expansion for the shear strain, which is uniformly valid in a neighbourhood behind the shear-wave front, is derived. This uniform asymptotic expansion reveals that for the shear strain there is a transition from O(1) disturbance to O(t -1/4 ) disturbance as the distance to the shear-wave front increases. We also find that the shear strain has three other jumps in terms of asymptotic orders. The first jump is from a larger O(t -2/3 ) disturbance to a smaller O(t -1 ) disturbance behind and ahead of the bar-wave front. The second jump is from a larger O(1) disturbance to a smaller O(t -1/2 ) disturbance behind and ahead of the bar-wave front. This also implies that in an asymptotic sense the initial singularity in the shear strain will be preserved and propagates with the shear-wave speed as time progresses. The third jump is from a smaller O(t -1 ) disturbance to a larger O(t -1/3 ) disturbance behind and ahead of a third wave front.

Dynamics and Condensation of Complex Singularities for Burgers' Equation II

The zero-viscosity limit of a meromorphic solution to Burgers' equation (BE) is found via an integral representation of the Mittag--Leffler expansion of the solution involving a "polar" measure. The weak zero-viscosity limit of this Borel measure (analogously to the zero-dispersion limit of the spectral measure in the Korteweg--de Vries (KdV) problem) corresponds to the asymptotic density of poles which characterizes their condensation on the imaginary axis. The resulting integral representation of the inviscid solution is computed by residues and is shown to match the characteristic solution up to the inviscid shock time t*. The continuum limit of the Mittag--Leffler expansion and the Calogero dynamical system (CDS) (which describes the time evolution of the poles) is a system of two integro-differential equations which provide a new representation of the solution to the inviscid BE. For t \leq t*, a uniform asymptotic expansion of the Fourier transform of the inviscid solution is obtained, thereby providing the analyticity properties of the inviscid solution.

Asymptotic expansions of the generalized Bessel polynomials

In this paper, we investigate the asymptotic behavior of the generalized Bessel polynomials y n ( z; a). Let z = α (n + 1) . We first derive infinite asymptotic expansions for y n ( z; a) when α lies in various regions of the complex plane, except when α is near ±i. Then we construct uniform asymptotic expansions for y n ( z; a) in neighborhoods of α = ± i. These expansions involve the Airy function and its derivative. Finally, we use these approximations to study the asymptotic behavior of the zeros of y n ( z; a) near α = i.

Dynamics and Condensation of Complex Singularities For Burgers' Equation I

Spatial analyticity properties of the solution to Burgers' equation with a generic initial data are presented, following the work of Bessis and Fournier [Research Reports in Physics: Nonlinear Physics, Springer-Verlag, Berlin, Heidelberg, 1990, pp. 252--257]. The positive viscosity solution is a meromorphic function with a countable set of conjugate poles confined to the imaginary axis. Their motion is governed by an infinite-dimensional Calogero dynamical system (CDS). The inviscid solution is a three-sheeted Riemann surface with three branch-point singularities. Exact pole locations are found independent of the viscosity at the inviscid shock time t* . For t \ne t* , the time evolution of the poles is obtained numerically by solving a truncated version of the CDS. A Runge--Kutta scheme is used together with a "multipole" algorithm to deal with the computationally intensive nonlinear interaction of the poles. Additionally, for t \leq t$, the small viscosity behavior of the poles is shown to be a perturbation of the conjugate inviscid branch-point singularities $\pm x_s(t)$. The numerical pole dynamics also provide the width of the analyticity strip which remains uniformly bounded away from zero, agreeing with asymptotic predictions. For small $\nu>0$ and t \geq t* , different saddle-point approximations of the solution are found within and outside the caustics $x=\pm x_s(t)$. The transition between the two regimes at $x=\pm x_s(t)$ is described by a uniform asymptotic expansion involving the Pearcey integral. The solution is computed for small viscosity using pole dynamics, finite differences (method of lines), and asymptotic methods (saddle-point method); numerical agreement is established.

The statistical mechanics of vesicles

Partially convex polygons, Exact solution, q-series asymptotics, Scaling function. I discuss two-dimensional lattice models of closed fluctuating membranes, or vesicles (11. The underlying mathematical model is that of self-avoiding polygons and their enumeration by perime- ter and area.. Physically, this model is of interest as its phase diagram contains a point at which tricritical scaling is observed [2]. By adding the constraint of partial directedness, one gets combinatorial models related to the partitioning of integers. These models are solvable in the sense that an explicit expression for the generating function can be given [3,4]. The derivation of this generating function involves a nonlinear functional equation, and the solution is given in terms of alternating q-series. The asymptotic analysis of these generating functions can be done with the help of a new contour integral representation. This leads to an explicit calculation of the scaling behaviour around the critical point in terms of the Airy function [5]. In particular, one gets a uniform asymptotic expansion of the involved q-series as

Uniform asymptotic expansions of the solutions of Stueckelberg's system with turning points

Uniform asymptotic expansions of the solutions of Stueckelberg's system with analytic potentials in the vicinity of the maximum of a potential barrier and in the vicinity of a potential-curve-crossing point are constructed. Bibliography: 11 titles.

Uniform high frequency approximation to scattering from infinite strip

A new technique for approximating the high frequency scattering amplitude from flat obstacles is used to solve the problem of acoustic plane wave scattering from an infinite soft strip. This method yields the leading order terms in a uniform high frequency asymptotic expansion for the scattering amplitude which reduce to the results found with the geometrical theory of diffraction in the regions in which that theory is valid. The asymptotic expressions derived here are compared with exact numerical solutions and are found to be accurate for all angles of incidence and reflection.

Error analysis in a uniform asymptotic expansion for the generalised exponential integral

Uniform asymptotic expansions are derived for the generalised exponential integral E p ( z), where both p and z are complex. These are derived by examining the differential equation satisfied by E p ( z), an equation which possesses a double turning point at z/ p = −1. The expansions, which involve the complementary error function, together approximate E p ( z) as ¦p¦ → ∞, uniformly for all non-zero complex z satisfying 0 ⩽ arg( z/ p) ⩽ 2 π. The error terms associated with the truncated expansions are shown to be solutions of inhomogeneous differential equations, and from these explicit and realistic bounds are derived. By employing the Maximum-Modulus Theorem the bounds are then simplified to make them more conducive to numerical evaluation.

Numerical algorithms for uniform Airy-type asymptotic expansions

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.

Uniform Asymptotic Expansions for Meixner Polynomials

Meixner polynomials m n (x;β,c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are \(j(x;\beta,c) = \frac{c^x(\beta)_x}{x!} \qquad \mbox{at}\quad x=0,1,2\ldots.\) Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper, we derive two infinite asymptotic expansions for m n (nα;β,c) as \(n\to\infty\) . One holds uniformly for \(0 < \epsilon\le \alpha\le 1+a\) , and the other holds uniformly for \(1-b\le \alpha\le M < \infty\) , where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results include all five asymptotic formulas recently given by W. M. Y. Goh as special cases.

Pole dynamics and oscillations for the complex Burgers equation in the small-dispersion limit

A meromorphic solution to the Burgers equation with complex viscosity is analysed. The equation is linearized via the Cole - Hopf transform which allows for a careful study of the behaviour of the singularities of the solution. The asymptotic behaviour of the solution as the dispersion coefficient tends to zero is derived. For small dispersion, the time evolution of the poles is found by numerically solving a truncated infinite-dimensional Calogero-type dynamical system. The initial data are provided by high-order asymptotic approximations of the poles at the critical time for the dispersionless solution via the method of steepest descents. The solution is reconstructed using the pole expansion and the location of the poles. The oscillations observed via the singularities are compared to those obtained by a classical stationary phase analysis of the solution as the dispersion parameter . A uniform asymptotic expansion as of the dispersive solution is derived in terms of the Pearcey integral in a neighbourhood of the caustic. A continuum limit of the pole expansion and the Calogero system is obtained, yielding a new integral representation of the solution to the inviscid Burgers equation.