Asymptotic methods

In solar physics many questions remain unanswered, such as the solar coronal heating problem in the solar atmosphere or magnetic field generation in the solar interior. However, there is a growing consensus that many different physical phenomena will combine to give a solution to these problems.
From the solar coronal heating point of view, one possible phenomenon to study is resonant absorption, which allows the natural transfer of wave energy to and from the background plasma. If the plasma is dissipative this energy can be converted into heat. One of the stumbling blocks when studying resonant absorption is that linear theory can breakdown around the resonant point. This causes a dilemma: use linear theory as an approximation regardless to try and find sensible solutions or try and solve the difficult, but realistic, nonlinear equations.
The present thesis will investigate the nonlinearity associated with resonant absorption. There are two different resonances in solar plasmas, one is the Alfve ́n resonance and the other is the slow resonance. Previous studies, in nonlinear theory, have concentrated mainly on the slow resonances as they are more affected by nonlinearity. We will study both the Alfve ́n and the slow resonances. The present thesis will analytically investigate these resonances in anisotropic, dispersive and dissipative plasmas - typical conditions of the solar upper atmosphere.
The second manifestation of nonlinearity is the generation of a mean shear flow outside the layer enclosing the resonant surface. We will derive the governing equations for this gener- ated flow at the Alfve ́n resonance. These flows (like all flows) are completely determined by the boundary conditions and we produce an example flow.
The thesis culminates by studying coupled resonances; when the distance between an Alfve ́n and slow resonance is so small (in comparison to the incoming wave) they act as if they interact with an incoming wave simultaneously. We derive the governing equations for the absorption of fast magnetoacoustic waves at the coupled resonance, and then numerically analyse the coeffi- cient of wave energy absorption to compare with the results found for single resonances. We find that the absorption of fast magnetoacoustic waves is far more efficient under coronal conditions compared with chromospheric conditions despite an increase in absorption due to the coupled resonance.

In the present paper , we obtain analytical equations by using Optimal Homotopy Asymptotic method compared with some other powerful and perturbation method (HPM) and simple computations with quite acceptable approximate solutions, which has close agreement with exact solutions.The accuracy and efficiency of OHAM approach illustrated by presenting four test examples that satisfy the ADM and HPM methods.

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 ≤ t * , a uniform asymptotic expansion of the Fourier transform of the inviscid solution is obtained, thereby providing the analyticity properties of the inviscid solution.

A mechanism is proposed for the generation of large-amplitude acoustically-driven streaming flows in which time-mean flow speeds are comparable to the instantaneous speed of fluid particles in a high-frequency sound-wave field. Motivated by streaming observed in high-intensity discharge (HID) lamps, two-dimensional flow of a density-stratified ideal gas in a channel geometry is analysed in the asymptotic limit of high-frequency acoustic-wave forcing. Predictions of streaming flow magnitudes based on classical arguments invoking Reynolds stress divergences originating in viscous boundary layers are orders of magnitude too small to account for the observed mean flows. Moreover, classical 'Rayleigh streaming' theory cannot account for the direction of the cellular mean flows often observed in HID lamps. In contrast, the mechanism proposed here, which invokes fluctuating baroclinic torques away from viscous boundary layers and thus is largely independent of viscous effects, can account both for the magnitude and the orientation of the observed streaming flows.

—The paper presents the analysis and design of Bessel beam launchers using a finite inward cylindrical traveling wave aperture field distribution. The launcher radiates an electric field whose normal or longitudinal component takes the form of a zeroth-order Bessel function. The nondiffractive behavior of the structure in a well-defined area close to the radiating aperture is analyzed by decomposing the radiated field in its geometrical optics (GO) and diffractive (D) contributions. A closed-form expression is provided for the GO contribution whereas an asymp-totic approximation is provided for the diffractive part. Such theoretical analysis allows a precise definition of the nondiffrac-tive region for the generated Bessel beam. At the same time, it also highlights and predicts the oscillating behavior of the longitudinal component of the electric field along the z-axis due to the diffraction from the edges of the aperture. The proposed analysis is validated by a prototype at 30 GHz made by a radial waveguide loaded with metallic gratings and centrally fed by a coaxial probe. Measurement results for the longitudinal component of the electric field are in excellent agreement with full-wave results. In addition , the nondiffractive behavior for the radiated beam is reported over a bandwidth larger than 6.5% around 30 GHz. This behavior is peculiar of the nonresonant first kind Hankel aperture field distribution used for the generation of the Bessel beam.

tA comprehensive multiphysics model of a cantilevered carbon nanotube (CNT) including geometric andelectrostatic nonlinearities is developed. The continuous model is reduced to a finite degree of freedomsystem by the Galerkin discretization and solved using the harmonic balance method (HBM) coupled withthe asymptotic numerical method (ANM). The influence of higher modes on the nonlinear dynamics ofthe considered resonator is investigated in order to retain the number of modes which will be used bythe HBM+ANM procedure. Several simulations are performed for a specific CNT design in order to obtaina wide range of frequency shifts with respect to the mass and position of an added particle. This modelis an intuitive way for designers to develop resonant nanosensors vibrating at large amplitudes for massdetection. Particularly, it is demonstrated that the mass and position of a particle can be determinedbased on the proposed model reduced to the first bending mode coupled with the analytical expressionsof the linear frequency shifts for the second and third modes.

In this study, we propose an efficient method so-called Successive Complementary Expansion Method (SCEM), that is based on generalized asymptotic expansions, for approximating to the solutions of singularly perturbed two-point boundary value problems. In this easy-applicable method, in contrast to the well-known method the Method of Matched Asymptotic Expansions (MMAE), any matching process is not required to obtain uniformly valid approximations. The key point: A uniformly valid approximation is adopted first, and complementary functions are obtained imposing the corresponding boundary conditions. An illustrative and two numerical experiments are provided to show the implementation and numerical properties of the present method. Furthermore, MMAE results are also obtained in order to compare the numerical robustnesses of the methods.

An important problem in engineering is the determination of the system input based on the system response. This type of problem is difficult to solve as it is often ill-defined, and produces inaccurate or non-unique results. Current reconstruction techniques typically involve the employment of optimization methods or additional constraints to regular-ize the problem, but these methods are not without their flaws as they may be sub-optimally applied and produce inadequate results. An alternative approach is developed that draws upon concepts from control systems theory, the equilibrium analysis of linear dynamical systems with time-dependent inputs, and asymptotic approximation analysis. This paper presents the theoretical development of the proposed method. A simple application of the method is presented to demonstrate the procedure. A more complex application to a continuous system is performed to demonstrate the applicability of the method.

After a brief historical survey of some work done on the linear theory of longitudinal vibrations and wave
propagation in rods and tubes of uniform cross-section, a simple mathematical model for rods and tubes of linear
elastic materials is proposed. Three suitably selected propagation modes (one extensional and two shear modes) with dispersion relations corresponding to mixed boundary conditions are coupled in order to approximately comply with
zero-stress boundary conditions. The coupling gives a set of partial differential equations in the mode amplitudes, with
time and a single space coordinate (along the axis of symmetry of the rod or tube) as independent variables. Then, the
model is generalized to a set of partial integral-differential equations in order to be able to describe vibrations and
wave propagation in rods and tubes made of linear hereditary-elastic solids. In this first part of the work, the focus is
in either very low frequency or very high frequency phenomena using a simple model with only two coupled modes.
The model allows a fairly elegant and comparatively powerful analytical approach to longitudinal vibrations and to
longitudinal pulse propagation in solid waveguides. Analytical formulae for group velocities are derived, as well as
asymptotic expressions for the propagation of mode amplitudes. The limitations and pitfalls of the model are assessed,
and new experiments and digital simulations are suggested to test some of its predictions.
Keywords: mechanical vibrations; wave propagation; elastic and hereditary-elastic materials; propagation modes
in rods and tubes.

Analytical expressions are found for the wavenumbers and resonance frequencies in flexible, orthotropic shells using the asymptotic methods. These expressions are valid for arbitrary circumferential orders n. The Donnell–Mushtari shell theory is used to model the dynamics of the cylindrical shell. Initially, an in vacuo cylindrical isotropic shell is considered and expressions for all the wavenumbers (bending, near-field bending, longitudinal and torsional) are found. Subsequently, defining a suitable orthotropy parameter ϵϵ, the problem of wave propagation in an orthotropic shell is posed as a perturbation on the corresponding problem for an isotropic shell. Asymptotic expressions for the wavenumbers in the in vacuo orthotropic shell are then obtained by treating ϵϵ as an expansion parameter. In both cases (isotropy and orthotropy), a frequency-scaling parameter (ηη) and Poisson's ratio (νν) are used to find elegant expansions in the different frequency regimes. The asymptotic expansions are compared with numerical solutions in each of the cases and the match is found to be good. The main contribution of this work lies in the extension of the existing literature by developing closed-form expressions for wavenumbers with arbitrary circumferential orders n in the case of both, isotropic and orthotropic shells. Finally, we present natural frequency expressions in finite shells (isotropic and orthotropic) for the axisymmetric mode and compare them with numerical and ANSYS results. Here also, the comparison is found to be good.► Using Donnell–Mushtari shell theory orthotropy is posed as a perturbation to isotropy. ► Asymptotic expansions are found for wavenumbers in an infinite orthotropic shell. ► Resonance frequencies for orthotropic shells are calculated. ► The calculated resonances are compared with literature and Ansys.

Analytical expressions are found for the coupled wavenumbers in an infinite fluid-filled cylindrical shell using the asymptotic methods. These expressions are valid for any general circumferential order (n). The shallow shell theory (which is more accurate at higher frequencies) is used to model the cylinder. Initially, the in vacuo shell is dealt with and asymptotic expressions are derived for the shell wavenumbers in the high- and the low-frequency regimes. Next, the fluid-filled shell is considered. Defining a relevant fluid-loading parameter μμ, we find solutions for the limiting cases of small and large μμ. Wherever relevant, a frequency scaling parameter along with some ingenuity is used to arrive at an elegant asymptotic expression. In all cases, Poisson's ratio νν is used as an expansion variable. The asymptotic results are compared with numerical solutions of the dispersion equation and the dispersion relation obtained by using the more general Donnell–Mushtari shell theory (in vacuo and fluid-filled). A good match is obtained. Hence, the contribution of this work lies in the extension of the existing literature to include arbitrary circumferential orders (n).

Inference procedures based on the Hellinger distance provide attractive al-ternatives to likelihood based methods for the statistician. The minimum Hellinger distance estimator has full asymptotic efficiency under the model together with strong robustness properties under model misspecification. However, the Hellinger distance puts too large a weight on the inliers which appears to be the main reason for the poor efficiency of the method in small samples. Here some modifications to the inlier part of the Hellinger distance are provided which lead to substantial improvements in the small sample properties of the estimators. The modified divergences are members of the general class of disparities and satisfy the necessary regularity conditions so that the asymptotic properties of the resulting estimators follow from stan-dard theory. In limited simulations the proposed estimators exhibit better small sample performance at the model and competitive robustness proper-ties in relation to...

Wireless networking is a method by which homes, telecommunications networks and enterprise (business) installations avoid the costly process of introducing cables into a building, or as a connection between various equipment locations. For such a reasons this technology has become popular. Though it is familiar, its wireless channel is vulnerable to the eavesdroppers during message delivery (security is the major problem). In the previous cases this problem was solved by cryptographic methods such as RSA public key cryptosystem. But due to expensive key distribution and improvement in decoding technology, the message transmitted is said to be unsecured. The problem can be overcome by using artificial noise generation. This paper investigates and studies how to deliver the message securely in the wireless network using artificial noise generation concept.

In this work, we investigate the limits of classical homogenization theories pertaining to homogenization of periodic composite materials at low scale separations and demonstrate the effectiveness of higher-order periodic homogenization in alleviating this limitation. Classical homogenization techniques are very effective for materials with large scale separation between the scale of the heterogeneity and the macro-scale dimension, but inaccurate at low scale separations. Literature suggests that asymptotic homogenization is capable of pushing the limit to smaller scale separation by taking on board higher-order terms of the asymptotic expansion. We show that the classical homogenization deviates from the actual solution for scale ratios below 10. Beyond this limit, the higher-order asymptotic homogenization solution still gives a very good approximation which becomes better as more higher-order terms are included. This results in a size-dependent macroscopic model, which indeed allows one to push the limitations of homogenization in the direction of less scale separation.

Linear rank statistics for the two-sample problem are considered. Under general conditions on the score function and the distribution of the observations large deviation asymptotics for these statistics under the alternative are obtained. After using the Sanov principle an extremal problem of

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 = 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 ≤ t * , the small viscosity behavior of the poles is shown to be a perturbation of the conjugate inviscid branch-point singularities ±xs(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 ν > 0 and t ≥ t * , different saddle-point approximations of the solution are found within and outside the caustics x = ±xs(t). The transition between the two regimes at x = ±xs(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.

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 t s 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 → 0 +. A uniform asymptotic expansion as → 0 + 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.

El campo de las funciones especiales ha tenido un gran desarrollo en las últimas décadas dado que son muchos los fenómenos que se pueden estudiar mediante el uso de las mismas como, por ejemplo, procesos estocásticos relacionados, investigación de operaciones, teoría cuántica, ecuaciones funcionales, vibración de placas, conducción del calor, elasticidad, y radiación. En este trabajo se considera una ampliación de las teorías presentadas por M. Dotsenko en 1991, quien introdujo la generalización de la función hipergeométrica de Gauss, denotada por 2R1τ (z), y estableció su representación en serie e integral. Es importante notar que en 1999 Nina Virchenko y luego, en el 2003, Leda Galué consideraron esta función, introduciendo un conjunto de fórmulas de recurrencia y de diferenciación. En este trabajo se establecen algunas representaciones simples para la función 2R1τ (a, b; c; τ; z), las cuales serán muy útiles en futuras investigaciones puesto que permiten simplificar cálculos en el momento de solucionar problemas que involucren esta función.

MSC: 33D15, 33D90, 33D60, 34M03,  62E15

The theory of thin wires developed in Dret and Meunier (Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 337:143–147, 2003) is adapted to phase-transforming materials with large elastic moduli in the sense discussed in James and Rizzoni (J Elast 59:399–436, 2000). The result is a one-dimensional constitutive model for shape memory wires, characterized by a small number of material constants. The model is used to analyze self-accommodated and detwinned microstructures and to study superelasticity. It also turns out that the model successfully reproduces the behavior of shape memory wires in experiments of restrained recovery (Tsoi et al. in Mater Sci Eng A 368:299–310, 2004; Tsoi in 50:3535–3544, 2002; S̆ittner et al. in Mater Sci Eng A 286:298–311, 2000; vokoun in Smart Mater Struct 12:680–685, 2003; Zheng and Cui in Intermetallics 12:1305–1309, 2004; Zheng et al. in J Mater Sci Technol 20(4):390–394, 2004). In particular, the model is able to predict the shift to higher transformation temperatures on heating. The model also captures the effect of prestraining on the evolution of the recovery stress and of the martensite volume fraction.