BIANALYTIC FUNCTIONS, BIHARMONIC FUNCTIONS AND ELASTIC PROBLEMS IN THE PLANE

Let the elastic body only be acted by gravity. By investigating the relations of bianalytic functions and biharmonic functions, the uniqueness and existence of the stress functions (Airy functions) are established in planar simple connected region. Moreover, the integral representation formula of the stress functions in the unit disk of the plane is obtained.

Effective transmittance of Fabry–Perot cavity under non-parallel beam incidence

The Fabry–Perot(FP) resonant cavity is widely used in laser and spectroscopic measurements due to its unique interference transfer function(ITF). In the ideal case of parallel incident light, the ITF of the FP resonant cavity can be expressed by the Airy function. However, in reality, it is difficult to achieve perfect parallelism with collimated beams. In this article, a theoretical model is established for non-parallel light incidence, which assumes that the non-parallel incident light is a cone-shaped beam, and the cone angle is used to quantify the non-parallelism of the beam. The transmittance function of the FP resonant cavity under non-parallel light incidence is derived. The accuracy of the model is experimentally verified. Based on this model, the effects of divergence angle, tilt angle and FP cavity parameters(reflectivity, cavity length)on the ITF are studied. The reasons for the decrease in peak value, broadening and asymmetry of the interference peak under non-parallel light incidence are explained. It is suggested that a fine balance between the interference peak and the collimation effect of the incident light should be considered in the design and application of FP resonant cavities, especially for tilted applications such as angle-scanned spectroscopy. The research results of this article have certain significance for the design and application of FP resonant cavities.

THE GENERALIZED AIRY FUNCTIONS

This paper is concerned with the second order linear differential equationd 2ydx 2-x λ y=0, which is called the generalized Airy equation of order λ , where the constant λ≥ 0 (not necessary positive integer) satisfies (-1) λ =± 1 . The properties of its two special linearly independent real-valued solutions, called the generalized Airy function of order λ , are systematically investigated.

SOME IDENTITY RELATIONS BETWEEN PLANE PROBLEMS FOR VISCO-AND ELASTICITY

In this paper.the boundary value problems of plane problems with a simply-ormultiply-connected domain for isotropic linear visca-elosticity are first established byterms of Airy stress function F(Xu t). Secondly some identity relations betweendisplacements and stresses for plane problems of sisco-and elasticity are discussed indetait and some meaningful conclusions are obtained As an example the deformationresponse for viscoelastic plate with a small circular hote at the center is analyzed undera uniasial uniform extension.

A GLOBALLY UNIFORM ASYMPTOTIC EXPANSION OF THE HERMITE POLYNOMIALS

In this article,the author extends the validity of a uniform asymptotic expansion of the Hermite polynomials Hn（√2n＋1α）to include all positive values of α. His method makes use of the rational functions introduced by Olde Daalhuis and Temme （SIAM J.Math.Anal.,（1994）,25：304-321）.A new estimate for the remainder is given.

A NOVEL SOLUTION OF TOROIDAL SHELLS UNDER AXISYMMETRIC LOADING

Several improvements are made for existing asymptotic expansions for the axisymmetric toroidal shells. The new expansions are numerically satisfactory and satisfy the accuracy of the theory of thin shells. All of them are expressed in terms of generalized Airy functions , instead of Bessel or Airy function for the homogeneous and Lommel function for the particular solutions, respectively, as in the existing work. In this paper, three particular solutions are given, one of which is just the solution obtaine d by Tumarkin(1959) and Clark(1963).

Approximate Analytical Solutions to the Heat and Stokes Equations on the Half-Line Obtained by Fokas’ Transform

In this paper, we evaluate the integrals that are solutions of the heat and Stokes’ equations obtained by Fokas’ transform method by deriving exact formulas. Our method is more accurate and efficient than the contour deformation and parametrization used by Fokas to compute these integrals. In fact, for the heat equation, our solution is exact up to the imaginary error function and for the Stokes equation, our solution is exact up to the incomplete Airy function. In addition, our solutions extend to the lateral boundary without convergence issues, allow for asymptotic expansions, and are much faster than those obtained by other methods.

Flow through a Variable Permeability Brinkman Porous Core

In this work, we consider the flow through composite porous layers of variable permeability, with the middle layer representing a porous core bounded by two Darcy layers. Brinkman’s equation is valid in the middle layer and has been reduced to an Airy’s inhomogeneous differential equation. Solution is obtained in terms of Airy’s functions and the Nield-Kuznetsov function.

ELASTIC CALCULATION OF STRESSES IN RINGS USING AIRY STRESS FUNCTION

Stress calculation formulae for a ring have been obtained by using Airy stress function of the plane strain field with the decomposition of the solutions for normal stresses of Airy biharmonic equation into two parts when it is loaded under two opposite inside forces along a diameter. One part should fulfill a constraint condition about normal stress distribution along the circumference at an energy valley to do the minimum work. Other part is a stress residue constant. In order to verify these formulae and the computed results, the computed contour lines of equi-maximal shear stresses were plotted and quite compared with that of photo-elasticity test results. This constraint condition about normal stress distribution along circumference is confirmed by using Greens’ theorem. An additional compression exists along the circumference of the loaded ring, explaining the divorcement and displacement of singularity points at inner and outer boundaries.

Global Asymptotics of Krawtchouk Polynomials——a Riemann-Hilbert Approach

In this paper, we study the asymptotics of the Krawtchouk polynomials Kn^N（z;p,q） as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c E （0, 1） as n →∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p; in particular, they are valid in regions containing the interval on which these polynomials are orthogonal, Our method is based on the Riemann-Hilbert approach introduced by Delft and Zhou.

Resurgence relation and global asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert approach

A global asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert approach is presented,with respect to the polynomial degree.The domains of uniformity are described in certain phase variables.A resurgence relation within the sequence of Riemann-Hilbert problems is observed in the procedure of derivation.Global asymptotic approximations are obtained in terms of the Airy function.The system of Hermite polynomials is used as an illustration.

SOME PROBLEMS ABOUT SHIP WAVES

Several problems about ship waves were discussed in the dissertation: (1) Transient ship waves from calmness to the generation of steady-state ship waves were described. (2) The procedure of the formation of the V-shaped steady-state ship waves were clearly shown, and the difference of ship waves on an inviscid fluid and on a viscous fluid was exmined. (3) With the Lighthill two-stage scheme, the algebraic expression for ship waves on a viscous fluid of finite depth was obtained. (4) Singularity on the two boundaries of the ship waves was treated.

Error Bounds for Uniform Asymptotic Expansions-Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe?cients and the remainders in the Taylor expansions of analytic functions.By using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.

Decay Mode Solutions for Kadomtsev-Petviashvili Equation

The decay mode solutions for the Kadomtsev-Petviashvili (KP) equation are derived by Hirota method (direct method).The decay mode solution is a new set of analytical solutions with Airy function.