Complete Asymptotic Expansion Research Articles

Asymptotics of the Solution of a Bisingular Optimal Boundary Control Problem in a Bounded Domain

A bisingular problem of optimal boundary control for solutions of an elliptic equation in a bounded domain with a smooth boundary is considered. The coefficient of the Laplacian is assumed to be small, and integral constraints are imposed on the control. A complete asymptotic expansion in powers of the small parameter is obtained for the solution of the problem.

The T-stress along a 3-D straight crack

The T-stress in a three dimensional elastic domain containing a straight crack is considered. First the complete asymptotic expansion in the vicinity of the crack front, including the T-stress, is presented explicitly. Having obtained the asymptotic expansion, we compute the dual solution associated with the T-stress, and prove it must contain logarithmic terms. This dual solution is used to extend the quasi dual function method (QDFM) for the extraction of the T-stress from finite element solutions and express it as a function along the crack front.Since the stress field tends to infinity at the crack front whereas the T-stress remains constant, its extraction from finite element solutions, containing numerical errors, is a non-trivial task. We herein develop accurate and efficient methods for extracting the T-stress function along the crack edge by the QDFM. Numerical examples are provided for the computation of the T-stresses from conventional and high order finite element methods.

Special properties of the stationary solution for two‐dimensional singularly perturbed parabolic problem, stability, and attraction domain

We consider a singularly perturbed elliptic Dirichlet problem in the case of a degenerate equation's multiple root. The existence of problem solution is proved. A complete asymptotic expansion of the solution is constructed and justified. It has distinctive features. The boundary layer becomes a three‐zone layer with different solution behavior in each of three zones. The solution of elliptic problem considered is a stationary solution of the corresponding parabolic problem. Asymptotic stability of this solution is proved and its attraction domain is found. We have considered the extended version of concept of stationary solution attraction domain.

The asymptotic expansion of the swallowtail integral in the highly oscillatory region

We consider the swallowtail integral Ψ(x,y,z):=∫−∞∞ei(t5+xt3+yt2+zt)dt for large values of |x| and bounded values of |y| and |z|. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the simplified saddle point method introduced in (López et al., 2009). The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of Ψ(x, y, z) for large |x| and fixed y and z. The asymptotic analysis requires the study of three different regions for argx separated by three Stokes lines. The expansion is given in terms of inverse powers of x13 and x12 and the coefficients are elementary functions of y and z. The accuracy and the asymptotic character of the approximations is illustrated with some numerical experiments.

Rigorous derivation of the asymptotic model describing a nonsteady micropolar fluid flow through a thin pipe

In this paper we study a time-dependent flow of an incompressible micropolar fluid through a pipe with arbitrary cross-section. The effective behavior of the flow is found by means of rigorous asymptotic analysis with respect to the small parameter representing the pipe’s thickness. The complete asymptotic expansion (up to an arbitrary order) of the solution is constructed with detailed analysis of the boundary layers provided. The convergence of the expansion is proved justifying the usage of the formally derived asymptotic model.

Asymptotic Expansions for Bernstein–Durrmeyer–Chlodovsky Polynomials

We define a Durrmeyer variant of the Bernstein–Chlodovsky polynomials and study its rate of convergence. The main result is a complete asymptotic expansion. As a special case we obtain a Voronovskaja-type formula.

Note on the coefficients of Ramanujan’s expansion for the logarithm of the factorial

Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique.

A Complete Asymptotic Expansion for the Quasi-interpolants of Gauß–Weierstraß Operators

We derive the complete asymptotic expansion for the quasi-interpolants of Gaus–Weierstras operators $$W_{n}$$ and their left quasi-interpolants $$W_{n}^{\left[ r\right] }$$ with explicit representation of the coefficients. The results apply to all locally integrable real functions f on $$\mathbb {R}$$ satisfying the growth condition $${f}\left( t\right) =O\left( \mathrm{{e}}^{ct^{2}}\right) $$ as $$\left| t\right| \rightarrow +\infty $$ , for some $$c>0$$ . All expansions are shown to be valid also for simultaneous approximation.

Zeros of the deformed exponential function

Let f(x)=∑n=0∞1n!qn(n−1)/2xn (0<q<1) be the deformed exponential function. It is known that the zeros of f(x) are real and form a negative decreasing sequence (xk) (k≥1). We investigate the complete asymptotic expansion for xk and prove that for any n≥1, as k→∞,xk=−kq1−k(1+∑i=1nCi(q)k−1−i+o(k−1−n)), where Ci(q) are some q series which can be determined recursively. We show that each Ci(q)∈Q[A0,A1,A2], where Ai=∑m=1∞miσ(m)qm and σ(m) denotes the sum of positive divisors of m. When writing Ci as a polynomial in A0,A1 and A2, we find explicit formulas for the coefficients of the linear terms by using Bernoulli numbers. Moreover, we also prove that Ci(q)∈Q[E2,E4,E6], where E2, E4 and E6 are the classical Eisenstein series of weight 2, 4 and 6, respectively.

Complete asymptotic expansions for the density function of [formula omitted]-distribution

The density function of t-distribution with n degrees of freedom is given by the following formula: fn(x)=Γ(n+12)πnΓ(n2)1+x2n−(n+1)∕2,x>0.Ding (1988) obtained the following asymptotic formula in terms of 1∕n: fn(x)=φ(x){1+x4−2x2−14n+3x8−28x6+30x4+12x2+396n2+O1n3},n→∞,which derives the known result limn→∞fn(x)=φ(x), where φ(x)=12πe−x2∕2 is the probability density function of the standard normal distribution. In this paper, we develop Ding’s result to produce a complete asymptotic expansion.

Off-diagonal Asymptotic Properties of Bergman Kernels Associated to Analytic Kähler Potentials

Abstract We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size $k^{-\frac 14}$. These improve the earlier results in the subject for smooth potentials, where an expansion exists in a $k^{-\frac 12}$ neighborhood of the diagonal. We obtain our results by finding upper bounds of the form $C^m m!^{2}$ for the Bergman coefficients $b_m(x, \bar y)$, which is an interesting problem on its own. We find such upper bounds using the method of [3]. We also show that sharpening these upper bounds would improve the rate of shrinking neighborhoods of the diagonal x = y in our results. In the special case of metrics with local constant holomorphic sectional curvatures, we obtain off-diagonal asymptotic in a fixed (as $k \to \infty$) neighborhood of the diagonal, which recovers a result of Berman [2] (see Remark 3.5 of [2] for higher dimensions). In this case, we also find an explicit formula for the Bergman kernel mod $O(e^{-k \delta } )$.

The tunneling effect for a class of difference operators

We analyze a general class of self-adjoint difference operators [Formula: see text] on [Formula: see text], where [Formula: see text] is a multi-well potential and [Formula: see text] is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30–35]).Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs.The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian [Formula: see text] is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance [Formula: see text] induced by [Formula: see text], we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first [Formula: see text] eigenvalues of [Formula: see text] converge to the first [Formula: see text] eigenvalues of the direct sum of harmonic oscillators on [Formula: see text] located at several wells. In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low-lying eigenvalues of [Formula: see text]. These are obtained from eigenfunctions or quasimodes for the operator [Formula: see text], acting on [Formula: see text], via restriction to the lattice [Formula: see text].Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrödinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted [Formula: see text]-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two “wells” (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrödinger operator in [22].

Asymptotics of the Solution of a Singular Optimal Distributed Control Problem in a Convex Domain

We consider an optimal distributed control problem in a planar convex domain with smooth boundary and a small parameter at the highest derivatives of an elliptic operator. The zero Dirichlet condition is given on the boundary of the domain, and the control is included additively in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. Solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with a coefficient. This structure of the optimality criterion makes it possible to strengthen, if necessary, the role of either the first or the second term of the criterion. In the first case, it is more important to achieve the desired state, while, in the second case, it is preferable to minimize the resource consumption. We study in detail the asymptotics of the problem generated by the sum of the Laplace operator with a small coefficient and a first-order differential operator. A feature of the problem is the presence of the characteristics of the limit operator which touch the boundary of the domain. We obtain a complete asymptotic expansion of the solution of the problem in powers of the small parameter in the case where the optimal control is an interior point of the set of admissible controls.

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Monotonic Nonlinearity

For a singularly perturbed parabolic equation $${{\epsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\epsilon )$$ in a rectangle, a problem with boundary conditions of the first kind is considered. It is assumed that, at the corner points of the rectangle, the function $$F$$ is cubic in the variable $$u$$ . A complete asymptotic expansion of the solution at $$\varepsilon \to 0$$ is constructed, and its uniformity in a closed rectangle is substantiated.

Generalised magnetic polarizability tensors

We present a new complete asymptotic expansion for the low‐frequency time‐harmonic magnetic field perturbation caused by the presence of a conducting (permeable) object as its size tends to zero for the eddy current regime of Maxwell's equations. The new asymptotic expansion allows the characterisation of the shape and material properties of such objects by a new class of generalised magnetic polarizability tensors, and we provide an explicit formula for their calculation. Our result will have important implications for metal detectors since it will improve small object discrimination, and for situations where the background field varies over the inclusion, this information will be useable, and indeed useful, in identifying their shape and material properties. Thus, improving the ability of metal detectors to locate landmines and unexploded ordnance, sort metals in recycling processes, and ensure food safety as well as enhancing security screening at airports and public events.

Rational invariant tori and band edge spectra for non-selfadjoint operators

We study semiclassical asymptotics for spectra of non-selfadjoint perturbations of selfadjoint analytic $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Complete asymptotic expansions are established for all individual eigenvalues in suitable regions of the complex spectral plane, near the edges of the spectral band, coming from rational flow-invariant Lagrangian tori.

АСИМПТОТИКА РЕШЕНИЯ СИНГУЛЯРНО ВОЗМУЩЕННОЙ ЗАДАЧИ ДИРИХЛЕ СО СЛАБОЙ ОСОБОЙ ТОЧКОЙ

The Dirichlet problem for a singularly perturbed linear homogeneous ordinary differential equation of second order with a nonsmooth coefficient in real axis is considered. Such problems can be seen in physics, engineering, continuum mechanics, hydrodynamics, etc. Object of the research is to develop the asymptotic technique of boundary functions of Vishik–Lusternik–Vasilyeva–Imanaliev for singularly perturbed differential equations in case when the corresponding non-perturbed equation has nonsmooth solution in the considered area. According to terminology of A.M. Ilyin, such problems are called bisingular. The possibility to use a generalized method of boundary functions for constructing a complete proportional asymptotic expansion of the boundary problem solution for a singularly perturbed linear ordinary differential equation of second order with a weak critical point or an integrable critical point is proved in the article. The constructed expansion of solution is asymptotic in the sense of Erdey. When constructing the proportional asymptotic expansion of the Dirichlet problem, the following methods were used: small parameter method, method of mathematical induction, classical method of boundary functions, and the principle of maximum. Using the principle of maximum, an assessment for the asymptotic expansion’s remainder term is obtained, i.e. the proportional complete asymptotic expansion of the solution by small parameter is proved. A specific example is given.

Asymptotic failure rates for a general class of frailty models

Abstract We elucidate the long-term behavior of failure rates for a broad class of frailty models in survival analysis. The class properly includes the proportional hazard frailty model, the additive frailty model, and the accelerated failure time frailty model. A complete asymptotic expansion is derived and compared with the corresponding result for the limiting behavior obtained by Finkelstein and Esaulova (2006a). Several examples are provided to facilitate the comparison and to illustrate both the applicability and the limitations of our approach.

Boundary Behaviour of Weil–Petersson and Fibre Metrics for Riemann Moduli Spaces

Abstract The Weil–Petersson and Takhtajan–Zograf metrics on the Riemann moduli spaces of complex structures for an $n$-fold punctured oriented surface of genus $g,$ in the stable range $g+2n&gt;2,$ are shown here to have complete asymptotic expansions in terms of Fenchel–Nielsen coordinates at the exceptional divisors of the Knudsen–Deligne–Mumford compactification. This is accomplished by finding a full expansion for the hyperbolic metrics on the fibres of the universal curve as they approach the complete metrics on the nodal curves above the exceptional divisors and then using a push-forward theorem for conormal densities. This refines a two-term expansion due to Obitsu–Wolpert for the conformal factor relative to the model plumbing metric which in turn refined the bound obtained by Masur. A similar expansion for the Ricci metric is also obtained.

Szegö Kernels and Asymptotic Expansions for Legendre Polynomials

We present a geometric approach to the asymptotics of the Legendre polynomials Pk,n+1, based on the Szegö kernel of the Fermat quadric hypersurface, leading to complete asymptotic expansions holding on expanding subintervals of [-1,1].