Asymptotic expansions for certain exponential-type operators connected with $$2x^{3/2}$$

In the present paper, we consider the complete asymptotic expansion of certain exponential-type operators connected with $$2x^{3/2}$$ 2 x 3 / 2 . Also, a modification of such exponential-type operators is provided, which preserve the function $$\mathrm{e}^{Ax}$$ e Ax .

Data Processing Method of Well Testing

Horner’s traditional method of processing well test data can be improved by a special transformation of the pressure curves, which reduces the time the converted curves reach the asymptotic regimes necessary for processing these data. In this case, to take into account the action of the «skin factor» and the effect of the wellbore, it is necessary to use a more complete asymptotic expansion of the exact solution of the conductivity equation at large values of time. At the same time, this method does not allow to completely eliminate the influence of the wellbore, since the used asymptotic expansion of the solution for small values of time is limited by the existence of a singular point, in the vicinity of which the asymptotic expansion ceases to be valid. To solve this problem, a new method of processing well test data is proposed, which allows completely eliminating the influence of the wellbore. The method is based on the introduction of a modified inflow function to the well, which includes a component of the boundary condition corresponding to the influence of the wellbore.

A complete asymptotic expansion for operators of exponential type with $$p\left( x\right) =x\left( 1+x\right) ^{2}$$

In the year 1978, Ismail and May studied operators of exponential type and proposed some new operators which are connected with a certain characteristic function $$p\left( x\right) $$ p x . Several of these operators were not separately studied by researchers due to its unusual behavior. The topic of the present paper is the local rate of approximation of a sequence of exponential type operators $$R_{n}$$ R n belonging to $$p\left( x\right) =x\left( 1+x\right) ^{2}$$ p x = x 1 + x 2 . As the main result we derive a complete asymptotic expansion for the sequence $$\left( R_{n}f\right) \left( x\right) $$ R n f x as n tends to infinity.

A Complete Asymptotic Expansion for Bernstein–Chlodovsky Polynomials for Functions on $$\mathbb {R}$$

We consider a variant of the Bernstein–Chlodovsky polynomials approximating continuous functions on the entire real line and study its rate of convergence. The main result is a complete asymptotic expansion. As a special case we obtain a Voronovskaja-type formula previously derived by Karsli [11].

Asymptotic behavior of acoustic waves scattered by very small obstacles

The direct numerical simulation of the acoustic wave scattering created by very small obstacles is very expensive, especially in three dimensions and even more so in time domain. The use of asymptotic models is very efficient and the purpose of this work is to provide a rigorous justification of a new asymptotic model for low-cost numerical simulations. This model is based on asymptotic near-field and far-field developments that are then matched by a key procedure that we describe and demonstrate. We show that it is enough to focus on the regular part of the wave field to rigorously establish the complete asymptotic expansion. For that purpose, we provide an error estimate which is set in the whole space, including the transition region separating the near-field from the far-field area. The proof of convergence is established through Kondratiev's seminal work on the Laplace equation and involves the Mellin transform. Numerical experiments including multiple scattering illustrate the efficiency of the resulting numerical method by delivering some comparisons with solutions computed with a finite element software.

Complete Asymptotics for Solution of Singularly Perturbed Dynamical Systems with Single Well Potential

We consider a singularly perturbed boundary value problem ( − ε 2 ∆ + ∇ V · ∇ ) u ε = 0 in Ω , u ε = f on ∂ Ω , f ∈ C ∞ ( ∂ Ω ) . The function V is supposed to be sufficiently smooth and to have the only minimum in the domain Ω . This minimum can degenerate. The potential V has no other stationary points in Ω and its normal derivative at the boundary is non-zero. Such a problem arises in studying Brownian motion governed by overdamped Langevin dynamics in the presence of a single attracting point. It describes the distribution of the points at the boundary ∂ Ω , at which the trajectories of the Brownian particle hit the boundary for the first time. Our main result is a complete asymptotic expansion for u ε as ε → + 0 . This asymptotic is a sum of a term K ε Ψ ε and a boundary layer, where Ψ ε is the eigenfunction associated with the lowest eigenvalue of the considered problem and K ε is some constant. We provide complete asymptotic expansions for both K ε and Ψ ε ; the boundary layer is also an infinite asymptotic series power in ε . The error term in the asymptotics for u ε is estimated in various norms.

Counting connected graphs with large excess

International audience We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion.

The Large Genus Asymptotic Expansion of Masur–Veech Volumes

We study the asymptotic behavior of Masur–Veech volumes as the genus goes to infinity. We show the existence of a complete asymptotic expansion of these volumes that depends only on the genus and the number of singularities. The computation of the 1st term of this asymptotics expansion was a long-standing problem. This problem was recently solved in [2] using purely combinatorial arguments and then in [3] using algebro-geometric insights. Our proof relies on a combination of both methods.

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

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 $O(e^{-k \delta } )$.