Parametric formal Gevrey asymptotic expansions in two complex time variable problems

Approximate solution of Cauchy's problem for laplace's equation applicable to the problem of shaping of dense spatially inhomogeneous beams of charged particles: PMM vol. 35, no. 4, pp. 656–668

We investigate the problem of tangential incidence of short waves onto a surface with an inflection point. Formal solutions of the corresponding equation are constructed near the inflection point in the form of a quasihomogeneous function series. The formal solution is joined with the geometrical optics solution far from the inflection point of the boundary. The problem is restated as a scattering problem for the Schrodinger equation; existence, uniqueness, and smoothness theorems are proved. The formal asymptotic expansions are proved.

In an earlier paper, the first author showed that certain normalized formal solutions of homogeneous linear partial differential equations with constant coefficients are multisummable, with a multisummability type that can be determined from a Newton polygon associated with the PDE. In this article, some of the results obtained there are extended in several directions: First of all, arbitrary formal solutions of inhomogeous PDE are considered, and it is shown that, in some sense, they can be computed completely explicitly. Secondly, the Gevrey order of these formal solutions is determined. Finally, formal power series are discussed that, in general, do not satisfy a PDE with constant coefficients, but instead may be considered as solutions of singularly perturbed ODE, or integro-differential equations of a certain form. Introduction In [3, 6], the first author introduced and studied normalized formal solutions of a Cauchy problem for general homogeneous linear partial differential equations in two variables having constant coefficients. Multisummability of these formal power series was then investigated in [4]. In detail, it has been shown that, under the assumption that the initial condition used is holomorphic near the origin, one can determine a multisummability type corresponding naturally to the PDE under consideration. The normalized formal solution then is multisummable in a given multidirection, provided that the initial condition can be continued into finitely many (small) sectors, and in every such sector is at most of a certain exponential growth that, in general, depends upon the sector. The multisummability type, the location of the sectors, and the corresponding ∗Institut fur Angewandte Analysis, Universitat Ulm, D–89069 Ulm, Germany †Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526 Japan

Consider a system of N ordinary first-order differential equations in N dependent variables, and let the independent variable s not appear explicitly. Let the system depend on a small parameter ε and possess a formal infinite power series expansion in ε, and suppose that the limiting system for ε = 0 exists and has only periodic solutions. Then a formal solution can be constructed involving infinite power series in ε and satisfying the equations over large domains of s (of order 1/ε). The true solutions of the system exist over such domains and are asymptotically represented as ε→0 by the formal solutions. The construction is based on the standard type of formal series solution of a ``reduced'' system of N − 1 equations in N − 1 dependent variables and with the new independent variable σ = ε s; the omitted variable is essentially an angle variable φ describing the phase around the simple, closed curves. If the original system is Hamiltonian, then one can define the usual action integral J = ∫ p·dq to all orders; the integral is taken around the phase ring. It is proved that J is an integral of the system and that the Poisson bracket of φ with J is unity, both to all orders. The usefulness of this particular integral is that it is computable locally. The reduced system, after elimination of another dependent variable by means of the constancy of J, can itself be put in Hamiltonian form; if its solutions are nearly periodic, the whole procedure can be reapplied. The present theory encompasses previous proofs of adiabatic invariance to all orders for particular systems such as the harmonic oscillator, the nonlinear oscillator, the charged particle gyrating tightly in a given electromagnetic field, and the longitudinal back-and-forth motion of such a particle trapped between two ``magnetic mirrors'' in a weak electric field. There are many other applications.

We consider the Cauchy problem for Kowalevskaya-type fractional linear partial differential equations with constant coefficients in two complex variables. We show that the solutions can be analytically continued into certain sectors and have at most exponential growth there if and only if the Cauchy data have an appropriate property. Applying this result to the study of formal power series solutions of non-Kowalevskian linear partial differential equations, we obtain a characterization of Borel summable solutions in terms of analytic continuation property and growth estimations of the Cauchy data. We also obtain a similar result in the case of non-Kowalevskian fractional equations.

This paper provides a formal exact analytical solution to a rat-hole with a sloping base in two and three dimensions for a highly frictional granular material. A rat-hole is the general term used to describe those stable cavities, which frequently occur in storage hoppers and stock piles, whose formation prevents further material falling through the outlet. Figure 1a depicts the typical geometric configuration, comprising upper and lower sloping surfaces that form a channel or cylindrical cavity. In granular industries this is a commonly occurring situation, for example, where the flow of material from a hopper ceases due to the formation of a stable almost cylindrical vertical cavity. Despite their practical importance, the only analytical solution applies to the perfectly cylindrical cavity, assumed infinite in length with no upper sloping surface. In order to determine analytical solutions to more realistic situations, it is necessary to make compromises with regard to both geometric and constitutive considerations. Here, for both two and three-dimensional rat-holes, we present analytical parametric solutions for the special case of a highly frictional granular material, where the angle of internal friction is equal to ninety degrees. In addition, we assume that the highly frictional granular material is at the point of yield on a sloping rigid base, and with an infinitesimal central outlet as shown in Fig. 1b. The solutions given here are bona fide exact solutions of the governing equations for a Coulomb-Mohr granular solid, and satisfy exactly the free surface conditions on the sloping upper surface and a frictional condition along the sloping rigid base. We emphasize that while all zero-stress boundary conditions are correctly satisfied, and the solutions constitute the only known exact analytical solutions for a realistic rat-hole geometry, the solutions for both geometries exhibit infinite values of the other stress component on the free surface. This feature arises as a consequence of assuming an angle of internal friction equal to ninety degrees, and throws doubt on the physical applicability of the formal exact solution.

Filtration problems are actual for the design of underground structures and foundations, strengthening of loose soil and construction of watertight walls in the porous rock. A liquid grout pumped under pressure penetrates deep into the porous rock. Solid particles of the suspension retained in the pores, strengthen the loose soil and create watertight partitions. The aim of the study is to construct an explicit analytical solution of the filtration problem. A one-dimensional model of deep bed filtration of a monodisperse suspension in a homogeneous porous medium with size-exclusion mechanism of particles retention is considered. Solid particles are freely transferred by the carrier fluid through large pores and get stuck in the throats of small pores. The mathematical model of deep bed filtration includes the mass balance equation for suspended and retained particles and the kinetic equation for the deposit growth. The model describes the movement of concentrations front of suspended and retained particles in an empty porous medium. Behind the concentrations front, solid particles are transported by a carrier fluid, accompanied by the formation of a deposit. The complex model has no explicit exact solution. To construct the asymptotic solution in explicit form, methods of nonlinear asymptotic analysis are used. The new coordinate transformation allows to obtain a parameter that is small at all points of the porous sample at any time. In this paper, a global asymptotic solution of the filtration problem is constructed using a new small parameter. Numerical calculations are performed for a nonlinear filtration coefficient found experimentally. Calculations confirm the closeness of the asymptotics to the solution in the entire filtration domain. For a nonlinear filtration coefficient, the asymptotics is closer to the numerical solution than the exact solution of the problem with a linear coefficient. The analytical solution obtained in the paper can be used to analyze solutions of problems of underground fluid mechanics and fine-tune laboratory experiments.

In this paper, we apply asymptotic–numerical methods for computing non‐linear equilibrium paths of elastic beam, plate and shell structures. The non‐linear branches are sought in the form of asymptotic expansions, and they are determined by solving numerically (FEM) several linear problems with a single stiffness matrix. A large number of terms of the series can be easily computed by using recurrence formulas. In comparison with a more classical step‐by‐step procedure, the method is rapid and automatic. We show, with some examples, that the choice of the expansion's parameter and the use of Padé approximants play an important role in the determination of the size of the domain of convergence.

A uniformly valid asymptotic expansion for integrals of the form \[ F(t,\nu ) = e^{\nu h(t)} \int_t^\infty {e^{ - \nu h(x)} } g(x)x^{\alpha - 1} dx, \] where $0 < \alpha \leqq 1$ and $h(x)$ has a zero of order $r \geqq 1$ at $x = 0$, is established. The result, which generalizes a well-known one for $h(x) = x$, also confirms the formal matched asymptotic expansion solution of the equivalent singular perturbation problem $ - y' + \nu h'(t)y = g(t)t^{\alpha - 1} $, $y(\infty ) = 0$. Comparable matched expansion results are derived for related integrals, including one from Bessel function theory, which do not satisfy differential equations.

We study a class of nonlinear difference equations admitting a 1-Gevrey formal power series solution which, in general, is not 1- (or Borel-) summable. Using right inverses of an associated difference operator on Banach spaces of so-called quasi-functions, we prove that this formal solution can be lifted to an analytic solution in a suitable domain of the complex plane and show that this analytic solution is an accelero-sum of the formal power series.

The equations which govern the flow at high Reynolds number in the vicinity of the trailing edge of a finite flat plate at incidence to a uniform supersonic stream are solved numerically using a finite-difference procedure. The critical order of magnitude of the angle of incidence α* for the occurrence of separation on one side of the plate is α* = O(R−¼) (Brown &amp; Stewartson 1970), where R is a representative Reynolds number for the flow, and results are computed for three such values of α* which characterize the possible behaviour of the flow above the plate. The final set of computations leads to a numerical value for the trailing-edge stall angle α*s, the angle of incidence which just causes the flow to separate at the trailing edge of the plate. Analytic solutions are available in the form of asymptotic expansions near the trailing edge in terms of the scaled variable of order R−⅜. A multi-layer-type of expansion which occurs in the case α* = αs* is presented in detail for comparison with the computed solution.

A hybrid finite-boundary element method for inviscid flows with free surface

We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter ϵ whose coefficients depend holomorphically on $(\epsilon,t)$ near the origin in $\mathbb{C}^{2}$ and are bounded holomorphic on some horizontal strip in $\mathbb{C}$ w.r.t. the space variable. In our previous contribution (Lastra and Malek in Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, arXiv:1403.2350 ), we assumed the forcing term of the Cauchy problem to be analytic near 0. Presently, we consider a family of forcing terms that are holomorphic on a common sector in time t and on sectors w.r.t. the parameter ϵ whose union form a covering of some neighborhood of 0 in $\mathbb{C}^{\ast}$ , which are asked to share a common formal power series asymptotic expansion of some Gevrey order as ϵ tends to 0. We construct a family of actual holomorphic solutions to our Cauchy problem defined on the sector in time and on the sectors in ϵ mentioned above. These solutions are achieved by means of a version of the so-called accelero-summation method in the time variable and by Fourier inverse transform in space. It appears that these functions share a common formal asymptotic expansion in the perturbation parameter. Furthermore, this formal series expansion can be written as a sum of two formal series with a corresponding decomposition for the actual solutions which possess two different asymptotic Gevrey orders, one stemming from the shape of the equation and the other originating from the forcing terms. The special case of multisummability in ϵ is also analyzed thoroughly. The proof leans on a version of the so-called Ramis-Sibuya theorem which entails two distinct Gevrey orders. Finally, we give an application to the study of parametric multi-level Gevrey solutions for some nonlinear initial value Cauchy problems with holomorphic coefficients and forcing term in $(\epsilon,t)$ near 0 and bounded holomorphic on a strip in the complex space variable.

A novel asymptotic representation of the analytic solutions to a family of singularly perturbed q-difference-differential equations in the complex domain is obtained. Such asymptotic relation shows two different levels associated to the vanishing rate of the domains of the coefficients in the formal asymptotic expansion. On the way, a novel version of a multilevel sequential Ramis-Sibuya type theorem is achieved.

In 1996, Braaksma and Faber established the multi-summability, on suitable multi-intervals, of formal power series solutions of locally analytic, nonlinear difference equations, in the absence of “level 1 + ”. Combining their approach, which is based on the study of corresponding convolution equations, with recent results on the existence of flat (quasi-function) solutions in a particular type of domains, we prove that, under very general conditions, the formal solution is accelero-summable. Its sum is an analytic solution of the equation, represented asymptotically by the formal solution in a certain unbounded domain.