Analytic Coefficients Research Articles

Wave Propagation at Computational Domain Boundaries

The author studies multiple-time initial value problems for linear wave equations that model a situation common in numerical simulation of solutions to hyperbolic partial differential equations formulated on unbounded spatial domains. It is shown that the restriction of the solution to a convex spatial domain is determined by data inside the domain alone at an earlier time, provided that the support of the data at some (perhaps ancient) initial time was contained in the domain. More precisely, it is shown that if the initial (time t{sub 0}) data has support in the open convex set {Omega}, if x is in the closure of {Omega}, and if t{sub 0} < t{sub 1} < t{sub 2}, then the solution at spacetime point (x,t{sub 2}) is determined uniquely by the data at the earlier time t{sub 1} in the spatial region consisting of the intersection of {Omega} with the a priori domain of dependence for the solution at (x,t{sub 2}). That is, at each point in the closure of {Omega} the solution may be advanced in time using only knowledge of the data at a preceding time in that portion of the domain of dependence lying inside {Omega}. Similar conclusions are shown tomore » hold for nonhomogeneous equations. Consequently, the results apply in an approximate sense to solutions of nonlinear equations that have small amplitude near and outside the boundary. These results are obtained by proving more general uniqueness results that localize the dependence of solutions u to partial differential equations P(x,D)u = f where P has analytic coefficients and real principal part.« less

A new form of the Hensel factorization for Taylor series with analytic coefficients

Let K be a complete ultrametric algebraically closed field, let D be a clopen bounded strongly infraconnected set in K and let H( D) be the algebra of the Analytic Elements in D. Let F( Y) = Σ ∞ j=0 a s Y s ϵ H( D)[[itY]]. We assume first: a n divides a 03., a n-1 in H( D) and second: lim s→∞ a n−j s a j a n s D = 0 whenever j=0…, n-1. We also assume there exists a slight subset A ⊂ D such that, for each fixed x ϵ A, the series F x ( Y) with coefficients in K has exactly n zeros in the disk d(0, r( x)) with r(x) = max 0≤J≤ n-1 |(a j/a n)| 1/( n-j) . Then we prove F( Y) factorizes in H( D)[[ Y]] in the form P( Y) G( Y) with P a n-degree monic polynomial in H( D)[ Y] such that the zeros of P x are those of F x in d(0, r( x)), whenever x ϵ D \\ A and G an element of H( D)[[ Y]]. This factorization is the one obtained with stronger hypothesis and with less precision on r( x) in a previous article.

Characteristic multiplier assignment in continuous-time linear periodic systems

A characteristic multiplier assignment problem for continuous-time periodic systems with analytic coefficients is considered. First, the necessary and sufficient condition for controllability is shown. Under this condition, an algorithm for assigning a set of characteristic multipliers is proposed. The algorithm is constructed by a simple repetitive procedure, and is suited to computer programming. The given feedback gain matrix is composed of a sequence of Dirac delta functions; such a feedback system can easily be implemented using proper approximation to the delta function. Lastly, the design method is applied to the attitude control problem of a spin-stabilized satellite, and some simulation results are given.

On local solvability of linear partial differential operators with multiple characteristics

Let WY D)=C,,,., c,(x)D” be a linear partial differential operator with analytic coefficients c,, D = -ia, and L2 an open subset of R”; we denote by G’“)(Q) the class of Gevrey functions of order S, 1 es < 03. The operator L is said to be @)-solvable in 52 if for every f in G:)(Q), the space of all the Gevrey functions of order s with compact support, there exists a solution U, in the space of ultradistributions Gtr(sZ), of the equation Lu =f: We shall speak similarly of (co )-solvable operators when, in the above sentence, the class CF of all the indefinitely differentiable functions with compact support takes the place of Gt’ (and 9’ of Gr”). Owing to these definitions of solvability, when an operator L is not (s)-solvable then it is also not (t)-solvable for any t, s < t < co; we can therefore obtain non-solvability results in the class of indefinitely differentiable functions from non-solvability results in some Gevrey classes. Vice versa, dealing with operators whose unsolvability in the C” category is known, we could study up to which orders s they remain (s)-unsolvable. This way of considering the problem of the solvability of differential operators was introduced by Rodino in [13] and then carried on by the author in [3]; this last work, in particular, is the starting point of the present paper (see also [4]). We shall study here necessary conditions for the local solvability (that is, in small neighborhoods of points in R”) in Gevrey classes of differential operators with analytic coefficients of the form

Transformations of ordinary linear differential equations with analytic coefficients

Transformations of ordinary linear differential equations with analytic coefficients

On the Boundary Value of a Solution of the Heat Equation

Let U be an open subset of R, and let P(x, Dx) be a linear differential operator with analytic coefficients defined on a neighborhood of the closure [£/] of U. Suppose that the boundary dU of U is smooth (i.e., non-singular and analytic) and that dU is non-characteristic with respect to P at each point in dU. Then it is well-known ([7], [11]) that the boundary value of a hyperfunction solution of the equation Pu = 0 on U is a well-defined hyperfunction. However, little is known about the characterization of a solution whose boundary value determines a hyperfunction near a characteristic boundary point. The purpose of this article is to discuss this problem for one n-l special case, i.e., the pair of the heat operator d/dt — A = d/dt — J^ d/dxj def j=i and the domain {(£, t > 0}. Our main result (Theorem 1 below) asserts that, (i) if a C°°-solution u(t, x) does not behave too wildly as t|0, and (ii) if u(t, x) uniformly tends to zero outside a compact set K a R' as *iO, then we can assign a compactly supported hyperfunction g(x) to u(t, x) so that the vanishing of g(x) entails the vanishing of u(t, x) itself. Furthermore we can find such a tame solution u(t, x) of the heat equation for any compactly supported hyperfunction g(x). [See Theorem 1 for the precise statement. Note also that a hyperfunction supported by a compact set, say L, is an analytic functional with the real carrier L.] Let us note the following two facts: First, if u(t, x) tends to infinity too rapidly as t j 0, then our procedure will not assign a hyperfunction g(x). (Cf. §2(i)) Second, we know (see [3], for example) that there exists a hyperfunction e(t, x) (xeR) supported by {(t, x)eR; t = 0, x ^ 0} satisfying the equation

On minimalization of time varying linear systems with well-posed boundary conditions

The notion of minimal realization of a pair of finite rank operators between Hilbert spaces is introduced. Constructions of such minimal realizations are made. A minimal realization of an integral operator with semi-separable kernel is constructed for the case that one of the realizations has analytic coefficients. A consequence for the characterization of minimal stability of an analytic minimal system is given.

Representation of solutions of strongly degenerate elliptic equations with analytic coefficients

q(O, ~=qo(x)#O; p(O, x)=po(~#O if p(z, x)~O. The case when p(z, x)~O, q~, ~=c 2 is constant is studied in [i]. It is proved that to each function v0(x) which is holomorphic in ~ in a neighborhood of zero there corresponds an analytic solution u(z, x) of (i) which at the point z = 0 has an essential singularity and can be represented in the form of a sum of Laplace integrals. The existence of an asymptotic solution is proved and its expansion is given. In [2] the representation of solutions as Laplace integrals is studied in general for even n < 2m and for any n~2m. In the present paper it will be proved that in the statements made above, to each function v0(x) which is holomorphic in ~ there corresponds a solution of (I) which is analytic in a neighborhood of z = 0 which at the point z = 0 has an essential singularity and can be represented in the form of a sum of Laplace integrals. The existence of an asymptotic solution will also be proved and recursion formulas will be given for calculating the coefficients of the asymptotic decomposition. We shall study (i) in the case not included in [2], i.e., we shall assume that n~3 is odd and either n < 2m or p(z, x) E 0~ We make the change of variables z=z~, u(z, x)=Vz~ul(zl, x). (3) Then (cf. [3]) for the function u1(z l, x) we get the differential equation

Remarks on transmission, antitransmission and antilocal properties for sums of stable generators

The antilocal property (to all directions)of an operator A states that, if f=Af=0 in a domain U, then / is identically zero on the whole space. (For the precise definition see §2). This property is known to hold for the class of fractional powers of the Laplacian: A―Ax, where i is a non-integral real number, by Goodman-Segal [6] when the space dimension is odd and by Murata [14] in general. Liess [13] showed this property for the fractional powers of the ellipticdifferentialoperator with analytic coefficients using the theory of pseudodifferential operators with analytic symbol. As is well known, the fractional power Aa/2is the generator of the isotropic stable process in case 0 0, then /=0 on (0, + oo) but not necessarily f=Q in (-co, 0). The result has been shown for aG(0, 1) and will be extended to fl£(0,1)W(1, 2) in §4, b), 2). [It is stillopen if the asymmetric Cauchy generators (the generalized case for a=l) have the similar property.] The proof is carried out on the similar line as in Liess [13], and so Sato-Kashiwara-Kawai theory is essentialin the proof. The firstaim of this note is to show what type of antilocal property holds for a finitesum of one-dimensional stable generators, for example, the sum o1 a one-sided a-stable generator to the right and a one-sided /3-stablegeneratoi to the left. The solution will be given in §4, b), 3). In fact the example citec

The partial stability of motion

It is proved that the problem of stability (asymptotic stability) with respect to some of the variables, for a linear system with periodic analytic coefficients, is equivalent to the same problem with respect to all the variables, either for the same system or for an auxiliary linear system with periodic but not necessarily continuous coefficients, in less dimensions than the original system. A constructive procedure is described for constructing this auxiliary system, and the necessary and sufficient conditions are established for partial stability (asymptotic stability), generalizing the results of the Floquet-Lyapunov theory. It is shown that the class of non-linear systems for which the problem of partial stability is solvable by linear approximation may be enlarged if, instead of the linear part of the original (non-linear) system, one considers a specially constructed linear approximating system which is equivalent to a certain non-linear subsystem of the original system. Constructive procedures are described for constructing such auxiliary systems, and a theorem on partial stability is proved. Well-known theorems on stability in the Lyapunov-critical cases are extended.

EXACT AND ASYMPTOTIC SOLUTIONS OF SYSTEMS WITH TURNING POINTS

A system of linear ordinary differential equations with analytic coefficients and small parameters on the derivative is considered. In a neighborhood of a turning point a new representation is constructed for the exact solution of the system in the form of a multiphase series. It is proved that this series converges uniformly with respect to the parameter. An expression is obtained for the Stokes constant at the maximal exponential. Bibligraphy: 10 titles.

Differential operators and univalent functions

The goal of the present paper is to obtain new, more general sufficient conditions for univalency and quasiconformal extendibility of analytic functions in the domain by means of the general nonlinear differential operators of the form , where F is an analytic function of its arguments (n ≥ 2). It is connected with the solution of the question about the existence of univalent solutions of nonlinear differential equations with analytic coefficients. The most natural conditions are obtained in the case of the operators F is nonlinear. In the paper the operators of such kind in simply connected domains with quasiconformal boundary are considered, in the case of the disk the operator with the linear part a1 f (n)/f′ + a2 f (n)(n = 2, 3) is also regarded.

On the analyticity of the locus of singularity of real analytic solutions with minimal dimension

Let P(x, D) be a linear partial differential operator with real analytic coefficients and let C ⊂ Rn be a germ of closed subset, say at the origin. We say that C is (the locus of) an irremovable singularity of a real analytic solution u of P(x, D)u = 0 if u is defined outside C on a neighborhood Ω of 0 but cannot be extended to the whole neighborhood Ω even as a hyperfunction solution of P(x, D)u = 0. This usage of the word “singularity” is the same as the one for the analytic functions in complex analysis, and is different of the usual usage of “singularities of solutions” in the theory of partial differential equations.

The asymptotic form of the Titchmarsh-Weyl m-function for second-order linear differential equations with analytic coefficients

The asymptotic form of the Titchmarsh-Weyl m-function for second-order linear differential equations with analytic coefficients

On minimality and stable minimality of time-varying linear systems with well-posed boundary conditions

For time-varying linear systems with well-posed boundary conditions the concept of stable minimality is introduced. A complete characterization is given of stably minimal systems with analytic coefficients.

On analytic microlocal hypoellipticity of linear partial differential operators of principal type

It is shown that if P is a linear partial differential uperator with analytic coefficients defined near a point xo in Rn and if P in Rn - 0 is such that: the principal symbol pm,(x, ξ) vanishes at (x0. ξ0). the differential of pm, with respect to ξ is different from zero at (x0, ξ0). the Poisson bracket {Pm, Pm} is zero at (x0. ξ0) and the Poisson bracket {pm, {pm.pm }} is different from zero at (x0, ξ0), then P is analytic hypoelliptic at (x0, ξ0). It is also proved that P is analytic hypoelliptic under the assumption that the first non-vanishing repeated Poisson bracket of pm, and pm, is of odd length and under some additional hypothesis on the commutators of the Hamilton fields of Re pm, and Im pm,

Estimates of the coefficients of asymptotic series for solving the cauchy problem in the case of analytic coefficients

Estimates of the coefficients of asymptotic series for solving the cauchy problem in the case of analytic coefficients

On free boundary problems in two dimensions

Let L be a linear elliptic operator in two dimensions with analytic coefficients and of second order, andu(x, y) a solution of Lu=0 in a simply connected domain ω with rectifiable boundary Γ. Suppose ψ(x, y) analytic on ω∪Γ and L ψ≠0 there.H is shown that ifu and ψ coincide with first derivatives on an open portion Γ0 of Γ, then Γ0 permits the representation λ=x (θ),y=y (θ) withx(θ),y(θ)analytic functions of a real parameter θ.

CONSTRUCTION AND INVESTIGATION OF SOLUTIONS OF DIFFERENTIAL EQUATIONS BY METHODS IN THE THEORY OF APPROXIMATION OF FUNCTIONS

The steady-state equation , the parabolic Cauchy problem , , and the hyperbolic problem , , , are considered, where is a matrix-valued positive selfadjoint second-order partial differential operator with analytic coefficients, and is an analytic function. Methods in the theory of weighted approximation of functions by polynomials on the line are used to construct polynomial representations of solutions of these problems of the form , where the polynomials , , are constructed in explicit form. Estimates of the rate of convergence are given. With the help of these estimates and Bernstein's inverse theorems in approximation theory, theorems are obtained on the smoothness and analyticity of solutions of degenerate systems whose coefficients are trigonometric polynomials.Bibliography: 9 titles.

Explicit invariant solutions for invariant linear differential operators

SynopsisLet F be a real analytic function on a real analytic manifold X. Let P be a linear differential operator on X such that , where Q is an ordinary differential operator with analytic coefficients whose singular points are all regular. For each (isolated) critical value z of F, we construct locally an F-invariant solution u of the equation Pu - v, v being an arbitrary F-invariant distribution supported in F−1(z). The solution u is constructed explicitly in the form of a series of F-invariant distributions.