On adjoint operators associated with boundary value problems

Since the theory of non-characteristic boundary value problems in the category of hyperfunctions was initiated by H. Komatsu-T. Kawai, and P. Schapira independently ([8], [11]), this theory has made a remarkable progress. ( [5], [6], [2] ) But the methods do not seem to be sufficient for more general problems, for example, hyperbolic mixed problems. Recently, a micro local formulation of non-characteristic boundary value problems was found by the auther ([7]), and it looks like very useful to understand and analyse general boundary value problems. So in this note we try to survey this formulation and the related results. Furthermore readers should refer to the results about mixed problems which were announced by P. Schapira at this symposium. ([12])

It has been shown that, under suitable hypotheses, boundary value problems of the form, $Ly+\lambda y=f,$ $BC y =0$ where $L$ is a linear ordinary or partial differential operator and $BC$ denotes a linear boundary operator, then there exists $\Lambda >0$ such that $f\ge 0$ implies $\lambda y \ge 0$ for $\lambda\in [-\Lambda ,\Lambda ]\setminus\{0\},$ where $y$ is the unique solution of $Ly+\lambda y=f,$ $BC y =0$. So, the boundary value problem satisfies a maximum principle for $\lambda\in [-\Lambda ,0)$ and the boundary value problem satisfies an anti-maximum principle for $\lambda\in (0, \Lambda ]$. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y=f,$ $BC y =0$ where $D_{0}^{\alpha}$ is a Riemann-Liouville fractional differentiable operator of order $\alpha$, $1<\alpha \le 2$, and $BC$ denotes a linear boundary operator, then there exists $\mathcal{B} >0$ such that $f\ge 0$ implies $\beta D_{0}^{\alpha -1}y \ge 0$ for $\beta \in [-\mathcal{B} ,\mathcal{B} ]\setminus\{0\},$ where $y$ is the unique solution of $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y =f,$ $BC y =0$. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of $\beta D_{0}^{\alpha -1}y.$ The boundary conditions are chosen so that with further analysis a sign property of $\beta y$ is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection r H on a (finite rank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of r H. The treatment is via a uniform procedure, even though in most cases no normal Cartan connection is available. The prolongations of a k th order linear differential operator between vector bundles arise by differentiating the given operator D : E ! F, and forming a new system comprising D along with auxiliary operators that capture some of this derived data. To exploit this effectively it is crucial to determine what part of this information should be retained, and then how best to manage it. With this understood, for many classes of operators the resulting prolonged operator can expose key properties of the original differential operator and its equation. Motivated by questions related to integrability and deformations of structure, a theory of overdetermined equations and prolonged systems was developed during the 1950s and 1960s by Goldschmidt, Spencer, and others (2, 17). Generally, results in these works are derived abstractly using jet bundle theory, and are severely restricted in the sense that they apply most readily to differential operators satisfying involutivity conditions. These features mean the theory can be difficult to apply. In the case that the given partial differential operator D : E ! F, has surjective symbol there is an effective algorithmic approach to this problem. The prolongations are constructed from the leading symbol �(D) : J k � 1 E ! F, where J k � 1 is the bundle of symmetric covariant tensors on M of rank k. At a point of M, denoting by K the kernel of �(D), the spaces K ` = ( J ` � 1 K) ( J k+` � 1 E), `� 0, capture spaces of new variables to be introduced, and the system closes up if K ` = 0 for

We consider the Fuchsian linear differential equation obtained (modulo a prime) for , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of and can be removed from and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the ‘depleted’ differential operator and it is shown to be equivalent to the symmetric fourth power of LE, the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order terms and . We conjecture that a linear differential operator equivalent to a symmetric (n − 1) th power of LE occurs as a left-most factor in the minimal order linear differential operators for all 's.

A Chebyshev spectral method (CSM) for the dynamic analysis of non-uniform Timoshenko beams under various boundary conditions and concentrated masses at their ends is proposed. The matrix-based Chebyshev spectral approach was used to construct the spectral differentiation matrix of the governing differential operator and its boundary conditions. A matrix condensation approach is crucially presented to impose boundary conditions involving the homogeneous Cauchy conditions and boundary conditions containing eigenvalues. By taking advantage of the standard powerful algorithms for solving matrix eigenvalue and generalized eigenvalue problems that are embodied in the MATLAB commands, chebfun and eigs, the modal parameters of non-uniform Timoshenko beams under various boundary conditions can be obtained from the eigensolutions of the corresponding linear differential operators. Some numerical examples are presented to compare the results herein with those obtained elsewhere, and to illustrate the accuracy and effectiveness of this method.

This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.

A mathematical model for the formation and development of defects in metals

The general boundary value problem is studied for a parabolic equation in spaces of insufficiently smooth and generalized functions. Starting from Green's formula, the generalized solution of a boundary value problem is defined, and two families (scales) of spaces are constructed in which the boundary value problem is studied: the spaces of solutions , and the spaces of right-hand sides . It is proved that the closure with respect to continuity of the boundary value problem operator establishes an isomorphism of the spaces and for .Bibliography: 35 titles.

Euler type partial differential operators on real analytic functions

Let E, F, H be Banach spaces, $$A,B\in L(E,F)$$ such that $$A+\lambda B$$ and its adjoint $$A^*+\lambda B^*$$ have a biorthogonal system of eigenvectors and associated vectors $$(y_i)_{i=1}^\infty$$ and $$(v_i)_{i=1}^\infty$$ . It is assumed that a closed finite codimensional subspace $$E_0$$ of E exists such that $$E_0$$ is dense in H and such that the restriction of B to $$E_0$$ has a continuous extension to H. It is shown that a subsystem $$(y_i)_{i=k+1}^\infty$$ is minimal in H if $$(v_i)_{i=1}^k$$ has a certain basis property. This abstract result can be applied to regular ordinary linear differential operators in Sobolev spaces whose boundary conditions may depend linearly on the eigenvalue parameter for proving minimality in $$L_p$$ . Explicit examples for Sturm–Liouville problems with one or both boundary conditions depending on $$\lambda$$ are considered.

Linear perturbations of differential of difference operators with polynomials as eigenfunctions

General nonlocal boundary value problems are considered for systems of impulsive equations with finite and fixed points of impulses. Sufficient conditions are established for the solvability and unique solvability of these problems, among them effective spectral conditions.

An analogue of the irreducibility is developed for linear differential operators, or, more generally, noncommutative polynomials, and is applied to a few simple examples. Introduction. The question of the irreducibility of an ordinary linear differential operator is of interest in the Picard-Vessiot theory (see Kolchin [1, ?22]). Indeed, the operator is irreducible if and only if the Galois group is linearly irreducible. In this note we develop an Eisenstein criterion to help answer this question. Because a linear differential operator is a special type of noncommutative polynomial, it is more natural to phrase the result in this more general context. The result, and indeed the proof with only slight changes in notation, is valid for partial differential operators, or more generally, noncommutative polynomials in several variables. Because the hypothesis that the ring be principal is unnatural in this setting, and because the reader may make the necessary changes if he so desires, we confine our attention to the ordinary case. 1. The irreducibility criterion. By a differential ring M we shall mean a ring (associative, commutative, with unit) together with two commuting operators a and 3 such that a is an automorphism of R, 3 is additive, and 6(ab)=aa3b+6a.b for every a, b in S. 1[6] will denote the noncommutative ring of differential operators with coefficients in R, so that 1[6] is the set of 0 ai6i, with ai E $, n E N, and right multiplication is defined by 6 * a-=oa * + 3a. Evidently if or=id, then W is an ordinary differential ring with derivation operator 6. By a differential ideal of R we mean an ideal which is closed under the two operators a and 6. The smallest differential ideal which contains the set Ec$S is denoted by [2]. Received by the editors October 7, 1971. AMS 1970 subject classifications. Primary 12H05, 12E05; Secondary 13B25, 16A02.

In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W^3_2(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed.

The paper considers a mixed initial-boundary value problem for a parabolic equation with nonhomogeneous boundary conditions. The classical approach to search for analytical solution of such problems in the first phase involves variable substitution, leading to a problem with homogeneous boundary conditions. Reference materials [1] give, as a rule, the simplest types of variable substitutions where new and old unknown functions differ by a term, linear in the spatial variable. The form of this additive term depends on the type of the boundary conditions, but is in no way related to the equation under consideration. Moreover, in the case of the second boundary-value problem, it is necessary to use a quadratic additive, since a linear substitution for this type of conditions may be unavailable. The courseware [2] - [4], usually, ends only with the first boundary-value problem generally formulated. The paper considers a substitution that takes into account, in principle, the form of a linear differential operator. Namely, as an additive term, it is proposed to use the parametrically time-dependent solution of the boundary value problem for an ordinary differential equation obtained from the original partial differential equation by the method of separation of the Fourier variables. The existence of the proposed substitution for boundary conditions of any type is proved by the example of a non-stationary heat-transfer equation with the heat exchange available with the surrounding medium. In this case, the additive term is a linear combination of hyperbolic functions. It is shown that, in addition to the insensitivity to the type of boundary conditions, the advantages of a new substitution in comparison with the traditional linear (or quadratic) one include a much simpler structure of the solution obtained. Just the described approach allows us to obtain a solution with a clearly distinguished stationary component, in case a stationarity occurs, for values of the time variable that exceeds relaxation time. This effect extends not only to parabolic equations, but is also observed in equations describing oscillatory processes with damping. It is also shown that in the simplest particular case of an equation with constant coefficients without heat exchange with the surrounding medium, the proposed substitution becomes appropriately linear and coincides with the classical one. The use of the proposed approach allows students majoring in engineering to expand a range of educational and research tasks while learning a course in partial differential equations.