A Tensor Equation of Elliptic Type

1953 ◽  
Vol 5 ◽  
pp. 524-535 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Riemannian Manifold ◽  
Differential Equations ◽  
Tensor Field ◽  
Differential Operators ◽  
Elliptic Type ◽  
Tensor Equation ◽  
Tensor Theory ◽  
Elliptic Differential Equations ◽  
Invariant Differential ◽  
Image Position

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be writtenin a notation explained below.

Limit Point Criteria for Differential Equations, II

1974 ◽  
Vol 26 (02) ◽  
pp. 340-351 ◽  
Author(s):  
Don Hinton
Keyword(s):  
Differential Equations ◽  
Weight Function ◽  
Limit Point ◽  
Differential Operators ◽  
Linearly Independent ◽  
Integrable Functions ◽  
Image Position ◽  
Continuous Weight ◽  
Finite Singularity

We consider here singular differential operators, and for convenience the finite singularity is taken to be zero. One operator discussed is the operator L defined by where q 0 > 0 and the coefficients q t are real, locally Lebesgue integrable functions defined on an interval (a, b). For a given positive, continuous weight function h, conditions are given on the functions qi for which the number of linearly independent solutions y of L(y) = λhy (Re λ = 0) satisfying.

Uniqueness theorems and the maximum-minimum principle for a type of non-linear partial differential equations

1938 ◽  
Vol 34 (4) ◽  
pp. 527-533
Author(s):  
W. H. J. Fuchs ◽  
P. Weiss
Keyword(s):  
Differential Equations ◽  
Minimal Surfaces ◽  
Minimum Principle ◽  
Linear Equations ◽  
Variational Problems ◽  
Elliptic Type ◽  
Principle Of Superposition ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

It is well known that solutions of partial linear differential equations of the second order and of elliptic type are uniquely determined by their boundary data, and that they assume their maximum and minimum values on the boundary. The usual proofs make use of the principle of superposition and are therefore not applicable to non-linear problems. But recently Pryce has proved the uniqueness theorem for the non-linear equations of minimal surfaces and of Born's electrostatics. These equations are the Euler equations of the variational problemk = + 1 corresponds to the case of minimal surfaces in n + 1 dimensions; k = − b−2, n = 3 corresponds to Born's electrostatics. Pryce's procedure depends essentially on the notion of conjugate variables in the calculus of variations for multiple integrals and can therefore be extended to a wide class of differential equations arising from variational problems (for several functions of several variables) as we show in § 3.

scholarly journals Poly-Sinc Solution of Stochastic Elliptic Differential Equations

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
Maha Youssef ◽  
Roland Pulch
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Polynomial Chaos ◽  
Polynomial Interpolation ◽  
Elliptic Type ◽  
Conformal Map ◽  
Partial Differential ◽  
Elliptic Differential Equations ◽  
Collocation Technique ◽  
First Time

AbstractIn this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a conformal map and when mixed with the polynomial interpolation, it yields an accurate approximation. The first step to solve SPDE is to use stochastic Galerkin method in conjunction with polynomial chaos, which implies a system of deterministic partial differential equations to be solved. The main difficulty is the higher dimensionality of the resulting system of partial differential equations. The idea here is to solve this system using a small number of collocation points in space. This collocation technique is called Poly-Sinc and is used for the first time to solve high-dimensional systems of partial differential equations. Two examples are presented, mainly using Legendre polynomials for stochastic variables. These examples illustrate that we require to sample at few points to get a representation of a model that is sufficiently accurate.

Limit Point Criteria for Differential Equations

1972 ◽  
Vol 24 (2) ◽  
pp. 293-305 ◽  
Author(s):  
Don Hinton
Keyword(s):  
Differential Equations ◽  
Limit Point ◽  
Differential Operators ◽  
Central Importance ◽  
Linearly Independent ◽  
Ordinary Differential Operators ◽  
Image Position

For certain ordinary differential operators L of order 2n, this paper considers the problem of determining the number of linearly independent solutions of class L2[a, ∞) of the equation L(y) = λy. Of central importance is the operator0.1where the coefficients pi are real. For this L, classical results give that the number m of linearly independent L2[a, ∞) solutions of L(y) = λy is the same for all non-real λ, and is at least n [10, Chapter V]. When m = n, the operator L is said to be in the limit-point condition at infinity. We consider here conditions on the coefficients pi of L which imply m = n. These conditions are in the form of limitations on the growth of the coefficients.

Oscillation criteria and the discreteness of the spectrum of self-adjoint, even order, differential operators

1991 ◽  
Vol 119 (3-4) ◽  
pp. 219-232 ◽  
Author(s):  
Ondřej Došlý
Keyword(s):  
Differential Equations ◽  
Spectral Properties ◽  
Differential Operators ◽  
Unified Approach ◽  
Oscillation Criteria ◽  
Even Order ◽  
Image Position ◽  
Oscillation Properties

SynopsisThis paper deals with the oscillation properties of self-adjoint differential equationsThe oscillation criteria are derived, which allows a unified approach to the investigation of (*) near a finite or infinite singularity. These criteria are used to study spectral properties of singular differential operators associated with (*).

scholarly journals Eigenspaces of the Laplace-Beltrami operator on a hyperboloid

1980 ◽  
Vol 79 ◽  
pp. 151-185 ◽  
Author(s):  
Jiro Sekiguchi
Keyword(s):  
Differential Equations ◽  
Symmetric Space ◽  
Differential Operators ◽  
Beltrami Operator ◽  
Poisson Integral ◽  
Riemannian Symmetric Space ◽  
Systems Of Differential Equations ◽  
Rank One ◽  
Joint Eigenfunctions ◽  
Invariant Differential

Ever since S. Helgason [4] showed that any eigenfunction of the Laplace-Beltrami operator on the unit disk is represented by the Poisson integral of a hyperfunction on the unit circle, much interest has been arisen to the study of the Poisson integral representation of joint eigenfunctions of all invariant differential operators on a symmetric space X. In particular, his original idea of expanding eigenfunctions into K-finite functions has proved to be generalizable up to the case where X is a Riemannian symmetric space of rank one (cf. [4], [5], [11]). Presently, extension to arbitrary rank has been completed by quite a different formalism which views the present problem as a boundary-value problem for the differential equations. It should be recalled that along this line of approach a general theory of the systems of differential equations with regular singularities was successfully established by Kashiwara-Oshima (cf. [6], [7]).

scholarly journals On generalized Whittaker functions on Siegel’s upper half space of degree 2

1991 ◽  
Vol 121 ◽  
pp. 171-184 ◽  
Author(s):  
S. Niwa
Keyword(s):  
Differential Equations ◽  
Representation Theory ◽  
Lie Groups ◽  
Half Space ◽  
Differential Operators ◽  
Whittaker Functions ◽  
System Of Differential Equations ◽  
Full Generality ◽  
Elementary Calculus ◽  
Invariant Differential

In [5], H. Maass showed that the dimension of a space of generalized Whittaker functions satisfying certain system of differential equations on Siegel’s upper half space H2 of degree 2 is three. First of all, we shall investigate the structure of a space of generalized Whittaker functions which are eigen functions for the algebra of invariant differential operators on H2. The theory of generalized Whittaker functions is discussed in Yamashita [12], [13], [14], [15] with full generality. But, we will get an outlook of the space of generalized Whittaker functions by using elementary calculus instead of representation theory of Lie groups.

scholarly journals Characterization of certain differential operators in the solution of linear partial differential equations

1976 ◽  
Vol 17 (2) ◽  
pp. 83-88 ◽  
Author(s):  
Rudolf Heersink
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Differential Operators ◽  
Holomorphic Functions ◽  
Linear Partial Differential Equations ◽  
Representation Of Solutions ◽  
Number Of Publications ◽  
Image Position ◽  
The Given

In this paper we consider differential equations of the formwhere the coefficients Ai are holomorphic functions in a domain G1 × G2 ⊂ C × C. We restrict our attention to those equations for which it is possible to represent the solutions in the formwhere g1(z1) and g2(z2) are arbitrary holomorphic functions in G1 and G2 respectively. The coefficients a1, k and a2, k depend on the given differential equation. Within the last ten years a number of publications have been devoted to this kind of representation of solutions.

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

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