Partial Differential Equations of Elliptic Type

2004 ◽  
pp. 197-311
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskii
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Type ◽  
Partial Differential

APPROXIMATION OF FUNCTIONS BY GAUSS-WEIERSTRASS INTEGRALS

2021 ◽  
Vol 2 ◽  
pp. 112-118
Author(s):  
Olga Shvai ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Modulus Of Continuity ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Elliptic Type ◽  
Partial Differential ◽  
Linear Positive Operator ◽  
Zygmund Class ◽  
Boundary Properties

When considering various schemes and algorithms for game problems of dynamics, researchers often have to deal with solutions of partial differential equations. A special place among the latter is occupied by the so-called equations of elliptic type (according to the corresponding classification), with the help of which natural and social processes can be described most fully and qualitatively. Moreover, the mathematical apparatus of partial differential equations of elliptic type makes it possible to get into the environment of deterministic phenomena and thus makes it possible to foresee their future. This fact undoubtedly increases the significance of the above type of equations among others in the sense of their application to mathematical modeling. At the same time, one of the most important concepts in applied mathematics is the concept of the modulus of continuity. The term "modulus of continuity" and its definition were introduced by Henri Lebesgue at the beginning of the last century in order to study various properties of continuous functions. Using the concept of the modulus of continuity and its properties, it is possible to investigate the belonging of the object under study to a certain class of functions: Hölder, Lipschitz, Zygmund, etc. This undoubtedly makes it possible to approximate functions of various kinds of operators most effectively. In this paper, using the example of the Gauss-Weierstrass integral as a solution to the corresponding differential equation of elliptic type, we study its rate of convergence in terms of the modulus of continuity of the second order to the function by which it was actually constructed. Namely, the boundary properties of the Gauss-Weierstrass integral were studied as a linear positive operator that realizes its best approximation on functions from the Zygmund class. The results obtained in this article can further be used to solve many problems in applied mathematics.

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.

Interior Estimates for Elliptic Partial Differential Equations in the ℒ(q,λ) Spaces of Strong Type

1984 ◽  
Vol 36 (3) ◽  
pp. 385-404
Author(s):  
Akira Ono
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Partial Differential Equations ◽  
The Other ◽  
Usual Sense ◽  
Elliptic Type ◽  
Strong Type ◽  
Partial Differential ◽  
Interior Estimates ◽  
Isomorphism Theorems

Recently the ℒ(q,λ) spaces have been investigated by many authors and the theory of these spaces has proved to be particularly important for research in partial differential equations (see for example [15], [16] and [18]).The equations of elliptic type in these spaces were first studied by C. B. Morrey [8], [9], who applied his well-known imbedding theorems, and afterwards by S. Campanato [3], [4] with the aid of isomorphism theorems and the so-called fundamental inequalities due to him.On the other hand, G. Stampacchia introduced the ℒ(q,λ) spaces of strong type [17], the structures of which are more general and complicated than those of ℒ(q,λ) Spaces in the usual sense, and greater part of them were characterized by him, L. C. Piccinini, Y. Furusho, the author and others (see [5], [11]-[14], [16] and [17]).

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