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Partial differential equations I — elliptic equations [PDF]

, 1986
In this chapter we start to examine some techniques used for the numerical solution of partial differential equations (PDEs) and, in particular, equations which are special cases of the linear second-order equation with two independent variables $$a\frac{{{\partial ^2}u}}{{\partial {x^2}}} + b\frac{{{\partial ^2}u}}{{\partial x\partial y}} + c\frac{
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Orthogonal Collocation for Elliptic Partial Differential Equations

SIAM Journal on Numerical Analysis, 1976
An $O(\rho ^4 )$ collocation method ($\rho $ the mesh size) is presented for solving elliptic partial differential equations on the unit square and a convergence proof is given. The method is shown to compare favorably with the Ritz–Galerkin method, and some numerical results demonstrate the effectiveness of the method.
P. M. Prenter, R. D. Russell
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On solvability of a quasi-elliptic partial differential equations

Journal of Elliptic and Parabolic Equations, 2019
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Nilufer R. Rustamova, Alik M. Najafov, Alik M. Najafov, Sain T. Alekberli   +3 more
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Elliptic Partial Differential Equations

2016
General existence theories for solutions of partial differential equations require using concepts from functional analysis and considering generalizations of classical derivatives based on a multidimensional integration-by-parts formula. The chapter introduces Sobolev spaces, discusses their main properties, states existence theories for elliptic ...
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Elliptic Partial Differential Equations

1984
In this chapter we review the main tools used to study elliptic partial differential equations (PDE): Sobolev spaces, variational formulations, and continuous dependence on the data.
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Partial Differential Equations of Elliptic Type

Physics Bulletin, 1971
C Miranda Berlin: Springer 1970 pp xii + 370 price DM 58 This is a translation of the second revised edition of the monograph published in 1955. The number of pages has risen from 222 to 370 of which 69 are occupied by a bibliography.
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Partial Differential Equations of Elliptic Type

2004
In the present chapter we consider the well-posedness of an abstract boundary-value problem for differential equations of elliptic type $$- \upsilon ''\left( t \right) + A\upsilon \left( t \right) = f\left( t \right)\left( {0 \leqslant t \leqslant T} \right),\upsilon \left( 0 \right) = {{\upsilon }_{0}},\upsilon \left( T \right) = {{\upsilon }_{T}}$$
Pavel E. Sobolevskii, Allaberen Ashyralyev   +1 more
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Convexity of Solutions to Some Elliptic Partial Differential Equations

SIAM Journal on Mathematical Analysis, 1993
The authors reprove a result of \textit{A. U. Kennington} [Indiana Univ. Math. J. 34, 687-704 (1985; Zbl 0566.35025)] which states that the concavity function \(C\) of solutions \(v\) to elliptic equations \(a^{ij}(Dv)v_{ij}=b(x,v,Dv)\) cannot have a local positive maximum in \(\Omega \times \Omega\). The new proof makes use of a stronger assumption \((
GRECO, ANTONIO, PORRU GIOVANNI
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Partial differential equations of multi-quasi-elliptic type

ANNALI DELL UNIVERSITA DI FERRARA, 1999
Summary: The theory of multi-quasi-elliptic operators and associated Sobolev spaces is revised in this article and possible directions for research are indicated concerning operators of principal type and nonlinear equations. Moreover, some new results concerning operators with anti-Wick symbols are presented.
BOGGIATTO, Paolo, RODINO, Luigi Giacomo
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Linear Elliptic Partial Differential Equations

2017
In earlier chapters, we described how to apply the finite element method to ordinary differential equations. For the remainder of this book, we will focus on extending this technique for application to partial differential equations. As with ordinary differential equations, we begin with a simple example to illustrate the key features.
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