Pseudo-differential Operators and Non-elliptic Boundary Problems

1966 ◽  
Vol 83 (1) ◽  
pp. 129 ◽  
Author(s):  
Lars Hormander
Keyword(s):  
Differential Operators ◽  
Elliptic Boundary ◽  
Boundary Problems ◽  
Pseudo Differential Operators

scholarly journals Recent developments on Pseudo-Differential Operators (I)

2015 ◽  
Vol 46 (1) ◽  
pp. 1-30 ◽  
Author(s):  
D.-C. Chang ◽  
W. RUNGROTTHEERA ◽  
B.-W. SCHULZE
Keyword(s):  
Boundary Value Problems ◽  
Differential Calculus ◽  
Differential Operators ◽  
Elliptic Boundary ◽  
Higher Order ◽  
Symbolic Calculus ◽  
Elliptic Boundary Value ◽  
The Past ◽  
Pseudo Differential Operators ◽  
Recent Developments

In recent years the analysis of (pseudo-)differential operators on manifolds with second and higher order corners made considerable progress, and essential new structures have been developed. The main objective of this series of paper is to give a survey on the development of this theory in the past twenty years. We start with a brief background of the theory of pseudo-differential operators which including its symbolic calculus on $\R^n$. Next we introduce pseudo-differential calculus with operator-valued symbols. This allows us to discuss elliptic boundary value problems on smooth domains in $\R^n$ and elliptic problems on manifolds. This paper is based on the first part of lectures given by the authors while they visited the National Center for Theoretical Sciences in Hsinchu, Taiwan during May-July of 2014.

Estimates for General Coercive Boundary Problems on a Half-Space for a Class of Elliptic Partial Differential Operators

1968 ◽  
Vol 20 ◽  
pp. 679-697
Author(s):  
Peter C. Greiner
Keyword(s):  
Boundary Value Problems ◽  
Half Space ◽  
Singular Integral ◽  
Differential Operators ◽  
Elliptic Boundary ◽  
Integral Operators ◽  
Boundary Problems ◽  
Elliptic Boundary Value ◽  
Partial Differential Operators ◽  
Boundary Operators

In recent years elliptic boundary value problems have been studied in great detail; see, for example, Agmon (1), Agmon, Douglis, and Nirenberg (2), Browder (4), Hormander (7), Schechter (10; 11; 12), Agranovich and Dynin (3). In all these cases the boundary problems considered were local or semilocal, i.e. the boundary operators involved are differential operators possibly having singular integral operators for coefficients (cf. (3)).

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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