Differential operators, their fundamental solutions and related integral representations in Clifford analysis

2006 ◽  
Vol 51 (5-6) ◽  
pp. 407-427 ◽  
Author(s):  
H. Begehr ◽  
H. Otto ◽  
Z. X. Zhang
Keyword(s):  
Clifford Analysis ◽  
Differential Operators ◽  
Fundamental Solutions ◽  
Integral Representations ◽  
Related Integral

scholarly journals On the Stability in Bonnet's Theorem of the Surface Theory

2007 ◽  
Vol 14 (3) ◽  
pp. 543-564
Author(s):  
Yuri G. Reshetnyak
Keyword(s):  
Differential Operators ◽  
Sobolev Class ◽  
Integrability Condition ◽  
Complete Integrability ◽  
Integral Representations ◽  
Completely Integrable Systems ◽  
Frobenius Theorem ◽  
Fundamental Tensor ◽  
The Stability ◽  
Codazzi Equations

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

On integral representations of correct restrictions and regular extensions of differential operators

2010 ◽  
Vol 81 (1) ◽  
pp. 94-96 ◽  
Author(s):  
T. Sh. Kal’menov ◽  
B. E. Kanguzhin ◽  
B. D. Koshanov
Keyword(s):  
Differential Operators ◽  
Integral Representations

scholarly journals An axially symmetric forced convection problem

1975 ◽  
Vol 20 (1) ◽  
pp. 1-17
Author(s):  
J. A. Belward
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Convection Heat Transfer ◽  
Fundamental Solutions ◽  
Integral Representations ◽  
Original Equation ◽  
Transfer Model ◽  
Axially Symmetric ◽  
Simple Diffusion ◽  
Diffusion Convection

AbstractA simple diffusion-convection heat transfer model is formulated which leads to an axially symmetric partial differential equation. The equation is shown to be closely related to a second one which is adjoint to the original equation in one variable can and be interpreted as a description of another diffusion-convection model. Fundamental solutions of the original equation are constructed and interpreted with reference to both models. Some boundary value problems are solved in series form and integral representations of the solutions are also given. The boundary value problems are shown to be equivalent to an integral equation and the correspondence between the two formulations is understood in terms of the two diffusion-convection problems. A Péclet number is defined in one of the boundary value problems and the behaviour of the solutions is studied for large and small values of this parameter.

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