scholarly journals A Carleman estimate for the linear shallow shell equation and an inverse source problem

2008 ◽  
Vol 23 (1/2) ◽  
pp. 367-380
Author(s):  
Shumin Li ◽  
Bernadette Miara ◽  
Masahiro Yamamoto
Keyword(s):  
Shallow Shell ◽  
Carleman Estimate ◽  
Inverse Source Problem ◽  
Shell Equation

scholarly journals Inverse parabolic problems of determining functions with one spatial-component independence by Carleman estimate

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Oleg Y. Imanuvilov ◽  
Yavar Kian ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Boundary Data ◽  
Stability Estimate ◽  
Carleman Estimate ◽  
Parabolic Problems ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Lateral Boundary ◽  
Spatial Component

Abstract For a parabolic equation in the spatial variable x = ( x 1 , … , x n ) {x=(x_{1},\ldots,x_{n})} and time t, we consider an inverse problem of determining a coefficient which is independent of one spatial component x n {x_{n}} by lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. Also, we prove similar results for the corresponding inverse source problem.

scholarly journals Size Estimates of the Inverse Inclusion Problem for the Shallow Shell Equation

2013 ◽  
Vol 45 (1) ◽  
pp. 88-100 ◽  
Author(s):  
M. Di Cristo ◽  
C.-L. Lin ◽  
S. Vessella ◽  
J.-N. Wang
Keyword(s):  
Shallow Shell ◽  
Inclusion Problem ◽  
Shell Equation ◽  
Size Estimates

Some Results on the Analytical Solution of Cylindrical Shells With Large Openings

1991 ◽  
Vol 113 (2) ◽  
pp. 297-307 ◽  
Author(s):  
M.-D. Xue ◽  
Y. Deng ◽  
K.-C. Hwang
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Stress Analysis ◽  
Shallow Shell ◽  
Cylindrical Shells ◽  
Geometric Description ◽  
Present Solution ◽  
Approximate Boundary Conditions ◽  
Shell Equation ◽  
Good Agreement

An analytical solution for cylindrical shells with large openings has been developed by the asymptotic method. In comparison with previous analytical solutions obtained by Eringen, Van Dyke and Lekkerkerker, the following three areas have been improved by the present method: 1) The modified Morley’s equation, which is applicable to ro/RT≫1, is used instead of Donnell’s shallow shell equation. 2) The accurate expression of boundary curve of hole is expanded in terms of powers of ρo = ro/R, while the previous approximate geometric description corresponds to the first term of the expansion. 3) The accurate boundary conditions for generalized forces are expanded in terms of powers of ρo and truncated after the terms of ρo3; the previous approximate boundary conditions correspond to the terms of order up to ρo, in the asymptotic expansions. The present solutions are in good agreement with Van Dyke’s solutions for small openings. The analytical results show that the error caused by the previous approximate boundary conditions is significant and has the order O(ρo); the error caused by the previous approximate geometric description of hole boundary has the order O(ρo2); and finally, the error caused by using Donnell’s shallow shell equation is very small for ρo ≤ 0.7. For openings with ρo > 0.25, say, noticeable differences exist between the present and the previous results. The stress analysis of cylindrical shells connected with nozzles has been developed by using the present solution. For increasing the accuracy of stress analysis in the nozzle, the exact description of the edge of nozzle is given and the near-edge boundary layer stress state in the nozzle is considered. The results obtained are in good agreement with those by Eringen for small openings and with those by F.E.M. and experiments for large openings.

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