Inverse problems of mixed type in linear plate theory

2004 ◽  
Vol 15 (2) ◽  
pp. 129-146 ◽  
Author(s):  
DOMINGO SALAZAR ◽  
REX WESTBROOK
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Existence And Uniqueness ◽  
Mixed Type ◽  
Plate Theory ◽  
Order Partial Differential Equation ◽  
General Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Sag Bending

The characterisation of those shapes that can be made by the gravity sag-bending manufacturing process used to produce car windscreens and lenses is modelled as an inverse problem in linear plate theory. The corresponding second-order partial differential equation for the Young's modulus is shown to change type (possibly several times) for certain target shapes. We consider the implications of this behaviour for the existence and uniqueness of solutions of the inverse problem for some frame geometries. In particular, we show that no general boundary conditions for the inverse problem can be prescribed if it is desired to achieve certain kinds of target shapes.

scholarly journals Determination of a control parameter in a parabolic partial differential equation

1991 ◽  
Vol 33 (2) ◽  
pp. 149-163 ◽  
Author(s):  
J. R. Cannon ◽  
Yanping Lin ◽  
Shingmin Wang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Global Solution ◽  
Existence And Uniqueness ◽  
Control Parameter ◽  
Parabolic Partial Differential Equation ◽  
Solution Pair ◽  
Image Position

AbstractThe authors consider in this paper the inverse problem of finding a pair of functions (u, p) such thatwhere F, f, E, s, αi, βi, γi, gi, i = 1, 2, are given functions.The existence and uniqueness of a smooth global solution pair (u, p) which depends continuously upon the data are demonstrated under certain assumptions on the data.

scholarly journals Mixed problem with integral boundary condition for a high order mixed type partial differential equation

2003 ◽  
Vol 16 (1) ◽  
pp. 69-79 ◽  
Author(s):  
M. Denche ◽  
A. L. Marhoune
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Mixed Problem ◽  
Mixed Type ◽  
Energy Inequality ◽  
Integral Boundary Conditions ◽  
High Order ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Integral Boundary

In this paper, we study a mixed problem with integral boundary conditions for a high order partial differential equation of mixed type. We prove the existence and uniqueness of the solution. The proof is based on energy inequality, and on the density of the range of the operator generated by the considered problem.

scholarly journals A three-point boundary value problem with an integral condition for a third-order partial differential equation

2005 ◽  
Vol 2005 (1) ◽  
pp. 33-43 ◽  
Author(s):  
C. Latrous ◽  
A. Memou
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Existence And Uniqueness ◽  
Integral Boundary Conditions ◽  
Integral Condition ◽  
Order Equation ◽  
Order Partial Differential Equation ◽  
Point Boundary ◽  
Third Order ◽  
Integral Boundary

We prove the existence and uniqueness of a strong solution for a linear third-order equation with integral boundary conditions. The proof uses energy inequalities and the density of the range of the operator generated.

scholarly journals Nonlinear diffusion in transparent media: The resolvent equation

2018 ◽  
Vol 11 (4) ◽  
pp. 405-432 ◽  
Author(s):  
Lorenzo Giacomelli ◽  
Salvador Moll ◽  
Francesco Petitta
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dirichlet Problem ◽  
Nonlinear Diffusion ◽  
Existence And Uniqueness ◽  
A Priori ◽  
Resolvent Equation ◽  
Uniqueness Of Solutions ◽  
Transparent Media ◽  
Bounded Variation Functions

AbstractWe consider the partial differential equationu-f=\operatornamewithlimits{div}\biggl{(}u^{m}\frac{\nabla u}{|\nabla u|}% \biggr{)}with f nonnegative and bounded and {m\in\mathbb{R}}. We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the {{\mathcal{H}}^{N-1}}-Hausdorff measure. Results and proofs extend to more general nonlinearities.

scholarly journals Mixed problem with nonlocal boundary conditions for a third-order partial differential equation of mixed type

2001 ◽  
Vol 26 (7) ◽  
pp. 417-426 ◽  
Author(s):  
M. Denche ◽  
A. L. Marhoune
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Mixed Problem ◽  
Mixed Type ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
Partial Differential ◽  
Integral Boundary ◽  
Equation Of Mixed Type

We study a mixed problem with integral boundary conditions for a third-order partial differential equation of mixed type. We prove the existence and uniqueness of the solution. The proof is based on two-sided a priori estimates and on the density of the range of the operator generated by the considered problem.

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