scholarly journals Existence results for a degenerated nonlinear elliptic partial differential equation

2005 ◽  
Vol 310 (2) ◽  
pp. 641-656 ◽  
Author(s):  
M. Amara ◽  
A. Obeid ◽  
G. Vallet
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Elliptic Partial Differential Equation ◽  
Existence Results ◽  
Partial Differential ◽  
Nonlinear Elliptic

A NUMERICAL METHOD FOR PROGRESSIVE LENS DESIGN

2004 ◽  
Vol 14 (04) ◽  
pp. 619-640 ◽  
Author(s):  
JING WANG ◽  
FADIL SANTOSA
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Method ◽  
Elliptic Partial Differential Equation ◽  
Computational Method ◽  
Necessary Condition ◽  
Perturbation Approach ◽  
Lens Design ◽  
Partial Differential ◽  
Nonlinear Elliptic

The problem of progressive lens design can be posed as a variational problem. The necessary condition is a fourth-order nonlinear elliptic partial differential equation. The partial differential equation can be linearized using a perturbation approach. A numerical method using a special type of splines, chosen for their smoothness properties, is devised to solve the resulting PDE. The computational method is shown to be both convergent and efficient.

scholarly journals About a Partial Differential Equation-Based Interpolator for Signal Envelope Computing: Existence Results and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Oumar Niang ◽  
Abdoulaye Thioune ◽  
Éric Deléchelle ◽  
Mary Teuw Niane ◽  
Jacques Lemoine
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Mathematical Problem ◽  
Existence Results ◽  
Test Results ◽  
Partial Differential ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Mode Decomposition ◽  
Characteristic Points

This paper models and solves the mathematical problem of interpolating characteristic points of signals by a partial differential Equation-(PDE-) based approach. The existence and uniqueness results are established in an appropriate space whose regularity is similar to cubic spline one. We show how this space is suitable for the empirical mode decomposition (EMD) sifting process. Numerical schemes and computing applications are also presented for signal envelopes calculation. The test results show the usefulness of the new PDE interpolator in some pathological cases like input class functions that are not so regular as in the cubic splines case. Some image filtering tests strengthen the demonstration of PDE interpolator performance.

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