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.

scholarly journals Partial Differential Equation-Based Approach for Empirical Mode Decomposition: Application on Image Analysis

2012 ◽  
Vol 21 (9) ◽  
pp. 3991-4001 ◽  
Author(s):  
Oumar Niang ◽  
Abdoulaye Thioune ◽  
Mouhamed Cheikh El Gueirea ◽  
Eric Delechelle ◽  
Jacques Lemoine
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Image Analysis ◽  
Empirical Mode Decomposition ◽  
Partial Differential ◽  
Mode Decomposition

Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Svatoslav Staněk
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Existence Results ◽  
Point Boundary ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Carathéodory Function ◽  
Fractional Differential

AbstractWe investigate the fractional differential equation u″ + A c D α u = f(t, u, c D μ u, u′) subject to the boundary conditions u′(0) = 0, u(T)+au′(T) = 0. Here α ∈ (1, 2), µ ∈ (0, 1), f is a Carathéodory function and c D is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives.

scholarly journals Smoothing the domain out for positive solutions

2000 ◽  
Vol 61 (3) ◽  
pp. 405-413 ◽  
Author(s):  
Yaping Liu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Nonlinear Partial Differential Equation ◽  
Irregular Boundary ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Analytical Tools ◽  
The Given

For a given nonlinear partial differential equation defined on a bounded domain with irregular boundary, the available analytical tools are very limited in relation to the study of positive solutions. In this paper wer first use weak convergence methods to show that for an elliptic equation of a certain type, classical positive solutions on nearby smooth domains approach a generalised positive solution on the given domain. The idea is then applied to sublinear elliptic problems to obtain existence and uniqueness results.

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