# Parabolic Behavior of a Hyperbolic Delay Equation

@article{Laurent2006ParabolicBO, title={Parabolic Behavior of a Hyperbolic Delay Equation}, author={Thomas Laurent and Brian Rider and Michael C. Reed}, journal={SIAM J. Math. Anal.}, year={2006}, volume={38}, pages={1-15} }

It is shown that the fundamental solution of a hyperbolic partial differential equation with time delay has a natural probabilistic structure, i.e., is approximately Gaussian, as $t \rightarrow \infty.$ The proof uses ideas from the DeMoivre proof of the central limit theorem. It follows that solutions of the hyperbolic equation look approximately like solutions of a diffusion equation with constant convection as $t \rightarrow \infty.$

## 8 Citations

### Self-Similar Behavior of a Nonlocal Diffusion Equation with Time Delay

- MathematicsSIAM J. Math. Anal.
- 2020

This work states that the solutions of a class of linear non-local measure-valued differential equations with time delay exhibit a parabolic like behaviour in the large times, that is precisely expressed in term of heat kernel.

### Energy estimates for delay diffusion-reaction equations

- Mathematics
- 2006

Center for Mathematics of University of Coimbra; Instituto Superior de Engenharia de Coimbra

### Modelling real-time dynamic substructuring using partial delay differential equations

- EngineeringProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2007

Real-time dynamic substructuring is a new component testing method for simulating the dynamics of complex engineering systems. The physical component is tested within a computer-generated ‘virtual’…

### An overview of LULU operators and discrete pulse transform for image analysis

- Computer Science, Mathematics
- 2013

An overview of LULU operators and discrete pulse transform (DPT) is presented, which describes the sequences into pulses with different magnitudes and allows a multi-resolution measure of roughness of images and sequences.

### 3D shape from focus using LULU operators and discrete pulse transform in the presence of noise

- Computer ScienceJ. Vis. Commun. Image Represent.
- 2013

### Comparison of LULU and Median Filter for Image Denoising

- Engineering, Computer Science
- 2013

The comparison between Median and LULU filters for impulse noise in images is performed using two different image quality metrics which are RMSE and PSNR.

### 3-D content generation using optical passive reflective techniques

- Computer Science2011 IEEE 15th International Symposium on Consumer Electronics (ISCE)
- 2011

This paper review the different object oriented filtering methods in 3D content generation based on SFF (Shape from Focus) concept and compare between them and a new method in shape recovery based on Discrete Pulse Transform is presented and analyzed.

## References

SHOWING 1-10 OF 10 REFERENCES

### Applied Theory of Functional Differential Equations

- Mathematics
- 1992

Preface. 1. Models. 2. General Theory. 3. Stability of Retarded Differential Equations. 4. Stability of Neutral Type Functional Differential Equations. 5. Stability of Stochastic Functional…

### Bifurcations and traveling waves in a delayed partial differential equation.

- MathematicsChaos
- 1992

The resulting mathematical model is a nonlinear first-order partial differential equation for the cell density u(t,x) in which there is retardation in both temporal (t) and maturation variables (x), and contains three parameters.

### A semilinear transport equation with delays

- Biology
- 2003

An analysis is provided for a model of the blood production system based on a cell population structured by a continuous variable corresponding to the maturity of individual cells. Cell maturity is…

### Normal Approximation and Asymptotic Expansions

- Mathematics
- 1976

Preface to the Classics Edition Preface 1. Weak convergence of probability measures and uniformity classes 2. Fourier transforms and expansions of characteristic functions 3. Bounds for errors of…

### Théorie des distributions

- Mathematics
- 1966

II. Differentiation II.2. Examples of differentiation. The case of one variable (n = 1). II.2.3. Pseudofunctions. Hadamard finite part. We calculate the derivative of a function f(x) which is equal…

### An Introduction to Probability Theory and Its Applications

- Mathematics
- 1950

Thank you for reading an introduction to probability theory and its applications vol 2. As you may know, people have look numerous times for their favorite novels like this an introduction to…

### Introduction to Functional Differential Equations Springer-Verlag, New York

- 1993