Adding police to a mathematical model of burglary

2010 ◽  
Vol 21 (4-5) ◽  
pp. 401-419 ◽  
Author(s):  
ASHLEY B. PITCHER
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuum Limit ◽  
The Other ◽  
Residential Burglary ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
The Continuum

We review the Short model of urban residential burglary derived from taking the continuum limit of two difference equations – one of which models the attractiveness of individual houses to burglary, and the other of which models burglar movement. This leads to a system of non-linear partial differential equations. We propose a change to the Short model and also add deterrence caused by the presence of uniformed officers to the model. We solve the resulting system of non-linear partial differential equations numerically and present results both with and without deterrence.

ON TAYLOR'S SCRAPING PROBLEM AND FLOW OF A SISKO FLUID

2009 ◽  
Vol 14 (4) ◽  
pp. 515-529 ◽  
Author(s):  
Abdul M. Siddiqui ◽  
Ali R. Ansari ◽  
Ahmed Ahmad ◽  
N. Ahmad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Viscous Fluid ◽  
Normal Stress ◽  
Partial Differential ◽  
Sisko Fluid ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
Special Case

The aim of the present investigation is to study the properties of a Sisko fluid flowing between two intersecting planes. The problem is similar to Taylor's scraping problem for a viscous fluid. We determine the solution of the complicated set of non‐linear partial differential equations describing the flow analytically. The analysis is carried out in detail reflecting the effects of varying the angle of the scraper on the flow. In addition, the tangential and normal stress are also computed. We also show the well known Taylor scraper problem as a special case.

Fluid and Structure Interaction in Cochlea’s Similar Geometry

2010 ◽  
Author(s):  
Mahmoud Hamadiche
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Wave ◽  
Transverse Wave ◽  
Basilar Membrane ◽  
Striking Similarity ◽  
Partial Differential ◽  
Material Parameters ◽  
Non Linear

A non linear mathematical model addressing the passive mechanism of the cochlea is proposed in this work. In this respect, the interaction between the basilar membrane seen as an elastic solid and fluids in both scala vestibuli and tympani is developed. Via the fluid/solid interface, a full fluid/solid interaction is taking into account. Furthermore a significant improvement of the existing models has been made in both fluid flow modelling and solid modelling. In the present paper, the flow is three dimensional and the solid is non homogeneous two dimensional membrane where the material parameters depend only on the axial distance. The problem formulation leads to a system of non linear partial differential equations. Solution of the linearized system of partial differential equations of the proposed approach is presented. The numerical results obvious a lower and upper limits of the cochlea resonance frequency versus the material parameters of the basilar membrane. It is shown that a monochromatic acoustic wave energises only a portion of the basilar membrane and the location of the excited portion depends on the frequency of the incident acoustic wave. Those results explain the ability of the cochlea in deciphering the frequency of sound with high resolution in striking similarity with the known experimental results. The mathematical model shows that the excited strip of the basilar membrane by a monochromatic acoustic wave is very small when a transverse wave exists in the basilar membrane. Thus, a transverse wave improves highly the resolution of the cochlea in deciphering the high frequency of the incident acoustic wave.

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