scholarly journals Large-Scale Dynamics of Self-propelled Particles Moving Through Obstacles: Model Derivation and Pattern Formation

2020 ◽  
Vol 82 (10) ◽  
Author(s):  
P. Aceves-Sanchez ◽  
P. Degond ◽  
E. E. Keaveny ◽  
A. Manhart ◽  
S. Merino-Aceituno ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Large Scale ◽  
Coupled System ◽  
Model Parameters ◽  
Partial Differential ◽  
Pattern Size ◽  
Non Local ◽  
Model Derivation ◽  
Macroscopic Equations

Abstract We model and study the patterns created through the interaction of collectively moving self-propelled particles (SPPs) and elastically tethered obstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations model for the interactions between the self-propelled particles and the obstacles, for which we assume large tether stiffness. The result is a coupled system of nonlinear, non-local partial differential equations. Linear stability analysis shows that patterning is expected if the interactions are strong enough and allows for the predictions of pattern size from model parameters. The macroscopic equations reveal that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive.

scholarly journals Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1309
Author(s):  
P. R. Gordoa ◽  
A. Pickering
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Waves ◽  
Elliptic Functions ◽  
Coupled System ◽  
Ultrasonic Waves ◽  
Partial Differential ◽  
Special Cases ◽  
Spherical Bubbles

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

scholarly journals Modeling transport of antibiotic resistant bacteria in aquatic environment using stochastic differential equations

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Ritu Gothwal ◽  
Shashidhar Thatikonda
Keyword(s):  
Antibiotic Resistance ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Aquatic Environment ◽  
Stochastic Partial Differential Equations ◽  
Resistant Bacteria ◽  
Model Parameters ◽  
Antibiotic Resistant Bacteria ◽  
Partial Differential ◽  
Antibiotic Resistant

Abstract Contaminated sites are recognized as the “hotspot” for the development and spread of antibiotic resistance in environmental bacteria. It is very challenging to understand mechanism of development of antibiotic resistance in polluted environment in the presence of different anthropogenic pollutants. Uncertainties in the environmental processes adds complexity to the development of resistance. This study attempts to develop mathematical model by using stochastic partial differential equations for the transport of fluoroquinolone and its resistant bacteria in riverine environment. Poisson’s process is assumed for the diffusion approximation in the stochastic partial differential equations (SPDE). Sensitive analysis is performed to evaluate the parameters and variables for their influence over the model outcome. Based on their sensitivity, the model parameters and variables are chosen and classified into environmental, demographic, and anthropogenic categories to investigate the sources of stochasticity. Stochastic partial differential equations are formulated for the state variables in the model. This SPDE model is then applied to the 100 km stretch of river Musi (South India) and simulations are carried out to assess the impact of stochasticity in model variables on the resistant bacteria population in sediments. By employing the stochasticity in model variables and parameters we came to know that environmental and anthropogenic variations are not able to affect the resistance dynamics at all. Demographic variations are able to affect the distribution of resistant bacteria population uniformly with standard deviation between 0.087 and 0.084, however, is not significant to have any biological relevance to it. The outcome of the present study is helpful in simplifying the model for practical applications. This study is an ongoing effort to improve the model for the transport of antibiotics and transport of antibiotic resistant bacteria in polluted river. There is a wide gap between the knowledge of stochastic resistant bacterial growth dynamics and the knowledge of transport of antibiotic resistance in polluted aquatic environment, this study is one step towards filling up that gap.

scholarly journals Introduction: Big data and partial differential equations

2017 ◽  
Vol 28 (6) ◽  
pp. 877-885 ◽  
Author(s):  
YVES VAN GENNIP ◽  
CAROLA-BIBIANE SCHÖNLIEB
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Large Scale ◽  
Data Science ◽  
Applied Mathematics ◽  
Mathematical Finance ◽  
Energy Functional ◽  
Continuous Change ◽  
Opinion Formation ◽  
Partial Differential

Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

scholarly journals Using analog computers in today's largest computational challenges

2021 ◽  
Vol 19 ◽  
pp. 105-116
Author(s):  
Sven Köppel ◽  
Bernd Ulmann ◽  
Lars Heimann ◽  
Dirk Killat
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Energy Efficient ◽  
Large Scale ◽  
Scientific Computing ◽  
Analog Computer ◽  
Partial Differential ◽  
Technology Platform

Abstract. Analog computers can be revived as a feasible technology platform for low precision, energy efficient and fast computing. We justify this statement by measuring the performance of a modern analog computer and comparing it with that of traditional digital processors. General statements are made about the solution of ordinary and partial differential equations. Computational fluid dynamics are discussed as an example of large scale scientific computing applications. Several models are proposed which demonstrate the benefits of analog and digital-analog hybrid computing.

scholarly journals On the influence of the initial data in a combustion problem

1980 ◽  
Vol 22 (2) ◽  
pp. 193-209 ◽  
Author(s):  
K. K. Tam
Keyword(s):  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Coupled System ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Approximate Expressions ◽  
Combustion Problem

AbstractThe combustion of a material can be modelled by two coupled parabolic partial differential equations for the temperature and concentration of the material. This paper deals with properties of the solution of these equations inside a cylinder or a sphere and under given initial conditions. Bounds for the variation of the temperature with the initial conditions are first established by considering a decoupled form of the equations. Then the coupled system is used to obtain approximate expressions for the temporal evolution of temperature and concentration.

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